Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5.R: Chapter 5 Review Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

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)dxbag(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=3x22x+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) 11(x32x2+4x)dx

Answer
43

10) 403t1+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) 4x21x6dx

Answer
43sin1(x3)+C

16) e2x1+e4xdx

In exercises 17 - 20, find the derivative.

17) ddtt0sinx1+x2dx

Answer
sint1+t2

18) ddxx314t2dt

19) ddxln(x)1(4t+et)dt

Answer
4lnxx+1

20) ddxcosx0et2dt

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)=6400t26505t+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: 0t0.5. 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

5.R: Chapter 5 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

  • Was this article helpful?

Support Center

How can we help?