Differentiation Review, Part 1 (Part 2 follows; there are answers at the end of each part.)

Transcription

1 Differentiation Review 1 Name Differentiation Review, Part 1 (Part 2 follows; there are answers at the end of each part.) Derivatives Review: Summary of Rules Each derivative rule is summarized for you below. Write an example that would best exemplify each rule. 1. Power rule f (x) = ax b then f '(x) = a bx b-1 2. Constant rule f (x) = c then f '(x) = 0 3. Product rule y = f (x) g(x) then y'= f (x) g'(x) + g(x) f '(x) 4. Quotient rule y = f (x) g(x) then y'= g(x) f '(x) - f (x) g'(x) [g(x)] 2 Only use it when the denominator is a function of x; not for something like y = 2x Chain rule y = f (g(x)) then y'= f '(g(x)) g'(x) Or y = f (u) and u = g(x) then dy = dy du du or y'= f '(u) g'(x)

2 6. Logarithmic functions y = ln x then y'= 1 x Differentiation Review 2 more generally if y = lnu then dy = 1 u du 7. Exponential functions y = e x then y'= e x more generally if y = e u then dy = eu du 8. Trigonometric functions (You need to memorize the derivatives below.) d du d du sinu = cosu cosu = -sinu (You do NOT need to memorize the derivatives below.) d tanu = sec2 u du d cot u = -csc2 u du Examples: d du secu = sec utanu d du cscu = -csc ucot u

3 Differentiation Review 3 Important Techniques for Differentiation 1. Simplify first!!! y = x 2-3 x rewrite as y = x 4x x y'= x æ y = ln 1 ö ç è x 2 rewrite as y = ln1- ln( x 2 ) ø = 0-2ln ( x ) = 2 ln( x) 2. Implicit differentiation: when you cannot easily solve for y. Don t forget to use the product rule when necessary: x 2 + 2xy + 3y 2 = 7 2x + 2x dy dy + 2y + 6y = 0 2x dy dy + 6y = -2x - 2y dy -2x - 2y = 2x + 6y = -x - y x + 3y 3. Logarithmic differentiation: take the natural log of both sides, use the laws of logs to simplify, and then use implicit differentiation to find the derivative. y = 4 x sin x ln y = ln(4 x ) + ln( sin x ) = x ln ln(sin x) 2 1 y dy = ln sin x cos x dy æ cos x ö = y ln 4 + è 2sin xø dy = 4 x æ sin x ln è 2 cot x ö ø

4 Differentiation Review 4 Derivative Review Questions Complete on separate paper. Find the derivative of the following functions. You might find it helpful to simplify before taking the derivative. Simplify your answers. 1. g(t) = 2 3t 2. h(x) = 2 2 (3x) 2 3. y = x x 3 4. f (x) = x - 1 x 5. f (x) = x 3 (5-3x 2 æ ) 6. s = 4-1 ö è t 2 ø (t 2-3t) 7. f (x) = x 2 + x -1 x h(x) = 4x 2 + x 3x f (x) = x h(x) = 2 x g(t) = t (1- t) f (t) = (t +1) t y = 4e x y = x 2 e x 15. y = x 2x 16. y = 3 xe3x e 17. y = 5 1+ e 2x 18. y = e x 1- xe x 19. y = ln x(x -1) x y = x ln x 21. f (x) = ln 4x 22. y = ln(x 2-2) y = ln ex 1+ e x 24. y = e-x sin x 25. y = cos3x 26. y = x 2 sin 1 x 27. y = cos 2 x - sin 2 x 28. y = cos x sin x 29. y = sin x 30. y = sin5px 31. y = cos x cos(x -1) 32. y = x 2 x -1

5 Differentiation Review y = 3sin 2 4x + x 34. y = 1 2 esin 2x 35. f (x) = (x - 4)3 x 2 (3x +1) If f and g are the functions whose graphs are shown, let u(x) = f (x) g(x) and v(x) = f (x) g(x). Find: y a. u (1) = 4 f(x) b. v (5) = 2 g(x) c. u (0) = 5 x If g is a differentiable function, find an expression for the derivative of each of the following functions. æ a. y = x 6 g(x) b. y = ln 2x 4 + 3xö ç c. y = 14g(x) è g(x) ø e 7x 38. Use implicit differentiation to find an equation of the (a.) tangent and (b.) normal line at the indicated point. 2y - y 3 = xy at (-7, 3) dy 39. Find of these equations. a. 3x 2 y - 2x = cos(y) b. 3x 5-2y 3 = p c. ey - 2x 3 = xy 2

6 Differentiation Review Using the graphs below, calculate the following derivatives. a. h'(1) if h(x) = f (x) g(x) b. j'(1) if j(x) = f (x) g(x) c. m'(1) if m(x) = f (g(x)) d. k'(1) if k(x) = f ( f (x)) e. n'(4) if n(x) = g( f (x)) 41. Using the table below, calculate the following derivatives. a) If h(x) = f (g(x)), find h (1). b) If k(x) = g(x) f (x), find k'(3). c) If j(x) = [ f (x)] 3, find j'(2). d) If m(x) = f (x), find m'(3). [ g(x) ] 2 x f (x) g(x) f (x) g (x)

7 Differentiation Review 7 ANSWERS to Differentiation Review, Part 1 1. g'(t) = h'(x) = 3t 9x 3. y'= 3x x 4 4. f '(x) = 1 2 x f '(x) = 15x 2 (1- x 2 ) 6. s'= 8t x 3 t 2 7. f '(x) = - x x x - 9 (x 2-1) 8. 2 x f '(x) = 2 (3x 2-2) h'(x) = (x +1) g'(t) = 2x 3(x 2-1) t (1- t) 12. f '(t) = 2t 2 + t +1 4 t y'= 8xe x y'= xe x (x + 2) 15. y'= ( ) 16. y'= ex x x y'= 1 x + 1 x y'= 4x 3(x 2-2) x x e 2x 17. y'= -10e2x (1+ e 2x ) 18. y'= ex (1+ e x ) 2 (1- xe x ) y'= ln x ln x 25. y'= -3sin3x 26. y'= -cos 1 x + 2x sin 1 x 28. y'= -csc 2 x or 29. y'= cos x 2 sin x y'= -1- cot 2 x 21. f '(x) = 1 2x 23. y'=1- ex 1+e x = 1 1+e x 24. y'= e -x (cos x - sin x) 27. y'= -2sin2x or y'= -4sin xcos x 30. y'= 5p cos(5px) -x sin x - 2cos x -(x -1)sin(x -1) - cos(x -1) 31. y'= 32. y'= x 3 (x -1) 2 sin 2x 33. y'= 24sin(4x)cos(4x) +1 =12sin(8x) y'= cos2x e 35. f '(x) = (x - 4)3 x 2 æ 3 (3x +1) 2 x x - 6 ö è 3x +1ø 36 a. 0 b c. DNE 37 a. y'= x 6 g'(x) + 6g(x)x 5 b. y'= 8x x 4 + 3x - g'(x) g(x) 38 a. y = - 1 (x + 7) + 3 b. y = 6(x + 7) a. y' = 2-6xy 3x 2 +sin y b. y' = 5x4 2y 2 c. y' = 6x2 + y 2 e y - 2xy c. y'= 14 g'(x) - 98g(x) e 7x 40. a. -6 b. 5 8 c. 4 d. -1 e a. 30 b. 77 c. 15 d. -14

8 1. For the functions below: I. Where is it discontinuous? Name the type. II. Where is it not differentiable? Name the type. Differentiation Review 8 Differentiation Review, Part 2 a. c. b. d.

9 Differentiation Review 9 2. Sketch the graph of the derivatives of these functions. a. b.

10 Differentiation Review 10 c. 3. The graph below shows the position of a particle on a coordinate line. (a) When does the particle change direction? (b) When does the particle move at its greatest speed? (c) Graph the particle s velocity. Think about the following questions: When is the particle s velocity positive? negative? zero? 4. The graph below shows the velocity of a particle moving on a coordinate line. (a) When does the particle change direction? (b) When does the particle move at its greatest speed? (c) When does the particle have its greatest acceleration? (d) Graph the particle s acceleration. Think about the following questions: When is the particle s acceleration positive? negative? zero?

11 5. The following graph shows the velocity of a skydiver after he jumps out of a plane. At some point during his fall, his parachute opens to slow his fall before he lands on the ground. Here, positive velocity indicates a downward speed. Differentiation Review The cost of extracting T tons of ore from a copper mine is C = f ( T), where C is in dollars. Suppose that f 1000 ( ) = 250. f 1000 ( ) = 200,000 and a) What are the units of f ( T)? b) What is the average cost per ton of extracting 1,000 tons? c) What is the marginal cost per ton when 1,000 tons have been extracted? Approximate answers are okay. (a) When did the skydiver s parachute open? (b) When did the skydiver land on the ground? (c) During freefall, an object reaches a terminal velocity where the object no longer gains speed due to air resistance. What is the skydiver s terminal velocity during his freefall? 6. Find the values of A and B that make this function both continuous and differentiable. ì y = Ax 3 + B x 4 í î 4x +12 x > 4 8. Let f ( t) be the number of centimeters of rainfall since midnight. t is the time since midnight, in hours. a) What are the units of f ( t)? Explain the practical meaning of the following statements: b) f ( 7) =1.5. c) f ( 7) =.15 d) f 9 ( ) = 0 9. The following table gives the position of a moving body as a function of time. The position is measured in feet from some fixed point. The time is in seconds. t (s) p (ft ) a) What is the body s average velocity between t = 0 and t=4? b) What is the average velocity between t = 2.5 and t=3.5? c) Estimate the instantaneous velocity of the body at t=2.75? d) Approximately when does the body change direction?

12 Differentiation Review 12

13 Differentiation Review 13 Answers to: Chapter 3: Derivatives Test Review, Part 2 1. a. i. nowhere, ii. x = 2, cusp b. i. nowhere, ii. x = 4, corner c. i. x = 2, jump; ii. x = 2, discontinuity d. i. nowhere, ii. x = -1, vertical tangent 2. a. b. c. 3. a. about t = 3, 5 b. between t = 3 and 5 c. 5. a. about t = 60 second b. about t = 104 seconds c. about 185 mph 6. A = 1/12, B = 68/3 7. a. $ / ton of ore b. 200 $ / ton c. 250 $ / ton 4. a. about t = 9.2 b. t = 7 c. greatest acceleration between t = 3 and 4, greatest acceleration is between t = 4 to 7 d. 8. a. cm/hr b. after 7 hours (at 7am) it has rained 1.5 cm c. at 7am it is raining at.15 cm / hour d. at 9 am it has stopped raining 9. a. 0 ft/sec b. -32 ft/sec c. -22 ft/sec d. about t = 2 sec