5.R: Chapter 5 Review Exercises
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In exercises 1 - 4, answer True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous over their domains.
1) If f(x)>0,f′(x)>0 for all x, then the right-hand rule underestimates the integral ∫baf(x)dx. Use a graph to justify your answer.
- Answer
- False
2) ∫baf(x)2dx=∫baf(x)dx
3) If f(x)≤g(x) for all x∈[a,b], then ∫baf(x)dx≤∫bag(x)dx.
- Answer
- True
4) All continuous functions have an antiderivative.
In exercises 5 - 8, evaluate the Riemann sums L4 and R4 for the given functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.
5) y=3x2−2x+1) over [−1,1]
- Answer
- L4=5.25,R4=3.25, exact answer: 4
6) y=ln(x2+1) over [0,e]
7) y=x2sinx over [0,π]
- Answer
- L4=5.364,R4=5.364, exact answer: 5.870
8) y=√x+1x over [1,4]
In exercises 9 - 12, evaluate the integrals.
9) ∫1−1(x3−2x2+4x)dx
- Answer
- −43
10) ∫403t√1+6t2dt
11) ∫π/2π/32sec(2θ)tan(2θ)dθ
- Answer
- 1
12) ∫π/40ecos2xsinxcosxdx
In exercises 13 - 16, find the antiderivative.
13) ∫dx(x+4)3
- Answer
- −12(x+4)2+C
14) ∫xln(x2)dx
15) ∫4x2√1−x6dx
- Answer
- 43sin−1(x3)+C
16) ∫e2x1+e4xdx
In exercises 17 - 20, find the derivative.
17) ddt∫t0sinx√1+x2dx
- Answer
- sint√1+t2
18) ddx∫x31√4−t2dt
19) ddx∫ln(x)1(4t+et)dt
- Answer
- 4lnxx+1
20) ddx∫cosx0et2dt
In exercises 21 - 23, consider the historic average cost per gigabyte of RAM on a computer.
Year | 5-Year Change ($) |
1980 | 0 |
1985 | −5,468,750 |
1990 | −755,495 |
1995 | −73,005 |
2000 | −29,768 |
2005 | −918 |
2010 | −177 |
21) If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.
- Answer
- $6,328,113
Solution: $6,328,113
22) The average cost per gigabyte of RAM can be approximated by the function C(t)=8,500,000(0.65)t, where t is measured in years since 1980, and C is cost in US dollars. Find the average cost per gigabyte of RAM for the period from 1980 to 2010.
23) Find the average cost of 1 GB RAM from 2005 to 2010.
- Answer
- $73.36
24) The velocity of a bullet from a rifle can be approximated by v(t)=6400t2−6505t+2686, where t is seconds after the shot and v is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: 0≤t≤0.5. What is the total distance the bullet travels in 0.5 sec?
25) What is the average velocity of the bullet for the first half-second?
- Answer
- 1911712 ft/sec, or about 1593 ft/sec