Calculus IDifferential Calculus Lecture Notes Veselin Jungic & Jamie Mulholland Department of Mathematics Simon Fraser University c Jungic/Mulholland, August 26, 2013 License is granted to print this document for personal/educational use. Contents Contents i Preface iii Greek Alphabet v 1 Review: Functions and Models 1 1.1 Exponential Functions & Inverse Functions and Logarithms . . . . . . . . . . . . . . . . . . . 2 2 Limits and Derivatives 11 2.1 The Tangent and Velocity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Calculating Limits Using the Limit Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 The Precise Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Limits at Infinity: Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Derivatives and Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.8 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Differentiation Rules 59 3.1 Derivatives of Polynomials and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 60 3.2 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6 Derivatives of Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.7 Rates of Change in the Natural and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . 90 3.8 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.9 Related rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.10 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 i ii CONTENTS 4 Applications of the Derivative 113 4.1 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3 How Derivatives Affect the Shape of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4 Indeterminate Forms and L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.5 Summary of Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.8 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5 Parametric Equations and Polar Coordinates 155 5.1 Curves Defined by Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.3 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.4 Conic Sections in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6 Review Material 185 6.1 Midterm 1 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2 Midterm 2 Review Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.3 End of Term Review Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.4 Final Exam Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.5 Final Exam Practice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Bibliography 221 Preface This booklet contains our notes for courses Math 150/151 - Calculus I at Simon Fraser University. Stu- dents are expected to use this booklet during each lecture by follow along with the instructor, filling in the details in the blanks provided, during the lecture. Definitions of terms are stated in orange boxes and theorems appear in blue boxes . Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet to accompany the example - just like the ones used by your instructor in the lecture. Clicking the link above will take you to the following website containing all the applets: http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html Try it now. No project such as this can be free from errors and incompleteness. We will be grateful to everyone who points out any typos, incorrect statements, or sends any other suggestion on how to improve this manuscript. Veselin Jungic Simon Fraser University
[email protected] Jamie Mulholland Simon Fraser University j
[email protected] August 26, 2013 iii iv Greek Alphabet lower case capital name pronunciation lower case capital name pronunciation α A alpha (al-fah) ν N nu (new) β B beta (bay-tah) ξ Ξ xi (zie) γ Γ gamma (gam-ah) o O omicron (om-e-cron) δ ∆ delta (del-ta) π Π pi (pie) ε E epsilon (ep-si-lon) ρ P rho (roe) ζ Z zeta (zay-tah) σ Σ sigma (sig-mah) η H eta (ay-tah) τ T tau (taw) θ Θ theta (thay-tah) υ Υ upsilon (up-si-lon) ι I iota (eye-o-tah) φ Φ phi (fie) κ K kappa (cap-pah) χ X chi (kie) λ Λ lambda (lamb-dah) ψ Ψ psi (si) µ M mu (mew) ω Ω omega (oh-may-gah) v Part 1 Review: Functions and Models 1 PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 2 1.1 Exponential Functions & Inverse Functions and Logarithms (This lecture corresponds to Sections 1.5 and 1.6 of Stewart’s Calculus.) 1. Reminder. For all a ∈ (0, 1) ∪ (1, ∞) and all x, y ∈ R: (a) a x+y = a x a y (b) a x−y = a x a y (c) (a x ) y = a xy (d) (ab) x = a x b x 2. Reminder. Sketch the graphs of the functions f(x) = 2 x and g(x) = 3 x . 3. Reminder. Sketch the graph of the function F(x) = _ 1 2 _ x . PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 3 4. Reminder. Evaluate f(2), f(−2), f( 1 2 ) and f( 3 2 ) if f(x) = 4 x . 5. BIGQuestion. What is 4 √ 2 ? PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 4 6. Reminder. Napier’s constant: e ≈ 2.718281828459045235360287471352 (John Napier, 1550-1617) PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 5 7. Definition. A function f is called a one-to-one function if it never takes on the same value twice; that is if x 1 ,= x 2 then f(x 1 ) ,= f(x 2 ) . 8. Example. Which of the following functions are one-to-one? (a) f(x) = x 2 (b) g(x) = x 3 (c) h(x) = e x (d) i(x) = sin x (e) j(x) = sin x, x ∈ _ − π 2 , π 2 ¸ PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 6 9. Horizontal Line Test. A function is one-to-one if and only if no horizontal line intersects its graph more than once. 10. Definition. Let f be one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by f −1 (y) = x ⇔ f(x) = y for any y ∈ B. PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 7 11. Example. Find a formula for the inverse of f(x) = x 3x + 1 . 12. Logarithmic Function. The inverse function of the exponential function f(x) = a x is called the logarithmic function with base a. 13. All You Need To Know. For any a > 0, a ,= 1, any x > 0, and any y ∈ R log a x = y ⇔ a y = x 14. Example. Determine log 2 (16), log 2 ( 1 8 ) and log 2 (1). 15. Example. Can you find log 2 (−32)? 16. Reminder. For all a ∈ (0, 1) ∪ (1, ∞) and any positive x and y: (a) log a (xy) = log a x + log a y (b) log a _ x y _ = log a x −log a y (c) log a (x r ) = r log a x (r is a real number) 17. Notation. log 10 x = log x log e x = ln x PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 8 18. Reminder. Sketch the graph of the function y = ln x 19. Example. Solve the equation e x 3 −3 −9 = 0 for x. 20. Inverse Trig Functions. Here we will limit our discussion to sin. 21. Definition. The inverse function of the sine function f(x) = sin x, − π 2 ≤ x ≤ π 2 , is called arcsine and is denoted by either sin −1 or arcsin. PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 9 22. All You Need To Know. For any −1 ≤ x ≤ 1, and any − π 2 ≤ y ≤ π 2 sin −1 x = y ⇔ sin y = x 23. Determine the following. (a) sin −1 ( √ 3 2 ) (b) sin(sin −1 ( 1 3 )) (c) sin −1 (sin( 3π 4 )) PART 1: FUNCTIONS AND MODELS LECTURE 1.4 EXP, LOG, AND INVERSES 10 24. Additional Notes Part 2 Limits and Derivatives 11 PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 12 2.1 The Tangent and Velocity Problems (This lecture corresponds to Section 2.1 of Stewart’s Calculus.) 1. Quote. If I were again beginning my studies, I would follow the advice of Plato and start with math- ematics. Galileo Galilei, Italian philosopher and astronomer, 1564-1642. 2. The Tangent Problem. Find an equation of the tangent line to a curve with equation y = f(x) at a given point P. 3. Three Questions. (a) What is the tangent line to a curve with equation y = f(x) at a given point P? (b) If a curve with equation y = f(x) and a point P on the curve are given, does the tangent exist? (c) If a curve with equation y = f(x) and a point P = (x 0 , f(x 0 )) are given and if the tangent line exists then an equation of is given by y −f(x 0 ) = m(x −x 0 ) . How do we calculate the slope m? 4. Hint. Find the slopes of the secant lines to the parabola y = x 2 through the points (1, 1) and: (a) (2, 4) PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 13 (b) (1.5, 1.5 2 ) (c) (1.1, 1.1 2 ) (d) (1.001, 1.001 2 ) 5. BIGQuestion. What if the second point is VERY, VERYclose to the point (1, 1)? 6. Velocity Problem. By definition avarage velocity = distance traveled time elapsed What if the period of time elapsed is very small? 7. Example. The position of the car is given by the values in the table. t 0 1 2 3 4 5 s 0 10 32 70 119 178 where t is in seconds and s is in feet. Find the average velocity for the time beginning when t = 2 and lasting (a) 3 seconds PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 14 (b) 2 seconds (c) 1 second 8. Question. What is the meaning of the number that we see on the car speedometer as we travel in city traffic? 9. Answer. The number represents the instantaneous velocity. PART 2: LIMITS AND DERIVATIVES LECTURE 2.1 TANGENT AND VELOCITY PROBLEMS 15 10. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 16 2.2 The Limit of a Function (This lecture corresponds to Section 2.2 of Stewart’s Calculus.) (This lecture corresponds to Section 2.3 of Stewart’s Calculus.) 1. Quote. “Black holes are where God divided by zero.” Steven Wright, American comedian, 1955- 2. Problem. Let f(x) = x 2 −x −2 x −2 . (a) Determine the domain of f. (b) Complete the table x f(x) x f(x) 1 3 1.9 2.1 1.99 2.01 1.999 2.001 1.9999 2.0001 PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 17 3. Definition. We write lim x→a f(x) = L and say ”the limit of f (x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. 4. Example. Guess the value of lim x→0 sin x x . 5. Problem. What can we say about lim x→0 [x[ x ? 6. Definition. We write lim x→a + f(x) = L and say ”the right-hand limit of f (x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a and x greater than a. PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 18 7. Example. Sketch the graph of the function f(x) = _ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ _ x + 1 if x ≤ −1 x 2 if x ∈ (−1, 0) 1 if x = 0 x 2 if x ∈ (0, 1] x + 1 if x > 1 Find (a) lim x→−1 − f(x) (b) lim x→−1 + f(x) (c) lim x→0 − f(x) (d) lim x→0 + f(x) (e) lim x→0 f(x) 8. Fact. lim x→a f(x) = L ⇐⇒ ( lim x→a − f(x) = L and lim x→a + f(x) = L) PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 19 9. Problem. Sketch the graph of f(x) = 1 (x + 1) 2 . 10. Definition. Let f be a function defined on both sides of a, except possibly at a itself. Then lim x→a f(x) = ∞ means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a, but not equal to a. 11. Examples. Sketch the graph of the following function. g(x) = x + 3 x −1 12. Read Example 10 in text regarding f(x) = tan (x) 13. Definition. The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true: lim x→a f(x) = ∞ lim x→a − f(x) = ∞ lim x→a + f(x) = ∞ lim x→a f(x) = −∞ lim x→a − f(x) = −∞ lim x→a + f(x) = −∞ PART 2: LIMITS AND DERIVATIVES LECTURE 2.2 THE LIMIT OF A FUNCTION 20 14. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 21 2.3 Calculating Limits Using the Limit Laws (This lecture corresponds to Section 2.3 of Stewart’s Calculus.) 1. Quote. “Laws are like sausages. It’s better not to see them being made.” Otto von Bismarck , German statesman, 1815 - 1898) 2. Example. Guess the value of lim t→0 √ t + 9 −3 t . 3. Limit Laws. Suppose that c is a constant and the limits lim x→a f(x) and lim x→a g(x) exist. Then (a) lim x→a (f(x) + g(x)) = lim x→a f(x) + lim x→a g(x) (b) lim x→a (f(x) −g(x)) = lim x→a f(x) − lim x→a g(x) (c) lim x→a (c f(x)) = c lim x→a f(x) (d) lim x→a (f(x) g(x)) = lim x→a f(x) lim x→a g(x) (e) lim x→a f(x) g(x) = lim x→a f(x) lim x→a g(x) if lim x→a g(x) ,= 0. (f) lim x→a [f(x)] p/q = [ lim x→a f(x)] p/q 4. Two Special Limit Laws. (a) lim x→a c = c (b) lim x→a x = a PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 22 5. Example. Evaluate lim x→2 (x 3 + 3x 2 −4x + 5). 6. Direct Substitution Property. If f is a polynomial or a rational function and a is in the domain of f, then lim x→a f(x) = f(a) . 7. Examples. Find the following limits. (a) lim x→−1 x + 1 x 3 + 1 (b) lim t→0 √ t + 9 −3 t (c) lim h→0 f(x + h) −f(x) h if f(x) = x 2 PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 23 8. Example. Find lim x→0 x 2 [x[ Reminder. lim x→a f(x) = L ⇐⇒ ( lim x→a − f(x) = L and lim x→a + f(x) = L) 9. Theorem. If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then lim x→a f(x) ≤ lim x→a g(x) . 10. Squeeze Theorem. If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and lim x→a f(x) = lim x→a h(x) = L then lim x→a g(x) = L . PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 24 11. Example. Show that lim x→0 _ x _ sin 1 x + cos 1 x __ = 0 . PART 2: LIMITS AND DERIVATIVES LECTURE 2.3 CALCULATING LIMITS 25 12. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 26 2.4 The Precise Definition of Limit (This lecture corresponds to Section 2.4 of Stewart’s Calculus.) 1. Quote. “There’s a delta for every epsilon, It’s a fact that you can always count upon. There’s a delta for every epsilon And now and again, There’s also an N.” (Tom Lehrer, American singer-songwriter, satirist, pianist, and mathematician, 1928 - .) 2. The , δ Game. Consider the function f(x) = 3x − 1 and the point x = 1. There are two players in this game: Player A and Player B. The game is played as follows. Player A chooses a number, say . The object of Player B is to find a number δ so that all values in the interval (1 −δ, 1 +δ) have image in the interval (f(1) −, f(1) + ). The winner is determined as follows: 1) If Player A can pick a number such that Player B cannot find such a δ then Player A wins. 2) If Player B can find a δ for any given by Player A then Player B wins. Who wins Player A or Player B? PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 27 3. Definition. Let f be a function defined on some open interval that contains the number a, except possibly at a iself. Then we say that the limit of f (x) as x approaches a is L, and we write lim x→a f(x) = L if for every number ε > 0 there is a δ > 0 such that [f(x) −L[ < ε whenever 0 < [x −a[ < δ . [link to applet] PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 28 4. Example. Prove the statement using the , δ definition. lim x→3 (2 −5x) = −13 PART 2: LIMITS AND DERIVATIVES LECTURE 2.4 THE PRECISE DEFINITION OF LIMIT 29 5. To Be Continued ... ... in Math 242. PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 30 2.5 Continuity (This lecture corresponds to Section 2.5 of Stewart’s Calculus.) 1. Quote. “If I were asked to name, in one word, the pole star round which the mathematical firmament revolves, the central idea which pervades the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this! ” (JJ Sylvester, English mathematician, 1814-1897) 2. Example. What is the difference between the two graphs? 3. Definition. A function f is continuous at a number a if lim x→a f(x) = f(a) . 4. Note. (a) a belongs to the domain of f (b) lim x→a f(x) exists (c) lim x→a f(x) = f(a) PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 31 5. Definition. If (1) f is defined on an open interval containing a, except perhaps at a, and (2) f is not continuous at a we say that f is discontinuous at a. 6. Example. Where are each of the following functions discontinuous? (a) f(x) = _ _ _ x 2 −4 x −2 if x ,= 2 5 if x = 2 (b) g(x) = _ 1 x −2 if x ,= 2 5 if x = 2 (c) h(x) = _ 1 if x ∈ [1, 2) 2 if x ∈ [2, 3) PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 32 7. Definition. A function f is continuous from the right at a number a if lim x→a + f(x) = f(a) and f is continuous from the left at a if lim x→a − f(x) = f(a) . 8. Definiton. A function f is continuous on an interval if it is continuous at every number in that interval. We understand continuous at the endpoint to mean continuous from the right or continuous from the left. 9. Example. Find the number c that makes f(x) continuous for every x. f(x) = _ ¸ _ ¸ _ x 4 −1 x 3 −1 if x ,= 1 c if x = 1 PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 33 10. Fact. The following types of functions are continuous on their domains: (a) polynomials (b) rational functions (c) root functions (d) trigonometric functions (e) inverse trigonometric functions (f) exponential functions (g) logarithmic functions 11. More Facts. If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: f + g, f −g, cf, fg, f g if g(a) ,= 0 . 12. Example. For which a, b ∈ R is the function f(x) = _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ √ 1 −x −1 ax if x ∈ (0, 1] 1 if x = 0 bx 4 + bx x 2 + x if x ∈ (−1, 0) continuous on (−1, 1]? PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 34 13. Theorem. If f is continuous at b and lim x→a g(x) = b then lim x→a f(g(x)) = f( lim x→a g(x)) = f(b) . 14. Example. Evaluate lim x→0 e √ 1−x−1 x . 15. Theorem. If g is continuous at a and f is continuous at g(a), then the composite function f ◦ g given by (f ◦ g)(x) = f(g(x)) is continuous at a. PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 35 16. Intermediate Value Theorem. Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) ,= f(b). Then there exists a number c in (a, b) such that f(c) = N. 17. Example. Use the Intermediate Value Theorem to prove that √ 2 exists, i.e., prove that there is c ∈ R such that c 2 = 2. PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 36 18. Example. Use the Intermediate Value Theorem to show that the equation e x = 2 −x has at least one real solution. PART 2: LIMITS AND DERIVATIVES LECTURE 2.5 CONTINUITY 37 19. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 38 2.6 Limits at Infinity: Horizontal Asymptotes (This lecture corresponds to Section 2.6 of Stewart’s Calculus.) 1. Quote. “Infinity is a floorless room without walls or ceiling. ” (Anonymous) 2. Problem. Sketch the graphs of the following functions (a) f(x) = 1 x (b) g(x) = e x (c) h(x) = tan −1 x (d) i(x) = 1 1 + x 2 PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 39 3. Definition. Let f be a function defined on some interval (a, ∞). Then lim x→∞ f(x) = L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. 4. Example. Evaluate (a) lim x→∞ 1 x (b) lim x→−∞ 1 x (c) lim x→−∞ e x (d) lim x→∞ tan −1 x (e) lim x→−∞ tan −1 x PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 40 5. Definition. The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim x→∞ f(x) = L or lim x→−∞ f(x) = L . 6. Fact. If r > 0 is a rational number, then lim x→∞ 1 x r = 0 . If r > 0 is a rational number such that x r is defined for all x then lim x→−∞ 1 x r = 0 . 7. Example. Evaluate (a) lim x→∞ 3x 3 −4x 2 −1 6x 3 + x + 2 (b) lim x→−∞ √ 3x 2 −5 −1 2x + 5 PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 41 (c) lim x→−∞ ( _ x 2 + ax − _ x 2 + bx) 8. Problem. Find the following limits. (a) lim x→∞ x 2 (b) lim x→∞ x 2 + 2x −1 x 3 + 3 (c) lim x→∞ x 4 + 5x 3 −1 x 2 + x + 1 (d) lim x→∞ e x (e) lim x→∞ e x x 2 PART 2: LIMITS AND DERIVATIVES LECTURE 2.6 LIMITS AT INFINITY 42 9. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 43 2.7 Derivatives and Rates of Change (This lecture corresponds to Section 2.7 of Stewart’s Calculus.) 1. Quote. ”The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.” (Marcel Proust, French author, 1871- 1922) 2. Definition. The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope m = lim x→a f(x) −f(a) x −a provided that this limit exists. 3. Note. If lim x→a f(x) −f(a) x −a exists then lim x→a f(x) −f(a) x −a = lim h→0 f(a + h) −f(a) h PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 44 4. Example. (a) Find the slope of the tangent line to the graph of f(x) = x 3 at the point i. x = 1 ii. x = 2 (b) Find the equation of the tangent line at each of the points above. 5. Example. (a) Find the slope of the tangent to the curve y = 1 √ x at the point where x = a. PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 45 (b) Find the equation of the tangent line at the point (1, 1). 6. The Most Important Definition in This Course. Definition of Derivative. The derivative of a function f at a number a, denoted by f (a), is f (a) = lim h→0 f(a + h) −f(a) h if this limit exists. 7. Note. If lim x→a f(x) −f(a) x −a exists then f (a) = lim x→a f(x) −f(a) x −a = lim h→0 f(a + h) −f(a) h 8. Example. Find the derivative of the function y = 1 x −1 at the point where x = 3. 9. Example. The following limit represents the derivative of some function f at some number a. State f and a. lim h→0 2 h+3 −8 h PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 46 10. Example. Let f(x) = [x[. Does f (0) exist? 11. Must Know! An equation of the tangent line to y = f(x) at (a, f(a)) is given by y −f(a) = f (a)(x −a) . 12. Example. Find the equation of the tangent line to f(x) = 1 x −1 at the point where x = 3. 13. Compare the derivatives at each of the points on the graph. PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 47 14. Reminder. By definition average velocity = displacement time 15. More Precisely... Suppose an object moves along a straight line according to an equation of motion s = f(t), where s is the displacement of the object from the origin at time t. The average velocity of the object in the time interval from t = a to t = a + h is given by average velocity = f(a + h) −f(a) h . 16. BIGQuestion. What if h is small? 17. Definition. We define the velocity (or instantaneous velocity) v(a) at time t = a as v(a) = lim h→0 f(a + h) −f(a) h . 18. Example. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after t seconds is given by H = 58t −0.83t 2 . (a) Find the velocity of the arrow when t = a. PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 48 (b) When will the arrow hit the moon? (c) With what velocity will the arrow hit the moon? 19. Rates of Change. Let f be a function defined on an interval I and let x 1 , x 2 ∈ I. Then the incre- ment of x is defined as ∆x = x 2 −x 1 and the corresponding change in y is ∆y = f(x 2 ) −f(x 1 ) . The average rate of change of y with respect to x over the interval [x 1 , x 2 ] is defined as ∆y ∆x = f(x 2 ) −f(x 1 ) x 2 −x 1 . 20. Must Know! The instantaneous rate of change of y with respect to x is defined as lim ∆x→0 ∆y ∆x = lim x2→x1 f(x 2 ) −f(x 1 ) x 2 −x 1 . PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 49 21. Example. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V = 100, 000 _ 1 − t 60 _ 2 0 ≤ t ≤ 60 . Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are the units? PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 50 22. Example. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f(p). (a) What is the meaning of the derivative f (8)? What are the units? (b) Is f (8) positive or negative? Explain. PART 2: LIMITS AND DERIVATIVES LECTURE 2.7 DERIVATIVES AND RATES OF CHANGE 51 23. Additional Notes PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 52 2.8 The Derivative as a Function (This lecture corresponds to Section 2.8 of Stewart’s Calculus.) 1. Quote. “I turn away with fear and horror from this lamentable sore of continuous functions without derivatives.” (Charles Hermite, French mathematician, 1822-1901.) 2. Reminder. The derivative of a function f at a number a, denote by f (a), is f (a) = lim h→0 f(a + h) −f(a) h if this limit exists. 3. Find the derivative of the function f(x) = x 2 at (i) x = 0, (ii) x = 1, (iii) x = 2, (iv) x = 10. PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 53 4. Problem. If a function f : I →R is given, find the set J ⊂ I such that f (x) exists for each x ∈ J. If J ,= ∅ then this new function f : J →R is called the derivative of f . 5. Example. Let f(x) = x 2 3 = 3 √ x 2 . (i) Determine the domain of f. (ii) Determine the formula for f (x). What is the domain of f ? PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 54 (iii) Sketch graphs of f and f . PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 55 6. The graph of f is given. Sketch the graph of f . 7. Notation. For y = f(x) it is common to write: f (x) = y = dy dx = df dx = d dx f(x) = Df(x) = D x f(x) Also, f (a) = dy dx ¸ ¸ ¸ ¸ x=a = dy dx _ x=a . 8. Definition. A function is differentiable at a if f (a) exists. It is differentiable on an open interval (a, b) [or (a, ∞) or (−∞, a) or (−∞, ∞)] if it is differentiable at every number in the interval. PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 56 9. Two Questions. (i) Is every continuous function differentiable? (ii) Is every differentiable function continuous? 10. Three Cases. A function f is not differentiable at a number a from its domain if: (i) The graph of f has a corner at the point (a, f(a)); (ii) f is not continuous at a; (iii) The graph of f has a vertical tangent line when x = a. PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 57 11. Higher Derivatives. Suppose that f is a differentiable function. The second derivative of f is the derivative of f . Notation. (f ) = f (y ) = y d dx _ dy dx _ = d 2 y dx 2 12. Example. Find f (x) if f(x) = x 2 . 13. Acceleration. The instantaneous rate of change of velocity with respect to time is called the acceleration of the object. a(t) = v (t) = s (t). 14. Example. The figure shows the graphs of three functions. One is the position function of a particle, one is its velocity, and one is its acceleration. Identify each curve. PART 2: LIMITS AND DERIVATIVES LECTURE 2.8 THE DERIVATIVE AS A FUNCTION 58 15. Additional Notes Part 3 Differentiation Rules 59 PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 60 3.1 Derivatives of Polynomials and Exponential Functions (This lecture corresponds to Section 3.1 of Stewart’s Calculus.) 1. Quote. “Young man, in mathematics you don’t understand things, you just get used to them.” (John von Neumann, Hungarian mathematician and polymath, 1903-1957) 2. Reminder. The derivative of a function f is the function f defined by f (x) = lim h→0 f(x + h) −f(x) h for all x for which this limit exists. Recall that we also use the notation d dx (f(x)) = f (x) for the derivative. 3. Must Know! (a) Derivative of a Constant. d dx (c) = 0 (b) We have already seen that the following are true: d dx (x) = 1, d dx (x 2 ) = 2x, d dx (x 3 ) = 3x 2 . You may be able to see a pattern. In fact, we have the following rule. The Power Rule. If n is any real number, then d dx (x n ) = nx n−1 PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 61 (c) Constant Multiple Rule. If c is a constant and f is a differentiable function, then d dx (cf(x)) = c d dx f(x) (d) Sum Rule. If f and g are differentiable functions, then d dx (f(x) + g(x)) = d dx f(x) + d dx g(x) (e) The Derivative of a Polynomial. If p(x) = a n x n + a n−1 x n−1 + . . . + a 1 x + a 0 where n is a nonnegative integer and a n ,= 0 then p (x) = na n x n−1 + (n −1)a n−1 x n−2 + . . . + a 1 . PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 62 4. Example. Find an equation of the tangent line to the curve y = 2x 3 −7x 2 + 3x + 4 at the point (1, 2). 5. Example. Find an equation for the straight line that passes through the point (1, 5) and it is tangent to the curve y = x 3 . 6. Fact. If f(x) = a x , a > 0, a ,= 1, is an exponential function then f (0) = lim h→0 a h −1 h exists. 7. Fact It is straightforward to show that if f(x) = a x then f (x) = f (0) a x . PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 63 8. Must Know! e is is the number such that lim h→0 e h −1 h = 1 . e ≈ 2.71828 9. Derivative of the Natural Exponential Function. If f(x) = e x is the natural exponential function then f (x) = f(x) . Thus d dx (e x ) = e x . 10. Example. Differentiate the function f(x) = 2x 3 + 3x 2 3 −e x+2 . PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 64 11. Example. At what point on the curve y = e x is the tangent line parallel to the line y = 2x? PART 3: DIFFERENTIATION RULES LECTURE 3.1 DERIVATIVES: POLYNOMIALS AND EXP 65 12. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 66 3.2 The Product and Quotient Rules (This lecture corresponds to Section 3.2 of Stewart’s Calculus.) 1. Quote. ”Five out of four people have trouble with fractions.” (Steven Wright, American comedian, 1955-) 2. Problem. Suppose we have two functions f(x) = 3 √ x 2 and g(x) = e x and we want to compute the derivative of their product d dx ( 3 √ x 2 e x ). How do we do this? 3. Product Rule. If f and g are both differentiable, then d dx [f(x)g(x)] = f(x) d dx [g(x)] + g(x) d dx [f(x)] . In Newton’s notation this is written as (fg) = f g + g f . 4. Examples. (a) Differentiate f(x) = 3 √ x 2 e x . PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 67 (b) Differentiate g(x) = (x + 1)(2x 2 −x + 1). 5. Quotient Rule. If f and g are differentiable, then d dx _ f(x) g(x) _ = g(x) d dx [f(x)] −f(x) d dx [g(x)] [g(x)] 2 . In Newton’s notation this is written as _ f g _ = g f −f g g 2 . 6. Examples. (a) Differentiate y = 2t 2 −1 t 3 + 1 . (b) Differentiate f(x) = e −x . PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 68 (c) If f(3) = 4, g(3) = 2, f (3) = −6, and g (3) = 5, find the following numbers. i. (f + g) (3) ii. (fg) (3) iii. _ f g _ (3) iv. _ f f −g _ (3) PART 3: DIFFERENTIATION RULES LECTURE 3.2 THE PRODUCT AND QUOTIENT RULES 69 7. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 70 3.3 Derivatives of Trigonometric Functions (This lecture corresponds to Section 3.3 of Stewart’s Calculus.) 1. Quote. ”Trigonometry is the mathematics of sound and music.” (Frank Wattenberg, American mathematician, 1952-) 2. Problem. What is the derivative of sin x? 3. Must Know! (a) d dx (sin x) = cos x (b) d dx (cos x) = −sin x (c) d dx (tan x) = sec 2 x (d) d dx (sec x) = sec xtan x (e) d dx (csc x) = −csc xcot x (f) d dx (cot x) = −csc 2 x PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 71 4. Problem. Prove that d dx (sin x) = cos x . 5. Trigonometric Limits. Above we used the very important results lim θ→0 sin θ θ = 1 and lim θ→0 cos θ −1 θ = 0. We now prove these results. PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 72 6. Examples. (a) Differentiate y = 1 + tan x x −cot x . (b) Find the points on the curve y = cos x 2 + sin x at which the tangent is horizontal. 7. A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ = π/3? PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 73 8. Examples. Evaluate (a) lim x→0 sin 2x x (b) lim θ→0 sin 2θ cos θ −1 PART 3: DIFFERENTIATION RULES LECTURE 3.3 DERIVATIVES OF TRIG FUNCTIONS 74 9. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 75 3.4 Chain Rule (This lecture corresponds to Section 3.4 of Stewart’s Calculus.) 1. Puzzle. A duck before two ducks, a duck behind two ducks, and a duck in the middle. How many ducks are there? 2. Reminder. The composition of the functions f and g is defined by (f ◦ g)(x) = f(g(x)) . 3. Example. Let f(u) = sin u and g(x) = 1 + x 2 . Find F = f ◦ g. 4. Chain Rule. If f and g are both differentiable and F = f ◦ g is the composite function defined b F(x) = f(g(x)), then F is differentiable and F is given by F (x) = f (g(x)) g (x) . In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then dy dx = dy du du dx . 5. Examples. (a) Let f(u) = sin u and g(x) = 1 + x 2 and let F = f ◦ g. Find the derivative of F. PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 76 (b) Find y = dy dx . if i. y = (2 −5x) 3 ii. y = (x + sin x) 5 (1 + e x ) 2 iii. Express the derivative dy/dx in terms of x if y = u 5 and u = (4x −1) 2 x . PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 77 6. Examples. Find f . (a) f(x) = _ 2 + 5x 2 (b) f(x) = (tan (x 2 )) 3 (c) f(x) = e cos x 7. Must Know! d dx (a x ) = a x ln a . PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 78 8. Examples. (a) A pebble droppped into a lake creates an expanding circular ripple. Suppose that the radius of the circle is increasing at the rate of 2 in./s. At what rate is its area increasing when its radius is 10 in.? (b) Suppose that f(0) = 0 and f (0) = 1. Calculate the derivative of f(f(f(x))) at x = 0. (c) Under certain circumstances a rumor spreads according to the equation p(t) = 1 1 + ae −kt where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants. i. Find lim t→∞ p(t). ii. Find the rate of spread of the rumor. PART 3: DIFFERENTIATION RULES LECTURE 3.4 CHAIN RULE 79 9. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 80 3.5 Implicit Differentiation (This lecture corresponds to Section 3.5 of Stewart’s Calculus.) 1. Dictionary. implicit adjective Definition: 1. implied: not stated, but understood in what is expressed Asking us when we would like to start was an implicit acceptance of our terms. 2. absolute: not affected by any doubt or uncertainty implicit trust 3. contained: present as a necessary part of something Confidentiality is implicit in the relationship between doctor and patient. 2. Problem. The curve x 3 + y 3 = 3xy is called the folium of Descartes. Find the equation of the tangent line at the point _ 3 2 , 3 2 _ . 3. Implicitly Defined Function. An equation in two variables x and y may have one or more solutions for y in terms of x or for x in terms of y. These solutions are functions that are said to be implicitly defined by the equation. 4. Example (a) x 2 + y 2 = 1 (b) x 3 + y 3 = 3xy PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 81 5. Implicit Differentiation. (a) Use the chain rule to differentiate both sides of the given equation, thinking of x as the indepen- dent variable. (b) Solve the resulting equation for dy dx . 6. Example. The curve x 3 + y 3 = 3xy is called the folium of Descartes. Find the equation of the tangent line at the point _ 3 2 , 3 2 _ . 7. Example. Suppose that water is being emptied from a spherical tank of radius 10 ft. If the depth of water in the tank is 5 ft and is decreasing at the rate of 3 ft/sec, at what rate is the radius r of the top surface of the water decreasing? PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 82 8. Differentiation of an Inverse Function. Suppose f is a one-to-one differentiable function and its inverse function f −1 is also differentiable. Use implicit differentiation to show that (f −1 ) (x) = 1 f (f −1 (x)) provided that the denominator is not 0. 9. Must Know! (a) d dx (sin −1 (x)) = 1 √ 1 −x 2 (b) d dx (cos −1 (x)) = − 1 √ 1 −x 2 (c) d dx (tan −1 (x)) = 1 1 + x 2 (d) d dx (cot −1 (x)) = − 1 1 + x 2 PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 83 10. Example. Determine the points on the circle (x −1) 2 + (y −2) 2 = 4 where the tangent line is horizontal or vertical. 11. Example. For the curve x 2 + y 2 = 5 find y by implicit differentiation. PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 84 12. Orthogonal Trajectories. (a) Two curves are called orthogonal if at each point of intersection their tangent lines are perpen- dicular. (b) Two families of curves are orthogonal trajectories of each other if every curve in one family is orthogonal to every curve in the other family. (c) Show that the given families of curves are orthogonal trajectories of each other: x 2 + y 2 = ax and x 2 + y 2 = by . PART 3: DIFFERENTIATION RULES LECTURE 3.5 IMPLICIT DIFFERENTIATION 85 13. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 86 3.6 Derivatives of Logarithmic Functions (This lecture corresponds to Section 3.6 of Stewart’s Calculus.) 1. Quote. “One real estate development company advertised that an investment with it would grow logarithmically.” (From Ed Barbeaus column, Fallacies, Flaws, and Flimflam, in College Math. Journal 36 (2005), 394-396.) 2. Must Know! d dx (log a x) = 1 xln a 3. When a = e this becomes d dx (ln x) = 1 x . 4. Examples. Differentiate (a) y = log 2 (3x 2 + e x ) (b) y = ln(x + √ x 2 −1) PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 87 (c) y = √ ln x (d) y = ln √ x (e) y = ln _ x 2 (x + 3) 4 _ 5. More Examples. Differentiate (a) y = 4 √ x 3 5 √ x 3 + 1 (2x + 1) 3 PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 88 (b) y = x x 2 (c) y = ln [x[ 6. Must Know! lim x→0 (1 + x) 1 x = e PART 3: DIFFERENTIATION RULES LECTURE 3.6 DERIVATIVE: LOGARITHMS 89 7. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 90 3.7 Rates of Change in the Natural and Social Sciences (This lecture corresponds to Section 3.7 of Stewart’s Calculus.) 1. Quote. ”If you want to see practical applied mathematics, read chemical engineering; if you want to see theoretical applied mathematics, read electrical engineering. And if you want to read pure math, read economics.” (Unknown blogger.) 2. Example (Physics). The equation of motion for a particle is given by s = 2t 3 −3t 2 −12t, t ≥ 0 where s is in meters and t is in seconds. (a) Find the velocity and acceleration as functions of t. (b) The graph of s = s(t) is shown. Sketch the graphs of the velocity and acceleration functions for 0 ≤ t ≤ 4. (c) When is the particle speeding up? Slowing down? (d) What does the expression s (t) = a (t) represent? [link to applet] PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 91 3. Exercise. Let v(t) be a function which gives the velocity of a particle at time t. Consider the speed function w(t) = [v(t)[. (a) The particle is speeding up when w (t) > 0. Show that is equivalent to the condition that v(t) and a(t) have the same sign. (b) Similarly, the particle is slowing down when w (t) < 0. Show that is equivalent to the condition that v(t) and a(t) have opposite signs. [Hint: Remove the absolute value signs by writing w(t) as a piecewise defined function. Then differ- entiate the piecewise function, paying careful attention to the conditions which define each piece.] PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 92 4. Example (Chemistry). If one molecule C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value [A] = [B] = a moles/L then [C] = a 2 kt akt + 1 where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x = [C], then dx dt = k(a −x) 2 (c) What happens to the concentration as t →∞? (d) What happens to the rate of reaction as t →∞? PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 93 5. Example (Economics). Suppose that the cost (in dollars) for a company to produce x pairs of new line of jeans is C(x) = 2000 + 3x + 0.01x 2 + 0.0002x 3 (a) Find the marginal cost function. (b) Find C (100) and explain its meaning. What does it predict? (c) Compare C (100) with the cost of manufacturing the 101st pair of jeans. PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 94 6. Example. The height of a certain cylinder is always twice its radius r. If its radius is changing, show that the rate of change of its volume with respect to r is equal to its surface area. PART 3: DIFFERENTIATION RULES LECTURE 3.7 RATES OF CHANGE IN SCIENCE 95 7. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 96 3.8 Exponential Growth and Decay (This lecture corresponds to Section 3.8 of Stewart’s Calculus.) 1. Quote. “It’s the whole issue with exponential growth, it’s very slow in the beginning but over the long term it gets ridiculous.” (Drew Curtis, Founder and chief administrator of Fark.com, 1973 - ) 2. Exponential Growth and Decay: A quantity q is said to be growing (or decaying) exponentially if q = Ae kt where A and k are constants. 3. Natural Growth Equation. The solution of the initial-value problem dy dt = ky, y(0) = y 0 is y(t) = y 0 e kt . PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 97 4. Example. Calculopolis had a population of 25000 in 1980 and the population of 30000 in 1990. What population can the Calculopolis planers expect in the year 2020 if the population grows at a rate proportional its size? PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 98 5. Radioactive Decay. Radioactive material is known to decay at a rate proportional to the amount present. This means, if N is the amount (mass) of radioactive material at time t then it must satisfy the model: dN dt = −kN, k > 0. 6. Example. It takes 8 days for 20% of a particular radioactive material to decay. How long does it take for 100 grams of material to decay to 50 grams? 40 grams? 0 grams? 7. Remark. Usually k is specified in terms of the half-life of the isotope τ = ln 2 k . This is the time required for half of any given quantity to decay. PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 99 8. Newton’s Lawof Cooling and Heating. If a warm object is put in cooler surroundings its tempera- ture will steadily decrease. A law of physics known as Newtons law of cooling says that the rate at which the object cools is proportional to the difference between its temperature and the surrounding temperature. This law is modeled by the differential equation: dT dt = k(T −M) where • T(t) is the temperature of the object at time t • M is the temperature of the surroundings (ambient temperature - which is constant) • k a constant (called the cooling constant) 9. Example. When a cold drink is taken from a refrigerator, its temperature is 5 ◦ C. After 25 minutes in a 20 ◦ C room its temperature has increased to 10 ◦ C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature be 15 ◦ C? PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 100 10. Continuously Compound Interest. Suppose an amount A 0 is invested at an interest rate r. If A(t) is the amount after t years then A(t) = A 0 (1 + r) t . If the interest is compound n times per year then after t years the investment is worth A(t) = A 0 _ 1 + r n _ nt . The formula for continuous compounding is A(t) = A 0 e rt . Example: If $1000 is borrowed at 19% interest, find the amounts due at the end of 2 years if the interest is compounded (a) annually (b) quarterly (c) monthly (d) daily (e) continuously PART 3: DIFFERENTIATION RULES LECTURE 3.8 EXPONENTIAL GROWTH AND DECAY 101 11. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 102 3.9 Related rates (This lecture corresponds to Section 3.9 of Stewart’s Calculus.) 1. Quote. “If you want to increase your success rate, double your failure rate. ” (Thomas John Watson, Sr., Founder of IBM, 1874 - 1956.) 2. A spherical balloon is being inflated. The radius r of the balloon is increasing at the rate of 0.2 cm/s when r = 5 cm. At what rate is the volume V of the balloon increasing at that moment? PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 103 3. The Method of Related Rates When two variables are related by an equation and both are functions of a third variable (such as time), we can find a relation between their rates of change. In this case, we say the rates are related, and we can compute one if we know the other. We proceed as follows: (a) Identify the independent variable (usually time) on which the other quantities depend and as- sign it a symbol, such as t. Also, assign symbols to the variable quantities that depend on t. (b) Find an equation that relates the dependent variables. (c) Differentiate both sides of the equation with respect to t (using the chain rule if necessary). (d) Substitute the given information into the related rates equation and solve for the unknown rate. PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 104 4. A rocket is launched vertically and is tracked by a radar station located on the ground 5 km from the launch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 3 ◦ per second when θ = 60 ◦ . What is the velocity of the rocket at that instant? [link to applet] PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 105 5. A man 6 ft tall walks with a speed of 8 ft/s away from a street light that is atop an 18-ft pole. How fast is the tip of his shadow moving along the ground when he is 100 ft from the light pole? [link to applet] PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 106 6. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? [link to applet] PART 3: DIFFERENTIATION RULES LECTURE 3.9 RELATED RATES 107 7. Additional Notes PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 108 3.10 Linear Approximation and Differentials (This lecture corresponds to Section 3.10 of Stewart’s Calculus.) 1. Quote. It is the mark of an instructed mind to rest satisfied with the degree of precision to which the nature of the subject admits and not to seek exactness when only an approximation of the truth is possible. (Aristotle, Greek philosopher, 384 BC - 322 BC.) 2. Problem. If f(1) = 4 and f (1) = 1 use the linear approximation to f(x) at x = 1 to approximate f(2). 3. Idea. Instead of evaluating f(x) evaluate L(x) where L is the tangent line to the graph of y = f(x) at a known point (a, f(a)) that is close to the point (x, f(x)). 4. Linear Approximation. The linear function L(x) = f(a) + f (a)(x −a) is called the linearization of f at a. For x close to a we have that f(x) ≈ L(x) = f(a) + f (a)(x −a) and this approximation is called the linear approximation of f at a. PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 109 5. Example. If f(1) = 4 and f (1) = 1 use the linear approximation to f(x) at x = 1 to approximate f(2). 6. Example. Use linear approximation to approximate √ 37. What is the accuracy of this approxima- tion? PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 110 7. Differential. Let f be a function differentiable at x ∈ R. Let ∆x = dx be a (small) given number. The differential dy is defined as dy = f (x)∆x . 8. Important! f(a + dx) ≈ L(a + dx) f(a + dx) ≈ f(a) + f (a)(a + dx −a) f(a + dx) ≈ f(a) + f (a)dx = f(a) + dy dy ≈ f (a +dx) −f (a) Small differential means good approximation. PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 111 9. Example. The equatorial radius of the earth is approximately 3960 mi. Suppose that a wire is wrapped tightly around the earth at the equator. Approximately how much must this wire be length- ened if it is to be strung all the way around the earth on poles 10 ft above the ground. (1 mi = 1760 yards = 1760 3 ft.) PART 3: DIFFERENTIATION RULES LECTURE 3.10 LINEAR APPROX. AND DIFFERENTIALS 112 10. Additional Notes Part 4 Applications of the Derivative Image Source: http://commons.wikimedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair.jpg 113 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 114 4.1 Maximum and Minimum Values (This lecture corresponds to Section 4.1 of Stewart’s Calculus.) 1. Quote. “I feel the need of attaining the maximum of intensity with the minimum of means. It is this which has led me to give my painting a character of even greater bareness.” (Joan Miro, Catalan-Spanish artist, 1893 - 1983) 2. Definition. A function f has an absolute maximum at c if f(c) ≥ f(x) for all x ∈ D, the domain of f . The number f(c) is called the maximum value of f on D. A function f has an absolute minimum at c if f(c) ≤ f(x) for all x ∈ D, the domain of f . The number f(c) is called the minimum value of f on D. 3. Definition. A function f has a local maximum at c if f(c) ≥ f(x) for all x in an open interval, in the domain, containing c . A function f has a local minimum at c if f(c) ≤ f(x) for all x in an open interval, in the domain, containing c . 4. Extreme Value Theorem. If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c, d ∈ [a, b]. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 115 5. Fermat’s Theorem. If f has a local maximum or minimum at c, and f (c) exists, then f (c) = 0. 6. Examples. Find all local extrema of (a) f(x) = 3x 4 −16x 3 + 18x 2 , −1 ≤ x ≤ 4 (b) f(x) = [x[, −1 < x < 1 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 116 7. Definition. A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) does not exist. 8. Problem. Find the maximum and minimum values of the function f(x) = x 2 + 4x + 7, −3 ≤ x ≤ 0 9. Closed Interval Method. To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: (a) Find the values of f at the critical numbers of f in (a, b). (b) Find the values of f at the endpoints of the interval. (c) The largest of the values from Step (a) and Step (b) is the absolute maximum value; the smallest of these values is the absolute minimum value. 10. Examples. Find the maximum and minimum values of the given functions on the indicated closed intervals. (a) f(x) = x + 4 x , x ∈ [1, 4] PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 117 (b) g(x) = 2 − 3 √ x, x ∈ [−1, 8] PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.1 MAXIMUM AND MINIMUM VALUES 118 11. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 119 4.2 The Mean Value Theorem (This lecture corresponds to Section 4.2 of Stewart’s Calculus.) 1. Quote. “The Mean Value Theorem is the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance.” (Edwin Purcell and Dale Varberg, American mathematicians) 2. Rolle’s Theorem. (Michel Rolle, French mathematician, 1652-1719) Let f be a function that satisfies the following three hypotheses: (a) f is continuous on the closed interval [a, b]. (b) f is differentiable on the open interval (a, b). (c) f(a) = f(b). Then there is a number c in (a, b) such that f (c) = 0. 3. Example. Check if the following functions satisfy the hypotheses of Rolle’s theorem. (a) f(x) = x 1/2 −x 3/2 on [0, 1]. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 120 (b) f(x) = 1 −x 2/3 on [−1, 1]. 4. The Mean Value Theorem. Let f be a function that satisfies the following hypotheses: (a) f is continuous on the closed interval [a, b]. (b) f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f (c) = f(b) −f(a) b −a or, equivalently, f(b) −f(a) = f (c)(b −a). PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 121 5. Example. A car is driving along a rural road where the speed limit is 70 km/h. At 3:00 pm its odometer reads 18075 km. At 3:18 its reads 18100 km. Prove that the driver violated the speed limit at some instant between 3:00 and 3:18 pm. 6. Show that the equation x 4 = x + 1 has exactly one solution in the interval [1, 2]. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 122 7. Must Know! If f (x) = 0 for all x in an interval (a, b), then f is constant on (a, b). 8. Fact. If f (x) = g (x) for all x in an interval (a, b), then f −g is constant on (a, b); that is, f(x) = g(x) + c where c is a constant. 9. Example. Prove the identity arcsin _ x −1 x + 1 _ = 2 arctan ( √ x) − π 2 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.2 THE MEAN VALUE THEOREM 123 10. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 124 4.3 How Derivatives Affect the Shape of a Graph (This lecture corresponds to Section 4.3 of Stewart’s Calculus.) 1. Quote. “The spread of civilization may be likened to a fire; First, a feeble spark, next a flickering flame, then a mighty blaze, ever increasing in speed and power.” (Nikola Tesla, American inventor and engineer, 1856 - 1943) 2. Increasing/Decreasing Test. (a) If f (x) > 0 on an interval, then f is increasing on that interval. (b) If f (x) < 0 on an interval, then f is decreasing on that interval. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 125 3. Example. Find the open intervals on the x-axis on which the function f(x) = 3x 4 −4x 3 −12x 2 + 5 is increasing and those on which is decreasing. 4. The First Derivative Test. Suppose that c is a critical number of a continuous function f. (a) If f changes from positive to negative at c, then f has a local maximum at c. (b) If f changes from negative to positive at c, then f has a local minimum at c. (c) If f does not change sign at c, then f has no local minimum or maximum at c. 5. Example. Find all local extrema of the function f(x) = 3x 4 −4x 3 −12x 2 + 5 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 126 6. Examples. Find all local extrema of f(x) = [x[ 7. Definition. If the graph of f lies above all of its tangent lines on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 127 8. Concavity Test. (a) If f (x) > 0 for all x ∈ I, then the graph of f is concave upward on I. (b) If f (x) < 0 for all x ∈ I, then the graph of f is concave downward on I. 9. Definition. A point P on a curve y = f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 128 10. The Second Derivative Test. Suppose f is continuous near c. (a) If f (c) = 0 and f (c) > 0 then f has a local minimum at c. (b) If f (c) = 0 and f (c) < 0 then f has a local maximum at c. 11. Example. Sketch the graph of the function f(x) = 3x 4 −4x 3 −12x 2 + 5 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.3 SHAPE OF A GRAPH 129 12. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 130 4.4 Indeterminate Forms and L’Hospital’s Rule (This lecture corresponds to Section 4.4 of Stewart’s Calculus.) 1. “Proof”. Let a = b. a 2 = ab (Multiply both sides by a.) a 2 + a 2 −2ab = ab + a 2 −2ab (Add a 2 −2ab to both sides.) 2(a 2 −ab) = a 2 −ab (Factor the left, and collect like terms on the right.) 2 = 1 (Divide both sides by a 2 −ab.) 2. Indeterminate Forms. (a) Indeterminate form of type 0 0 . Example. Evaluate lim x→0 sin kx x for k ∈ R. (b) Indeterminate form of type ∞ ∞ . Example. Evaluate lim x→∞ ax + 1 bx + 1 for a, b ∈ R. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 131 3. L’Hospital’s Rule. Suppose that f and g are differentiable and g (x) ,= 0 near a (except possibly at a.) Suppose that lim x→a f(x) = 0 and lim x→a g(x) = 0 or that lim x→a f(x) = ±∞and lim x→a g(x) = ±∞ Then lim x→a f(x) g(x) = lim x→a f (x) g (x) if the limit on the right side exists (or is ∞or −∞). 4. Examples. Find (a) lim x→0 e x −1 sin 2x (b) lim x→∞ e x x 2 + x (c) lim x→∞ ln x √ x PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 132 5. Indeterminate Form 0 ∞. Find lim x→∞ xln _ x −1 x + 1 _ . 6. Indeterminate Form∞−∞. Find lim x→0 _ 1 x − 1 sin x _ . 7. Indeterminate Form 0 0 , ∞ 0 , 1 ∞ . 8. Examples. Find (a) lim x→0 (cos x) 1/x 2 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.4 L’HOSPITAL’S RULE 133 9. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 134 4.5 Summary of Curve Sketching (This lecture corresponds to Section 4.5 of Stewart’s Calculus.) 1. Puzzle. Connect the nine dots with four (only four) straight lines without ever lifting your pen or pencil from the paper. 2. Guidlines (a) Domain (b) Intercepts: For the x-intercepts set y = 0 and solve for x. For the y-intercept calculate f(0). (c) Symmetry: i. Even function - symmetric about the y-axis. ii. Odd function - symmetric about the origin. iii. Periodic functions. (d) Asymptotes: i. Horizontal Asymptotes: y = L if lim x→∞ f(x) = L or if lim x→−∞ f(x) = L. ii. Vertical Asymptotes: x = a if at least one of the following is true lim x→a + f(x) = ∞ lim x→a − f(x) = ∞ lim x→a + f(x) = −∞ lim x→a − f(x) = −∞. iii. Slant Asymptotes: y = mx + b if lim x→∞ (f(x) −(mx + b)) = 0. (e) Intervals of Increase and Decrease: f (x) > 0 on an interval I means f increasing ¸on I. f (x) < 0 on an interval I means f decreasing ¸on I. (f) Local Maximum and Minimum Values: - First Derivative Test - Second Derivative Test (g) Concavity and Points of Inflection: f (x) > 0 on an interval I means f concave up ( on I. f (x) < 0 on an interval I means f concave down ) on I. (h) Sketch the curve. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 135 3. Examples. Sketch graphs of the following functions. (a) f(x) = 2 + x −x 2 (x −1) 2 PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 136 (b) f(x) = x 2 e x PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 137 4. In this example we will focus just on asymptotes in the guidelines outlined in (2). Determine the asymptotes of the function f(x) = x 2 + x −1 x −1 . PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.5 SUMMARY OF CURVE SKETCHING 138 5. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 139 4.6 Optimization Problems (This lecture corresponds to Section 4.7 of Stewart’s Calculus.) (This lecture corresponds to Section 4.7 of Stewart’s Calculus.) 1. Quote. “There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.” (Nikolai Lobachevski, Russian mathematician, 1792 - 1856) 2. Examples. (a) Find the dimensions of the right circular cylinder with greatest volume that can be inscribed in a right circular cone of radius 8 cm and height 12 cm. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 140 (b) A painting in an art gallery has height 3 m and is hung so that its lower edge is about 1 m above the eye of an observer. How far from the wall should the observer stand to get the best view? (i.e. the observer wants to maximize the angle θ subtended at the eye by the painting.) [link to applet] For an interesting paper connecting this problem to exponential functions click here. [2] PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 141 (c) The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 142 (d) Maya is 2 km offshore on a boat and wishes to reach a coastal village which is 6 km down the straight shoreline form the point on the shore nearest to the boat. She can row at 2 km/hour and run 5 km/hour. Where she should land her boat to reach the village in the least amount of time? [link to applet] PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.6 OPTIMIZATION PROBLEMS 143 3. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 144 4.7 Newton’s Method (This lecture corresponds to Section 4.8 of Stewart’s Calculus.) 1. Quote. “Sometimes, close enough is good enough.” (Math Girl, Episode I - Differentials Attract) 2. Newton’s Method. (a) Problem. Find a solution, say x = r, to f(x) = 0. (b) Idea. i. Let x 1 be a ”good” estimate of r. ii. Consider the tangent line L to the curve y = f(x) at the point (x 1 , f(x 1 )). Look at the x- intercept of L, call it x 2 . If x 1 is close to r then x 2 seems to be even closer to r, and we use x 2 as a second approximation to r. How do we find x 2 in terms of x 1 and f? iii. Repeat this procedure to get a third approximation x 3 from our second approximation x 2 : If we keep repeating this process we obtain a sequence of approximations for r: x 1 , x 2 , x 3 , . . . . iv. If the numbers x n become closer and closer to r as n becomes large then we say that the sequence converges to r and we write lim n→∞ = r. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 145 (c) Method. i. Begin with an initial guess x 1 . ii. Calculate x 2 = x 1 − f(x 1 ) f (x 1 ) . iii. If x n is known then x n+1 = x n − f(x n ) f (x n ) . iv. If x n and x n+1 agree to k decimal places then x n approximates the root r up to k decimal places and f(x n ) ≈ 0. 3. Example. Solve 5x + cos x = 5 for x ∈ [0, 1] correct to 6 decimal places. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 146 4. Example. Use Newton’s method to find √ 2 accurate to eight decimal places. 5. Example. Use Newton’s method to solve x 1/3 = 0 by taking x 0 = 1. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 147 6. Example. Let f(x) = x 3 + 3x + 1. (a) Show that f has exactly one root in the interval (− 1 2 , 0). Explain. (b) Use Newton’s method to approximate the root that lies in the interval ( −1 2 , 0). Stop when the next iteration agrees with the previous one at two decimal places. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.7 NEWTON’S METHOD 148 7. Additional Notes PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 149 4.8 Antiderivatives (This lecture corresponds to Section 4.9 of Stewart’s Calculus.) 1. Quote. “In school I had a friend named Cos Davis. We nicknamed him the Antiderivative of Sin Davis. He actually liked it.” (Posted by SLD on May 14, 2004, 10:39 PM, at Stupid math or science jokes thread (again), http://www.iidb.org/vbb/archive/index.php/t-85561&e=747 ”Wouldn’t the antiderivative of Sin Davis be Negative Cos Davis?” (Posted by LVLLN on May 15, 2004, 03:30 PM) 2. Problem. For a given function f find all functions F with the property that F = f . 3. Definition. A function F is called an antiderivative of f on an interval I if F (x) = f(x) for all x ∈ I. 4. Example: Find an antiderivative of the following. (a) f(x) = x 2 (b) f(x) = 1 x 5. Example. Show that all antiderivatives of f(x) = x n , n ,= −1, are given by F(x) = 1 n + 1 x n+1 + C where C is an arbitrary constant. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 150 6. Theorem. If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant. 7. Table of Antiderivative Formulas. (f = Function, F = Particular antiderivative) f F f F cg(x) cG(x) sin x −cos x g(x) + h(x) G(x) + H(x) sec 2 x tan x x n , n = −1 1 n + 1 x n+1 sec xtan x sec x 1/x ln |x| 1 √ 1 −x 2 sin −1 x e x e x 1 1 + x 2 tan −1 x cos x sin x PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 151 8. Example. Find all antiderivatives of f(x) = x 3 + 3 √ x − 4 x 2 9. Example. (a) Find f if f (t) = 2 cos (3t) + 5 sin (4t) + 3e t . (b) Which of the functions in (a) satisfies f(0) = 0? 10. Example. Find f such that f (x) = 2e −2x + 5, f(0) = 3/2 and f (0) = 1. PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 152 11. Rectilinear Motion. (a) Position function - s = f(t) (b) Velocity function - v(t) = s (t) (c) Acceleration function - a(t) = v (t) 12. Example. The skid marks made by an automobile indicate that its brakes are fully applied for a distance of 160 ft before it came to a stop. Suppose that the car in question has a constant deceleration of 20 ft/s 2 under the condition of the skid. How fast was the car traveling when its brakes were applied? PART 4: APPLICATIONS OF THE DERIVATIVE LECTURE 4.8 ANTIDERIVATIVES 153 13. Additional Notes PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 4.8 ANTIDERIVATIVES 154 Part 5 Parametric Equations and Polar Coordinates 155 PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 156 5.1 Curves Defined by Parametric Equations (This lecture corresponds to Sections 10.1 & 10.2 (Tangents part only) of Stewart’s Calculus.) 1. Quote. “When you get to the top of the mountain, keep climbing.” (Zen proverb) 2. Motivation: Particle moving in plane 3. Problem. In the xy-plane draw the set P = ¦(t 2 , t) : t ∈ R¦. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 157 4. Vocabulary. Let I be an interval and let f and g be continuous on I. (a) The set of points C = ¦(f(t), g(t)) : t ∈ I¦ is called a parametric curve. (b) The variable t is called a parameter. (c) We say that the curve C is defined by parametric equations x = f(t), y = g(t). (d) We say that x = f(t), y = g(t) is a parametrization of C. (e) If I = [a, b] then (f(a), g(a)) is called the initial point of C and (f(b), g(b)) is called the terminal point of C. 5. Example. Find two parametrizations of the unit circle x 2 + y 2 = 1 . 6. Example. Find parametric equations for the ellipse x 2 a 2 + y 2 b 2 = 1 . PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 158 7. Example. Sketch the graph of the curve defined by parametric equations x = sin t, y = sin 2 t, −∞< t < ∞ 8. Some neat examples: Lissajous Curve: x = cos at, y = sin bt, t ∈ [0, 2π] Knotty Curve: x = t + 2 sin 2t, y = t + 2 cos 5t, t ∈ [−2π, 2π] PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 159 9. Cycloid. The curve traced by a point P on the edge of a rolling circle is called a cycloid. The circle rolls along a straight line without slipping or stopping. Find parametric equations for the cycloid if the line along which the circle rolls is the x-axis but always tangent to it, and the point P begins at the origin. 10. Derivatives of Parametric Curves. The derivative to the parametric curve x = f(t), y = g(t) is given by dy dx = dy dt dx dt = g (t) f (t) . PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.1 PARAMETRIC EQUATIONS 160 11. Example. Find the slope of the tangent line to the curve x = cos t, y = sin t, 0 ≤ t ≤ 2π. at the point corresponding to t = π/4. 12. Example. Determine the points on the cycloid where the tangent line is horizontal or vertical. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 161 5.2 Polar Coordinates (This lecture corresponds to Section 10.3 of Stewart’s Calculus.) 1. Ancient math joke: Q: What’s a rectangular bear? A: A polar bear after a coordinate transform. 2. Motivation: Given a point in the plane how can we describe its position? 3. Polar Coordinate System. (a) Choose a point in the plane. Call it O, which we also call the pole. (b) Choose a ray starting at O. Call it the polar axis. (Usually taken as positive x axis.) (c) Take any point P, except O, in the plane. Measure the distance d(O, P) and call this distance r. (d) Measure the angle between the polar axis and the ray starting at O and passing through P going from x in counterclockwise direction. Let θ be this measure in radians. (e) There is a bijection between the plane and the set R + [0, 2π) = ¦(r, θ) : r ∈ R + and θ ∈ [0, 2π)¦ This means that each point in the plane is uniquely determined by a pair (r, θ) ∈ R + [0, 2π). (f) r and θ are called polar coordinates of P. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 162 4. Example. Plot the points whose polar coordinates (r, θ) are given. (a) (1, π/4) (b) (2, 5π/4) (c) (2, −π/3) (d) (−1, 5π/6) 5. Example. Plot the three points whose polar coordinates are (1, π/2), (1, 5π/2), and (−1, 3π/2). 6. Example. Plot the point given by the polar coordinates (3, π/3). Then find two other pairs of polar coordinates of this point, one with r > 0 and one with r < 0. 7. Example. Find the connection between polar and Cartesian coordinates. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 163 8. Example. Convert the point (2, π/6) from polar to Cartesian coordinates. 9. Example. Plot the point whose polar coordinates are (2 √ 2, 3π/4). Find the Cartesian coordinates of this point. 10. Example. Represent the point with Cartesian coordinates (−1, 1) in terms of polar coordinates. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 164 11. Example. The cartesian coordinates of a point are (−2 √ 3, −2). Find polar coordinates (r, θ) of this point, where r > 0 and 0 ≤ θ < 2π). 12. Example. Sketch the region in the plane consisting of points whose polar coordinates satisfy: 1 ≤ r < 2, π 4 ≤ θ < 3π 4 . 13. Polar Curves: The graph of a polar equation r = f(θ) consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Some examples of polar curves are: PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 165 14. Example. What curve is represented by the polar equation r = 3? 15. Example. Sketch the graph of the curve r = 2 sin θ. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 166 16. Example. Sketch the graph of the curve r = 2 cos 3θ. 17. Derivatives of Polar Curves. Suppose that r = f(θ) is a differentiable function of θ. Then from the parametric equations x = r cos θ y = r sin θ it follows that dy dx = dy dθ dx dθ = dr dθ sin θ + r cos θ dr dθ cos θ −r sin θ 18. Example. Find an equation of the tangent line to the curve r = 2 cos 3θ when θ = 2π/3. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 167 19. Example. Find an equation of the tangent line to the curve r = 1 + cos θ if θ = π/6. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 168 20. Consider the curve given by the polar equation r = 1 + 2 sin θ. Find the point(s) on the curve which are furthest from the x-axis. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.2 POLAR COORDINATES 169 21. Additional Notes PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 170 5.3 Conic Sections (This lecture corresponds to Section 10.5 of Stewart’s Calculus.) 1. Quote. “If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me? ” (St. Augustine) 2. Objective: We are going to give geometric descriptions of parabolas, el- lipses and hyperbolas and derive their equations in Cartesian coordinates. These types of curves are called conic sections, or just conics, because they arise from intersecting a cone and a plane. 3. Reminder. • The distance d between two points (x 1 , y 1 ) and (x 2 , y 2 ) is given by d = _ (x 2 −x 1 ) 2 + (y 2 −y 1 ) 2 . • The midpoint M between two points (x 1 , y 1 ) and (x 2 , y 2 ) is given by M = _ x 1 + x 2 2 , y 1 + y 2 2 _ . 4. Geometric Definition of a Parabola. A parabola is the set of points in a plane that are equidistant from a fixed point F and a fixed line. 5. Terminology: • The point F is called the focus and the line is called the directrix. • The line through the focus perpendicular to the directrix is called the axis of the parabola. • The intersection of the parabola and the axis is called the vertex. (Note: The vertex is halfway between the focus and the directrix.) PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 171 6. Deriving the Equation for a Parabola: We get a simple equation for a parabola in Cartesian coordinates, if we start by making the axis of the parabola the y-axis. Then we place the vertex of the parabola at the origin O, and the x-axis parallel to the directrix. With the focus at (0, p), the equation of the directrix is . PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 172 7. Theorem. An equation of the parabola with focus (0, p) and directrix y = −p is y = ax 2 where a = 1 4p . When p > 0 the parabola opens upward, and when p < 0 it opens downward. 8. If we interchange the x and y in the equation x 2 = 4py, we obtain y 2 = 4px which is the equation of a parabola opening to the right if p > 0, or to the left if p < 0. (Interchanging x and y means reflecting about the line y = x.) All four cases for the parabola and the equation are shown below. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 173 9. Example. Find the vertex, focus, and directrix of the parabola and sketch its graph. 2y 2 = 5x 10. Example. Find an equation of the parabola. Then find the focus and directrix. 11. Geometric Definition of an Ellipse. An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. (F 1 and F 2 are called foci.) PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 174 12. Deriving the Equation for an Ellipse: To get a simple equation for an ellipse, let the foci be at (−c, 0) and (c, 0). This means the origin is halfway between the foci. Let the sum of the distances from a point on the ellipse to both foci be the constant 2a > 0. x 2 1 F (!c, 0) P (x, y) y F (c, 0) PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 175 13. Theorem. The ellipse x 2 a 2 + y 2 b 2 = 1, a ≥ b > 0 has foci (±c, 0), where c 2 = a 2 −b 2 , and vertices (±a, 0). 14. Terminology: • The points (a, 0) and (−a, 0) are both on the ellipse, and are called vertices. • The line segment joining the vertices is called the major axis. The length is 2a. • The line segment joining the center to one of the vertices is called the semimajor axis. The length of the semimajor axis is a which is also called the long radius of the ellipse. 15. If we interchange x and y then the equation for the ellipse becomes x 2 b 2 + y 2 a 2 = 1, where a ≥ b > 0. The foci are (0, ±c), and the vertices are (0, ±a). (Remember, interchanging x and y is the same as reflecting in the diagonal line y = x.) x !"# %& !"# '& !!(# "& !(# "& !"# !'& y !"# !%& PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 176 16. Example. Find the vertices and the foci of the ellipse and sketch its graph. 100x 2 + 36y 2 = 225 17. Geometric Definition of a Hyperbola. A hyperbola is the set of points in a plane the difference of whose distances from two fixed points F 1 and F 2 is a constant. (F 1 and F 2 are called foci.) P 1 2 F (c, 0) (!c, 0) F Notice the similarity to the definition of the ellipse. The only change here is that the sum of distances has become a difference. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 177 18. This similarity to the ellipse means that the method of construction of an equation for the hyperbola is very similar to that of the ellipse. If we let the foci lie on the x-axis at (±c, 0), and the distance difference be 2a then the equation of the hyperbola is: x 2 b 2 − y 2 a 2 = 1, where c 2 = a 2 + b 2 . The x-intercepts (±a, 0) are called the vertices and the lines y = ±(b/a)x are the asymptotes. !!"# %& !'# %& !!'# %& !"# %& 19. If the foci of the hyperbola are on the y-axis, then we switch x and y to get the equation y 2 a 2 − x 2 b 2 = 1, with foci (0, ±c) where c 2 = a 2 + b 2 and vertices (0, ±a) and asymptotes y = ±(a/b)x. !"# !%& !"# '& !"# !'& !"# %& PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 178 20. Conics That Are Shifted: So far, we have been constructing formulas for the conics by placing the origin at the simplest possible location. What happens if we shift the conics? In general, we replace x and y by x −h and y −k , for constants h and k. 21. Example. Find an equation of the ellipse with foci (2, −2) and (4, −2), and with vertices (1, −2) and (5, −2). PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.3 CONIC SECTIONS 179 22. Additional Notes PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.4 CONIC IN POLAR COORDS. 180 5.4 Conic Sections in Polar Coordinates (This lecture corresponds to Section 10.6 of Stewart’s Calculus.) 1. Math Joke. Q: Why do pirates work in polar coordinates? A: So their work involves lots or AARRR!!!! 2. Objective: We’re going to look at parabolas, ellipses and hyperbolas again, but this time in polar form. To begin with, we can give a more unified description of all three types of conic sections in terms of a focus and directrix. This will allow us to see that conic sections can be easily described in polar coordinates. 3. Definition. Let F be a fixed point, called the focus, and l be a fixed line, called the directrix. Suppose e is a fixed positive number, called the eccentricity. The set of all points P in the plane such that [PF[ [Pl[ = e is a conic section. If e < 1 the conic is an ellipse. If e = 1 the conic is a parabola. If e > 1 the conic is a hyperbola. This definition of the conic sections is equivalent to the geometric definitions given earlier. The proof of this is in the text, and involves some algebra. Notice that when e = 1, we have [PF[ = [Pl[ for the definition of the parabola, as before. 4. Using the definition for conic sections given above find equations in polar coordinates which describe the curves. P x = d F x y d PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.4 CONIC IN POLAR COORDS. 181 5. If the directrix is chosen to be on the left of the focus, at x = −d, then the conic section is r = ed 1 −e cos θ . See the first figure below. ! # !$ ! F % % # !$ directrix ! F % % # $ directrix % F ! directrix 6. If the directrix is chosen parallel to the x-axis then the conic section is as shown in the second two figures of the diagram above. 7. Theorem. A polar equation of the form r = ed 1 ±e cos θ or r = ed 1 ±e cos θ represents a conic section with eccentricity e. The conic is an ellipse if e < 1, a parabola if e = 1, or a hyperbola if e > 1. 8. Example. Find a polar equation for an ellipse, with eccentricity 1 2 , that has its focus at the origin and whose directrix is the line y = 4. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.4 CONIC IN POLAR COORDS. 182 9. Example. A conic is given by the polar equation r = 3 2 + 2 cos θ . Find the eccentricity, identify the conic, locate the directrix, and sketch the conic. 10. Example. PART 5: PARMETRIC EQNS, POLAR COORDS LECTURE 5.4 CONIC IN POLAR COORDS. 183 11. Additional Notes PART 6: REVIEW MATERIAL CONIC IN POLAR COORDS. 184 Part 6 Review Material 185 PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 186 6.1 Midterm 1 Review Package 1. Make sure you know the definitions of the terms: function, one-to-one, inverse of a function, compo- sition, even function, odd function, limit of a function, continuous function, asymptote, squeeze law, intermediate value property, derivative, etc. 2. Write sin 2 (e x 2 −x ) as a composition of (elementary) functions. 3. Compute the following limits. (a) lim x→2 (x 2 −1) 3 (b) lim x→0 e x 2 −1 + sin x x + 1 (c) lim x→3 − x 2 −9 [x −3[ PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 187 (d) lim x→0 1 − √ 1 −x 2 x (e) lim h→0 f(4 + h) −f(4) h where f(x) = 1 √ x (f) lim x→∞ ln _¸ ¸ ¸ ¸ 2 −3x 3 x 3 + 5x −4 ¸ ¸ ¸ ¸ _ PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 188 4. True or False. Justify your answers. (a) If f(s) = f(t) then s = t. (b) If f is an odd function and f(3) = 6 then f(−3) = −6. (c) If x 1 < x 2 and g is a decreasing function then g(x 1 ) > g(x 2 ). (d) If f and g are functions then f ◦ g = g ◦ f. (e) lim x→4 _ 2x x −4 − 8 x −4 _ = lim x→4 _ 2x x −4 _ − lim x→4 _ 8 x −4 _ . (f) If lim x→5 f(x) = 2 and lim x→5 g(x) = 0, then lim x→5 f(x) g(x) does not exist. PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 189 (g) If lim x→5 f(x) = 0 and lim x→5 g(x) = 0, then lim x→5 f(x) g(x) does not exist. (h) If g(1) = −1 and g(2) = 5 then there exists a number c between 1 and 2 such that g(c) = 0. (i) If 1 ≤ f(x) ≤ x 2 + 2x + 2 for all x near −1, then lim x→−1 f(x) = 1. (j) If the line x = 1 is a vertical asymptote of y = f(x), then f is not defined at 1 . (k) The equation x + ln (x + 1) = x 4 −1 has a root in the interval (0, 2). (l) If f is continuous on [1, 5] such that f(2) = 8 and the only solutions of the equation f(x) = 6 are x = 1 and x = 4 then f(3) > 6. (m) If f (r) exists, then lim x→r f(x) = f(r). PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 190 5. Determine the constant c that makes f continuous on (−∞, ∞). f(x) = _ c 2 + sin (xπ) if x < 2 cx 2 −4 if x ≥ 2 . PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 191 6. Is there a number b such that lim x→−2 3x 2 + bx + b + 3 x 2 + x −2 exists? If so, find the value of b and the value of the limit. 7. Evaluate lim x→2 √ 6 −x −2 √ 3 −x −1 PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 192 8. Let f(x) = 1 x 2 −x . Calculate f (x) directly from the definition of derivative. Find the tangent line to the curve y = f(x) at the point (2, f(2)). 9. For what value of x does the graph of f(x) = e x −2x have a horizontal tangent? 10. Where does the normal line to the parabola y = x − x 2 at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch. PART 6: REVIEW MATERIAL MIDTERM 1 REVIEW PACKAGE 193 Answers: Only answers are provided here. You are expected to provide fully worked out solutions. If you need help with solving any of these problems please visit the Calculus Workshop. 1. Know the precise statements of definitions and theorems as found in the textbook and the notes. 2. f ◦ g ◦ h ◦ k where f(x) = x 2 , g(x) = sin x, h(x) = e x , h(x) = x 2 −x. 3. (a) 27 (b) e −1 = 1 e (c) −6 (d) 0 (e) −1 16 (f) ln 3 4. (a) F (b) T (c) T (d) F (e) F (f) T (g) F (h) F (i) T (j) F (k) T (l) T (j) T. 5. c = 2 6. b = 15; limit is −1 7. 1 2 8. (a) −3/4 (b) y = (−3/4)x + 2 9. x = ln 2 10. (−1, −2) PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 194 6.2 Midterm 2 Review Package 1. Compute the following derivatives. You do not need to simplify your answers. (a) f (x) if f(x) = (2x 6 −4x + 3) 4 . (b) g (x) if g(x) = sec x xe x . (c) h (x) if h(x) = 3 + 2 sin x x 3 + 1 (d) y if y = x 2 log 3 (x 2/3 ) PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 195 (e) ds dt if s = 2 t 2 (f) h (51) (t) if h(t) = ln (t 2 ). (Compute the first few derivatives to find a pattern.) (g) dy dx ¸ ¸ ¸ ¸ x=0 if 2 _ x y _ −ln (x + y) = 0 (h) y if y = x cos x . PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 196 2. Suppose f is a differentiable such that f(g(x)) = x and f (x) = 1 + [f(x)] 2 . Show that g (x) = 1 1 + x 2 . 3. True or False. Justify your answers. (a) If f and g are differentiable then the derivative of f(x)g(x) is f (x)g (x). (b) The function f(x) = [x[ is differentiable for all real numbers. (c) If f is differentiable, then d dx _ f(x) = f (x) 2 √ x . (d) d dx (10 x ) = x10 x−1 . PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 197 4. The graph of f is given below. Sketch the graph of f and f (x) on coordinate axes below. You do NOT need to justify your answers. PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 198 5. If a hemispherical bowl of radius 10 in. is filled with water to a depth of x inches, the volume of water is given by V = π(10 − x 3 )x 2 . Find the rate of increase of volume per inch increase in depth. 6. Use logarithmic differentiation to find the derivative y of the following function (you do not need to simplify your answer) y = √ x 2 + 1 (3 −4x) 5 2(3x −1) 1/4 (x −2) 4 PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 199 7. Consider the curve defined by y 2 = x 3 + 5x 2 . The graph of the curve is shown below. (a) Show that the point (−1, 2) is on the curve. (b) Use implicit differentiation to find dy dx . (c) Find the equation of the tangent line to the curve at the point (−1, 2). PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 200 8. The position of a particle moving along a straight line is given by the function s(t) = t 3 − 15 4 t 2 + 3t + 2, t ≥ 0 (a) What is the particles starting position? (b) When is the particle speeding up? When is it slowing down? When is it stopped? (c) Find the total distance the particle travels in the time interval 0 ≤ t ≤ 4. PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 201 9. The position at time t ≥ 0 of a particle moving along a coordinate line is s(t) = 10 cos (t + π/4). (a) What is the particles starting position (t = 0)? (b) What are the points farthest to the left and right of the origin reached by the particle. (c) Find the particles velocity and acceleration at the points in (b). (d) When does the particle first reach the origin? What are its velocity and acceleration at this point? PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 202 10. At 12:00 an apple pie is removed from the oven and placed on a table to cool. The temperature of the room is 24 ◦ C. At 12:20 the temperature of the pie is 36 ◦ C and is decreasing at a rate of 2 ◦ /min. What was the temperature of the pie when it was brought out of the oven? 11. The designer of a 30-ft-diameter spherical hot-air balloon wishes to suspend the gondola 8 ft below the bottom of the balloon with suspension cables tangent to the surface of the balloon. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (−12, −9) and (12, −9). How wide must the gondola be? PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 203 12. A girl facing North is standing next to a river which flows East. She tosses a stick into the water exactly 4 meters North of where she stands. The river carries the stick East at the constant rate of 3 m/s. How fast is the stick moving away from the girl after 2 seconds? 13. A cup of coffee, cooling off in a room at temperature 20 ◦ C, has cooling constant k = 0.09min −1 . (a) How fast is the coffee cooling (in degrees per minute) when its temperature is T = 80 ◦ C? (b) Use linear approximation to estimate the change in temperature over the next 6 seconds when T = 80 ◦ C. (c) The coffee is served at a temperature of 90 ◦ . How long should you wait before drinking it if the optimal temperature is 65 ◦ C? Note: You may leave your answers in the exact form, i.e., as an expression that contains powers of e and/or logarithms. PART 6: REVIEW MATERIAL MIDTERM 2 REVIEW PACKAGE 204 Answers: Only answers are provided here. You are expected to provide fully worked out solutions. If you need help with solving any of these problems please visit the Calculus Workshop. 1. (a) f (x) = 4(2x 6 −4x + 3) 3 (12x 5 −4) (b) g (x) = _ sec x x 2 e x _ (xtan x −x −1) (c) h (x) = 2(x 3 + 1) cos x −(3x 2 )(3 + 2 sin x) (x 3 + 1) 2 (d) y = 2xlog 3 (x 2/3 ) + _ 2 3 ln 3 _ x = _ 2x 3 ln 3 _ (2 ln x + 1) (e) ds dt = 2 ln (2)t2 t 2 (f) h (51) (t) = 2 (50!)t −51 (g) dy dx ¸ ¸ ¸ ¸ x=0 = 1 (h) x cos x _ cos x x −ln (x) sin x _ 2. Use implicit differentiation on f(g(x)) = x and solve for g(x). 3. (a) False (b) False (c) False (d) False 5. dV dx = πx(20 −x) 6. y = √ x 2 + 1 (3 −4x) 5 2(3x −1) 1/4 (x −2) 4 _ x x 2 + 1 − 20 3 −4x − 3 4(3x −1) − 4 x −2 _ 7. (b) y = x(3x + 10) 2y (c) y = −7 4 x + 1 4 8. (a) 2 (b) stopped at t = 1/2 and 2. Speeding up when 1 2 < t < 5 4 or 2 < t. Slowing down when 0 < t < 1 2 or 5 4 < t < 2. (c) 155/8 9. (a) 10/ √ 2 (b) Point furthest to the left are s = −10 and these occur when t = (2n + 1)π − π 4 , where n ∈ Z. Point furthest to the right are s = 10 and these occur when t = 2nπ − π 4 , where n ∈ Z. (c) For leftmost points v = 0 and a = 10. For rightmost points v = 0 and a = −10. (d) t = π/4. v(π/4) = −10 and a(π/4) = 0. 10. 12e 10/3 + 24 degrees. 11. 3 feet 12. 9 √ 13 ≈ 2.5 m/s 13. (a) −5.4 ◦ C/min (b) −0.54 ◦ C (c) t = −1 0.09 ln _ 45 70 _ PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 205 6.3 End of Term Review Notes 1. Special Limits: lim x→0 sin x x = 1 lim x→0 (1 + x) 1/x = e lim n→∞ _ 1 + 1 n _ n = e 2. Definition of Derivative: The derivative of a function f at a number a, denote by f (a), is f (a) = lim h→0 f(a + h) −f(a) h if this limit exists. 3. Differentiation Rules: (Part 3) (a) General Formulas: d dx (c) = 0 d dx [cf(x)] = cf (x) d dx [x n ] = nx n−1 d dx [f(x) ±g(x)] = f (x) ±g (x) d dx [f(x)g(x)] = f(x)g (x) + f (x)g(x) (product rule) d dx _ f(x) g(x) _ = f (x)g(x)−f(x)g (x) [g(x)] 2 (quotient rule) d dx [f(g(x))] = f (g(x))g (x) (chain rule) (b) Exponential and Logarithmic Functions: d dx (e x ) = e x d dx (a x ) = a x ln a d dx (ln [x[) = 1 x d dx (log a x) = 1 x ln a (c) Trigonometric functions: d dx (sin x) = cos x d dx (cos x) = −sin x d dx (tan x) = sec 2 x (d) Inverse Trigonometric Functions: d dx (sin −1 x) = 1 √ 1−x 2 d dx (cos −1 x) = − 1 √ 1−x 2 d dx (tan −1 x) = 1 1+x 2 (e) Hyperbolic Functions: d dx (sinh x) = cosh x d dx (cosh x) = sinh x d dx (tanhx) = sech 2 x (f) Inverse Hyperbolic Functions: d dx (sinh −1 x) = 1 √ 1+x 2 d dx (cosh −1 x) = 1 √ x 2 −1 d dx (tanh −1 x) = 1 1−x 2 PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 206 4. Natural Growth Equation: (Lecture 3.8) The solution of the initial-value problem dy dt = ky, y(0) = y 0 is y(t) = y 0 e kt . Radioactive Decay problems: Usually k is specified in terms of the half-life of the isotope τ = ln 2 k . This is the time required for half of any given quantity to decay. Newton’s Law of Cooling/Heating problems: The temperature T of an object is modeled by: dT dt = k(T −M) −→ T(t) = Ae kt + M where • M is the temperature of the surroundings (ambient temperature - which is constant) • k a constant (called the heating/cooling constant) 5. Linear Approximation and Differentials: (Lecture 3.10) The linear function L(x) = f(a) + f (a)(x −a) is called the linearization of f at a. For x close to a we have that f(x) ≈ L(x) = f(a) + f (a)(x −a) and this approximation is called the linear approximation of f at a. The differential dy is defined as dy = f (x)∆x = f (x)dx . PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 207 6. L’Hospital’s Rule: (Lecture 4.4) Suppose that f and g are differentiable and g (x) ,= 0 near a (except possibly at a.) Suppose that lim x→a f(x) = 0 and lim x→a g(x) = 0 or that lim x→a f(x) = ±∞and lim x→a g(x) = ±∞ Then lim x→a f(x) g(x) = lim x→a f (x) g (x) if the limit on the right side exists (or is ∞or −∞). 7. Example. Compute the following limit: lim x→0 (ln x) 2 x 8. Newton’s Method for approximating solutions to f(x) = 0: (Lecture 4.8) i. Begin with an initial guess x 1 . ii. Calculate x 2 = x 1 − f(x 1 ) f (x 1 ) . iii. If x n is known then x n+1 = x n − f(x n ) f (x n ) . iv. If x n and x n +1 agree to k decimal places then x n approximates the root r up to k decimal places and f(x n ) ≈ 0. 9. Example. (a) Show that the equation e x = 5x has exactly two solutions. (b) Use Newton’s Method to find the two solutions to the equation in (a) to three decimal places. PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 208 10. Increasing/Decreasing Test. (a) If f (x) > 0 on an interval, then f is increasing ¸on that interval. (b) If f (x) < 0 on an interval, then f is decreasing ¸on that interval. 11. Definition. A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) does not exist. 12. The First Derivative Test. Suppose that c is a critical number of a continuous function f. (a) If f changes from positive to negative at c, then f has a local maximum at c. (b) If f changes from negative to positive at c, then f has a local minimum at c. (c) If f does not change sign at c, then f has no local minimum or maximum at c. 13. Concavity Test. (a) If f”(x) > 0 for all x ∈ I, then the graph of f is concave upward ( on I. (b) If f”(x) < 0 for all x ∈ I, then the graph of f is concave downward ) on I. 14. Definition. A point P on a curve y = f(x) is called an inflection point if f is continuous there the curve changes from concave upward ( to concave downward ) or from concave downward ) to concave upward ( at P. 15. The Second Derivative Test. Suppose f” is continuous near c. (a) If f (c) = 0 and f”(c) > 0 then f has a local minimum at c. (b) If f (c) = 0 and f”(c) < 0 then f has a local maximum at c. 16. The Mean Value Theorem. (Lecture 4.2) ) Let f be a function that satisfies the following hypothe- ses: (a) f is continuous on the closed interval [a, b]. (b) f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f (c) = f(b) −f(a) b −a or, equivalently, f(b) −f(a) = f (c)(b −a). 17. Closed Interval Method for finding Absolute Extrema: To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: (a) Find the values of f at the critical numbers of f in (a, b). (b) Find the values of f at the endpoints of the interval. (c) The largest of the values from Step 1 and Step 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. 18. Example. Find the absolute maximum and absolute minimum values of f(x) = e −x − e −2x on the interval [0, 1]. PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 209 19. Derivatives of Parametric Curves: (Lecture 5.1) The derivative to the parametric curve x = f(t), y = g(t) is given by dy dx = _ dy dt _ _ dx dt _ = g (t) f (t) . The second derivative is given by d 2 y dx 2 = d dx _ dy dx _ = d dt _ dy dx _ dx dt . 20. Derivative of Polar Curves: (Lecture 5.2) Suppose that r = f(θ) is a differentiable function of θ. Then from the parametric equations x = r cos θ y = r sin θ it follows that dy dx = dy dθ dx dθ = dr dθ sin θ + r cos θ dr dθ cos θ −r sin θ 21. Example. True or False. Justify your answers. (a) If f(x) is differentiable at x = a then f(x) is continuous at x = a. (b) If f (c) = 0 then f has a local max/min at c. (c) If f (x) < 0 for 1 < x < 6, then f is decreasing on (1, 6). (d) If f (2) = 0 then (2, f(2)) is an inflection point of f. (e) If f (x) = g (x) for 0 < x < 1, then f(x) = g(x) for 0 < x < 1. (f) There exists a function f such that f(x) > 0, f (x) < 0, and f (x) > 0 for all x. (g) If f is periodic then f is periodic. PART 6: REVIEW MATERIAL END OF TERM REVIEW NOTES 210 (h) If f is even then f is even. (i) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. (j) The equations r = 2, x 2 + y 2 = 4 and x = 2 sin (3t), y = 2 cos (3t) (0 ≤ t ≤ 2π 3 ) all have the same graph. (Answers: 7 0 21 (a) T (b) F (c) T (d) F (e) F (f) T (g) T (h) F (i) F (j) T ) PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 211 6.4 Final Exam Checklist (Page numbers refer to Stewart’s Calculus [4].) Definitions Chapter 2: 1. Limit (page 87) 2. Left-hand (right-hand) limit (page 92) 3. Infinite limit (pages 93 and 94) 4. Vertical asymptote (page 94) 5. Function continuous at a number (page 118) 6. Function continuous from the right (left) at a number (page 120) 7. Function continuous on an interval(page 120) 8. Limit at infinity (page 131) 9. Horizontal asymptote (page 131) 10. Tangent line (page 143) 11. The derivative of a function at a number (page 146) 12. Function differentiable on a set (page 157) 13. Vertical tangent line (page 159) Chapter 3: 14. The number e (page 179) 15. Linear approximation (page 251) 16. Differential (page 253) 17. Hyperbolic functions (page 257) Chapter 4: 18. Absolute maximum/minimum (page 274) 19. Local maximum/minimum (page 274) 20. Critical number (page 277) 21. Function concave upward/downward (page 293) 22. Inflection point (page 294) 23. Antiderivative of a function (page 344) Chapter 10: 24. Parameter, parametric equations, parametric curve (page 636) 25. Initial and terminal points (page 637) 26. The Cycloid (page 639) PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 212 27. Polar coordinate system, pole, polar axis, polar coordinates (page 654) Theorems/ Formulas/ Procedures Chapter 2: 1. lim x→a f(x) = L if and only if lim x→a − f(x) = L and lim x→a+ f(x) = L’ (page 92, page 104) 2. Limit Laws (Page 99-106) 3. Direct Substitute Property (page 101) 4. Theorem about the monotonicity and limits. (page 105) 5. The Squeeze Theorem (page 105) 6. Continuity and combinations of functions (page 121) 7. Continuity of polynomials and rational functions (page 122) 8. Continuity and the combination of functions (page 123) 9. Continuity and the composition of functions (page 124, 125) 10. The Intermediate Value Theorem (page 125) 11. Limits at infinity of the power functions with negative rational exponents (page 133) 12. Relationship between differentiable and continuous functions (page 158) Chapter 3: 13. Derivative of a constant function (page 174) 14. Power rule (pages 175 and 176) 15. The Constant Multiple Rule (page 177) 16. The Sum Rule (page 177) 17. The Difference Rule (page 178) 18. Derivative of the natural exponential function (page 180) 19. The Product Rule (page 185) 20. The Quotient Rule (page 187) 21. Derivatives of Trigonometric Functions (pages 193 and 194) 22. The Chain Rule (page 199) 23. Derivative the exponential function f(x) = a x (page 203) 24. Derivatives of Inverse Trigonometric Functions (page 214) 25. Derivatives of Logarithmic Functions (page 218 and 220) 26. e as a limit (page 222) 27. The solution of the initial-value problem (page 237) 28. Hyperbolic Identities (page 258) 29. Derivatives of Hyperbolic Functions (page 259) 30. Derivatives of Inverse Hyperbolic Functions (page 261) PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 213 Chapter 4: 31. The Extreme Value Theorem (page 275) 32. Fermat’s Theorem (page 276) 33. The Closed Interval Method (page 278) 34. Rolle’s Theorem (page 284) 35. The Mean Value Theorem (page 285) 36. The relationship between two functions implied by the equality of their derivatives (pages 288) 37. Increasing/Decreasing test (page 290) 38. The First Derivative Test (page 291) 39. Concavity Test (page 293) 40. The Second Derivative Test (page 295) 41. L’Hospital’s Rule (page 302) 42. Newton’s Method (page 339) 43. The relationship among antiderivatives of a function (page 344) Examples Chapter 2: 1. Function with neither left-hand nor right-hand limits at the given point (page 90) 2. Function with the left-hand and right hand limits that are not equal (page 91) 3. Function with infinite left-hand and right hand limits (page 93) 4. Function with an infinite number of vertical asymptotes (page 95) 5. Function F = f g so that the limits of F and f at a exist and the limit of g at a does not exist. (page 106) 6. Function F = f + g so that the limit of F at a exists and the limits of f and g at a do not exist. 7. Function F = fg so that the limit of F at a exists and the limits of f and g at a do not exist. 8. Function with a removable (infinite, jump) discontinuity. (page 119-120) 9. Function with the graph that intersects its horizontal asymptote (page 131) 10. Function with two horizontal asymptotes (page 132) 11. Function that is continuous but not differentiable (page 157) 12. Function with a vertical tangent line (page 159) 13. Function with a “corner” (page 159) Chapter 3: 14. Function that is the same as its derivative (page 180) [Can you find THREE different functions with this property?] PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 214 Chapter 4: 15. Function f with no minimum, no maximum but such that f (a) = 0 for some a. (page 275) 16. Function with a local minimum that is not the global minimum. (page 275) 17. Function(s) that does not satisfied the hypothesis of the Extreme Value Theorem. (page 275-276) 18. Function with a critical number but no maximum or minimum. (page 277) 19. Sketch the graph of a function that is increasing for x < 0, decreasing for 0 < x < 1, and increas- ing for x > 1. Also, suppose this function has a horizontal asymptote when x → −∞ and a slant asymptote when x →∞. 20. Function that is concave upward (concave downward) (page 293) 21. Function with an inflection point at which the first derivative equals 0. 22. Function with a local minimum at which the second derivative equals 0. Limits Are you able to do most of the following: 1. Exercises 2.3: 1-32, 37-50 2. Exercises 2.6: 1-57 3. Exercises 3.3: 39-48 4. Exercises 4.4: 1-66 Differentiation Are you able to do most of the following: 1. Exercises 2.7: 5-40 2. Exercises 2.8: 21-32, 37-40, 43-46, 50-57 3. Exercises 3.2: 3-35, 41-50 4. Exercises 3.3: 1-34, 49-51 5. Exercises 3.4: 1-54, 59-78 6. Exercises 3.5: 1-30, 35-40, 42-60, 73-80 7. Exercises 3.6: 1-34, 37-50 8. Exercises 3.11: 30-47 PART 6: REVIEW MATERIAL FINAL EXAM CHECKLIST 215 Applications 1. Tangent lines (throughout the text) 2. Velocity (throughout text) 3. The Intermediate Value Theorem (Section 2.5: Exercises 48-54) 4. Rates of change (Section 2.7: Exercises 41-48, Section 3.7: Exercises 7-26) 5. Exponential growth and decay (Section 3.8: all exercises 1-20) 6. Related rates (Section 3.9: all exercises) 7. Linear Approximation and Differentials (Section 3.10: Exercises 11-22, 33-40) 8. Maximum and minimum values (Section 4.1: Exercises 47-62, 69-71) 9. Mean Value Theorem (Section 4.2: Exercises 15-31) 10. Graphs of functions (Section 4.3 and Section 4.5) 11. Optimization (Section 4.7) 12. Newton’s method (Section 4.8: Exercises 5-8, 11-22) 13. Antiderivative (Section 4.9: Exercises 25-48, 65-69) Miscellaneous 1. Section 10.1: Exercises 1-28 2. Section 10.2: Exercises 1-30 3. Section 10.3: Exercises 15-46, 55-64 (page 648-649) PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 216 6.5 Final Exam Practice Questions Please refer to the final exam checklist (section 6.4) for an indication of what you will need to know for the final exam (in short, the material from any section we covered this term will be on the exam, with the exception of lectures 2.4, 5.3 and 5.4). To prepare for the exam you should: (i) read all sections of the textbook again; (ii) go through all the homework questions again and make sure you can do every single one of them, (iii) work through more problems from the textbook; (iv) work through the final exam checklist (this can be done in conjunction with (i) through (iii)). Only once you have done all this should you attempt the following questions. The following is a list of practice exam questions. This will give you an idea of the types of questions you will be asked on the exam. Instructions: No calculators, books, papers, or electronic devices shall be allowed within the reach of a student during the examination. Leave answers in ”calculator ready” expressions: such as 3 + ln 7 or e √ 2 . Questions 1-3 are on the statements of definitions and theorems and on your ability to give ex- amples of functions with specified properties.. 1. Define the following terms. (a) Limit (b) Function continuous at a number (c) Function continuous on an interval (d) Tangent line (e) The derivative of a function at a number (f) Function differentiable a on a set (g) The number e (h) Differential (i) Absolute maximum and absolute minimum (j) Local maximum and local minimus (k) Critical number (l) Function concave upward and concave downward (m) Inflection point (n) Antiderivative of a function 2. State the following theorems. (a) The Squeeze Theorem (b) The Intermediate Value Theorem (c) Fermat’s Theorem (d) Extreme Value Theorem (e) Rolle’s Theorem (f) The Mean Value Theorem (g) L’Hospital’s Rule 3. Give an example for each of the following. (a) Function with an infinite number of vertical asymptotes. PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 217 (b) Function F = f g so that the limits of F and f at a exist and the limit of g at a does not exist. (c) Function with a removable discontinuity. (d) The most general form of a function with the property that its second derivative is the zero function. (e) Function that is continuous but not differentiable at a point. (f) Function with a critical number but no maximum or minimum. (g) Function with a local minimum at which the second derivative equals 0. Questions 4-16 are short answer questions. The questions are given in no particular order. 4. Find the derivative y = dy dx of each of the following: (a) y = cos −1 (x 2 ) −ln (1 + x 3 ) [Note: Another notation for cos −1 is arccos.] (b) y = x sin(x) . (c) arctan _ y x _ = 1 2 ln (x 2 + y 2 ). 5. Let f(x) = tan x. Find f (x), the second derivative of f. 6. Find the tangent line to the curve y + xln y −2x = 0 at the point (1/2, 1). 7. Evaluate lim x→∞ 2x 3 + 3x −1 1 −2x 2 + 5x 3 . 8. Evaluate lim x→1 x 2 −1 x 2 −3x + 2 . 9. Let f(x) = 2x x 2 + 3 . (a) Find the equation of the tangent line to the curve y = f(x) at x = 1. (b) Use linear approximation to give an approximate value for f(1.2). 10. A particle moves along the x-axis so that its position at time t is given by x = t 3 −4t 2 + 1. (a) At t = 2, what is the particle’s speed? (b) At t = 2, in what direction is the particle moving? (c) At t = 2, is the particle’s speed increasing or decreasing? 11. The curve y = xe −x has one inflection point. Find the x-coordinate of this point. 12. Find an equation of the slant asymptote to the curve y = x 3 + 2x 2 x 2 + 3x + 2 . 13. Find a number x 0 between 0 and π such that the tangent line to the curve y = sin x at x = x 0 is parallel to the line y = −x/2. 14. Evaluate lim x→0 xsin 1 x . 15. Evaluate lim x→0 e x −1 −x x 2 . 16. Evaluate lim x→0 x sin x . Questions 17-33 are full-solution problems. Justify your answers and show all your work. The questions are given in no particular order. PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 218 17. Let f(x) = 5 3x −1 . Calculate f (2) directly from the definition of derivative. 18. A water-trough is 10m long and has a cross-section which is the shape of an isosceles trapezoid that is 30cm wide at the bottom, 80cm wide at the top, and has height 50cm. If the trough is being filled with water at the rate of 0.2 m 3 /min, how fast is the water level rising when the water is 30cm deep? [Recall: The area of an isosceles trapezoid as shown in the diagram is A = 1 2 (a + b)h.] 19. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50m. 20. At 2:00 p.m.a car’s speedometer reads 30 mi/h. At 2:10 p.m. it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h 2 . 21. A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other is bent into an circle. How should the wire be cut so that the total area enclosed is minimum. 22. A turkey is put into an oven that has a constant temperature of 200 ◦ C. A thermometer embedded in the turkey registers its temperature. When the turkey is put into the oven, the thermometer reads 20 ◦ C, and 30 minutes later it reads 30 ◦ C. The turkey will be ready to eat when the thermometer reads 80 ◦ C. How many minutes after being put into the oven will the turkey be ready to eat? Assume that the turkey’s temperature satisfies Newton’s law of cooling/heating. 23. Sketch the graph of the function f(x) = −2x 2 + 5x −1 2x −1 . 24. Let f(x) = 2x 3 −6x 2 + 3x + 1. (a) First show that f has at least one zero in the interval [2, 3] and then use the first derivative of f to show that there is exactly one root of f between 2 and 3. (b) Use Newton’s method to approximate the root of f in the interval [2, 3] by starting with x 1 = 5/2 and finding x 2 . 25. Find the dimensions of the largest rectangle that can be inscribed inside a semicircular region of radius 5 such that one side of the rectangle is parallel to the base of the semicircular region. 26. (a) A metal storage tank with fixed volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal? (b) Suppose the metal for the hemisphere costs twice as much as the metal for the lateral sides. What are the dimensions for the tank that minimizes cost? (Recall: The volume of a sphere of radius r is 4 3 πr 3 and the surface area is 4πr 2 .) 27. (a) Show that Newton’s Method applied to the equation x 2 −a = 0 yields the iterative formula x n+1 = 1 2 _ x n + a x n _ and thus provides a method for approximating the square root √ a which uses only addition and multiplication. (b) Approximate √ 3 by taking x 1 = 3/2 and calculating x 2 . PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 219 28. Find f if f (x) = 2 + cos x, f (0) = −1 and f(π/2) = 0. 29. Sketch the curve which is given by the parametric equations x = cos (πt), y = sin (πt), 1 ≤ t ≤ 2. Clearly label the initial and terminal points and describe the motion of the point (x(t), y(t)) as t varies in the given interval (i.e. indicate the direction the point is traveling). 30. A curve called the folium of Descartes is defined by the parametric equations x = 3t 2 (t + 1) 3t 2 + 3t + 1 , y = −3t(t + 1) 2 3t 2 + 3t + 1 , −∞< t < ∞. (a) Show that a Cartesian equation of this curve is x 3 + y 3 = 3xy. (b) Find the point on the curve corresponding to t = −1/2. (c) Find the equation of the tangent line to the curve at the point corresponding to t = −1/2. (d) Find the values of the parameter t which correspond to the point (0, 0) on the curve. (e) Find equations of the tangent lines to the curve at the point (0, 0). 31. Consider the curve given by the parametric equations x = 2 sin t, y = 4 + cos t, 0 ≤ t ≤ 2π. Determine the points on the curve which are closest to the origin and those which are furthest away. 32. Sketch the curve with the polar equation r = 2 cos 4θ. 33. Consider the curve given by the polar equation r = 1 + 2 sin (3θ), 0 ≤ θ ≤ 2π. (a) Determine the points on the curve which are furthest from the origin. (b) Find the slope of the tangent line to each of the points found in part (a). PART 6: REVIEW MATERIAL FINAL EXAM PRACTICE QUESTIONS 220 Answers: 4. (a) y = − 2x √ 1 −x 4 − 3x 2 1 + x 3 (b) y = x sin x ( sin x x + ln x cos x) (c) y = x + y x −y 5. f (x) = 2 sec 2 (x) tan (x) 6. y = 4 3 x + 1 3 7. 2/5 8. −2 9. (a) y = 1 4 x + 1 4 (b) f(1.2) ≈ 1 4 (1.2) + 1 4 = 11 20 = 0.55 10. (a) −4 (b) to the left (c) speed is decreasing 11. x = 2 12. y = x −1 13. 2π/3 14. 0 (Hint: use Squeeze Theorem since sin(1/x) is bounded) 15. 1/2 16. 1 17. −3/5 18. 1 30 m/min = 10 3 cm/min 19. 5 8 π ≈ 2 m 3 20. Hint: use Mean Value Theorem 21. The length of wire used to make the square should be 40 π+4 22. t = ln (2/3) 2 ln (17/18) ≈ 3.55 hours ≈ 3 hours 33 minutes 24. (b) x2 = 16/7 ≈ 2.285714286 25. base is 5 √ 2 and height is 5 √ 2 26. (a) hemisphere (b) r = 1 2 3 3V/π and h = 3 3V/π. 27. (b) 7 4 28. f(x) = −cos x + x 2 −x + π/2(1 −π/2) 29. Consists of points on the unit semicircle in quadrants 3 and 4. Points are moving counterclockwise along curve with initial point (−1, 0) and terminal point (1, 0). 30. (b) (3/2, 3/2) (c) y = −x + 3 (d) t = −1, 0 (e) There are two lines: the first is when t = −1 and the tangent is the horizontal line y = 0, the second occurs when t = 0 and is the vertical line x = 0. 31. closest point is (0, 3), furthest point is (0, 5) 33. (a) The three points are (r, θ) = (3, π 6 ), (3, 5π 6 ), (3, 3π 2 ). (b) slopes of tangents at these points are − √ 3, √ 3, and 0, respectively. Bibliography [1] C. Adams, A. Thompson, and J. Hass. How to Ace Calculus: The Streetwise Guide. W.H. Freeman and Company, 1998. [2] B. ´ Curgus. An exceptional exponential function. The College Mathematics Journal, 37(5):344–354, 2006. http://myweb.facstaff.wwu.edu/curgus/Papers/ExcExpFinal.pdf. [3] D. Ebersole, D. Schattschneider, A. Sevilla, and K. Somers. A Companion to Calculus. Brooks/Cole Sengage Learning, 2nd edition, 2005. [4] J. Stewart. Calculus: Early Transcendentals. Brooks/Cole Sengage Learning, 7th edition, 2011. 221