Applied Calculus
7th Edition
ISBN: 9781337291248
Author: Waner, Stefan.
Publisher: Cengage Learning,
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Chapter 9.3, Problem 58E
To determine
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In Exercises 1–4, use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width.
1. ƒ(x) = x2 between x = 0 and x = 1.
2. ƒ(x) = x3 between x = 0 and x = 1.
3. ƒ(x) = 1/x between x = 1 and x = 5.
4. ƒ(x) = 4 - x2 between x = -2 and x = 2.
In Exercises 27–32, use a graphingutility to graph the function on the closed interval [a, b].Determine whether Rolle’s Theorem can be applied to f on theinterval and, if so, find all values of c in the open interval (a, b)such that f '(c= ' 0.)
f(x)=|x|-1,[-1,1]
In Exercises 1–14, to establish a big-O relationship, find wit-
nesses C and k such that [f(x) k.
1. Determine whether each of these functions is O(x).
a) f(x) = 10
c) f(x) = x² +x+ 1
e) f(x) = [x]
b) f(x) — Зх +7
d) f(x) = 5 log x
f) f(x) = [x/2]
%3D
%3D
Chapter 9 Solutions
Applied Calculus
Ch. 9.1 - Prob. 1ECh. 9.1 - Prob. 2ECh. 9.1 - Prob. 3ECh. 9.1 - Prob. 4ECh. 9.1 - Prob. 5ECh. 9.1 - Prob. 6ECh. 9.1 - Prob. 7ECh. 9.1 - Prob. 8ECh. 9.1 - Prob. 9ECh. 9.1 - Prob. 10E
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In a...Ch. 9.2 - Simple Harmonic Motion and Damped Harmonic Motion...Ch. 9.2 - Prob. 68ECh. 9.2 - Prob. 69ECh. 9.2 - Prob. 70ECh. 9.2 - Prob. 71ECh. 9.2 - Prob. 72ECh. 9.2 - Prob. 73ECh. 9.2 - Prob. 74ECh. 9.2 - Prob. 75ECh. 9.2 - Prob. 76ECh. 9.2 - Prob. 77ECh. 9.2 - Prob. 78ECh. 9.2 - Prob. 79ECh. 9.2 - Prob. 80ECh. 9.2 - If the value of a stock price is given by...Ch. 9.2 - Prob. 82ECh. 9.2 - Prob. 83ECh. 9.2 - Prob. 84ECh. 9.3 - Prob. 1ECh. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - Prob. 15ECh. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.3 - Prob. 22ECh. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - Prob. 27ECh. 9.3 - Prob. 28ECh. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Prob. 33ECh. 9.3 - Prob. 34ECh. 9.3 - Prob. 35ECh. 9.3 - Prob. 36ECh. 9.3 - Prob. 37ECh. 9.3 - Prob. 38ECh. 9.3 - Prob. 39ECh. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.3 - Prob. 44ECh. 9.3 - Prob. 45ECh. 9.3 - Prob. 46ECh. 9.3 - Prob. 47ECh. 9.3 - Prob. 48ECh. 9.3 - Prob. 49ECh. 9.3 - Prob. 50ECh. 9.3 - Prob. 51ECh. 9.3 - Prob. 52ECh. 9.3 - Prob. 53ECh. 9.3 - Prob. 54ECh. 9.3 - Prob. 55ECh. 9.3 - Prob. 56ECh. 9.3 - Prob. 57ECh. 9.3 - Prob. 58ECh. 9.3 - Prob. 59ECh. 9.3 - Prob. 60ECh. 9.3 - Prob. 61ECh. 9.3 - Prob. 62ECh. 9.3 - Prob. 63ECh. 9.3 - Prob. 64ECh. 9.3 - Prob. 65ECh. 9.3 - Prob. 66ECh. 9.3 - Prob. 67ECh. 9.3 - Tides The depth of water at my favorite surfing...Ch. 9.3 - Prob. 69ECh. 9.3 - Prob. 70ECh. 9.3 - Prob. 71ECh. 9.3 - Prob. 72ECh. 9.3 - Prob. 73ECh. 9.3 - Prob. 74ECh. 9.3 - Prob. 75ECh. 9.3 - Prob. 76ECh. 9 - Prob. 1RECh. 9 - Prob. 2RECh. 9 - Prob. 3RECh. 9 - Prob. 4RECh. 9 - Prob. 5RECh. 9 - Prob. 6RECh. 9 - Prob. 7RECh. 9 - Prob. 8RECh. 9 - In Exercises 9-14, find the derivative of the...Ch. 9 - Prob. 10RECh. 9 - Prob. 11RECh. 9 - Prob. 12RECh. 9 - Prob. 13RECh. 9 - Prob. 14RECh. 9 - Prob. 15RECh. 9 - Prob. 16RECh. 9 - Prob. 17RECh. 9 - Prob. 18RECh. 9 - Prob. 19RECh. 9 - Prob. 20RECh. 9 - Prob. 21RECh. 9 - Prob. 22RECh. 9 - Prob. 23RECh. 9 - Prob. 24RECh. 9 - Prob. 25RECh. 9 - Prob. 26RECh. 9 - Prob. 27RECh. 9 - Prob. 28RECh. 9 - Prob. 29RECh. 9 - Prob. 30RECh. 9 - Prob. 31RECh. 9 - Prob. 32RECh. 9 - Prob. 1CSCh. 9 - Prob. 2CSCh. 9 - Prob. 3CSCh. 9 - Prob. 4CS
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- Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some ce R for which f(c) = K. You will have to select an appro- priate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere. 61. f(x) = x³ + 2, K = –15 62. f(x) = -2x²+ 4, K = 0 1 63. f(x) = sinx, K : 64. f(x) — sinх, К %— 2 65. f(x) — 1Зх + 11, К %3D 1 66. f(x) 3D 12 — Зx, К %3D 2arrow_forwardIn Exercises 49–51, sketch a graph of ƒ and identify the points c such that f'(c) does not exist. In which cases is there a corner at c?arrow_forwardEach of Exercises 15–30 gives a function f(x) and numbers L, c, and ɛ > 0. In each case, find an open interval about c on which the inequal- ity |f(x) – L| 0 such that for all x satisfying 0 0, L= 2m, c = 2, — тх, ɛ = 0.03 28. f(x) = mx, ɛ = c > 0 L = 3m, c = 3, m > 0, L = (m/2) + b, 29. f(x) c = 1/2, m> 0, ɛ = c > 0 = mx + b, 30. f(x) 3D тх + b, m> 0, L%3Dm+ b, с %3D 1, &%3D 0.05arrow_forward
- Answer the following questions about the functions whose derivatives are given in Exercises 1–5: a. What are the critical points of ƒ? b. On what open intervals is ƒ increasing or decreasing? c. At what points, if any, does ƒ assume local maximum and minimum values? 1. ƒ′(x) = x(x - 1) 2. ƒ′(x) = (x - 1)2(x + 2) 3. ƒ′(x) = (x - 1)(x + 2) 4. ƒ′(x) = (x - 1)2(x + 2)2 5. ƒ′(x) = (x - 1)e-xarrow_forwardIn Exercises 23–30, find T, centered at x = a for all n. 26. ƒ (x) = e" , a = -2 %3|arrow_forwardIn Exercises 102–103, find a. (fog)(x); b. the domain of (fo g). x + 1 * - 2" 103. f(x) = Vx – 1, g(x) = x + 3 102. f(x) = 8(x)arrow_forward
- Use the Extreme Value Theorem to show that each function f in Exercises 49–54 has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are contin- uous everywhere. 49. f(x) = xª – 3x² – 2, [a, b] = [–2, 2] 50. f) — х1 — 3х? — 2, [а, b] — [0, 2] 51. f) %3D х* - Зx? - 2, [а, b] %—D [-1, 1] 52. f(x) = 3 – 2r² +x³, [a, b] = [-1, 2] 53. f(x) = 3 – 2x2 +x³, [a, b] = [0, 2] 54. f(x) = 3 – 2x?+x³, [a, b] = [–1, 1] %3Darrow_forwardThis is exercise 6 of Section 7.5 page 460. Find the integral tan(Va) dx | +Carrow_forwardSolve 2)Realize Roll's theorem of the function f(x) = x + 1/x on [1/2, 2]. 3)Realize Mean Value theorem of the function f(x) = x + 1/x on [1/2, 3].arrow_forward
- Section 2.4.3. Evaluate each of the following integrals. 277 (24) sin³ (+2eit) dt 0arrow_forwardSection 3.7 p. 343/345 # 350 Evaluate the following integrals. If the integral is not convergent, answer “divergent." 99 1 dx 1 xln x 350arrow_forward(§5.4, Q42) Calculate So™ f(x) dx, where sin x, x Tarrow_forward
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