Chapter 4 Review Exercises

1. If $f(x)<0,f^{\prime }(x)<0$ for all $x$, then the right-hand rule underestimates the integral ${\int }_a^{b}f(x)$. Use a graph to justify your answer.

2.  ${\int }_a^{b}f{(x)}^{2}dx={\int }_a^{b}f(x)dx{\int }_a^{b}f(x)dx$

3.  If $f(x)\le g(x)$ for all $x\in [a,b]$, then ${\int }_a^{b}f(x)\le {\int }_a^{b}g(x)$.

4.  All continuous functions have an antiderivative.

5.  $y=3x^2-2x+1$ over $[-1,1]$

6.  $y=\sqrt{x}+\frac{1}{x}$ over $[1,4]$

7.  ${\int }_{-1}^1(x^3-2x^2+4x)dx$

8.  ${\int }_0^4\frac{3t}{\sqrt{1+6t^2}}dt$

9.  $\int \frac{dx}{{(x+4)}^3}$

10.  $\frac{d}{dx}{\int }_1^{x^3}\sqrt{4-t^2}dt$

11.  $\frac{d}{dx}{\int }_1^{\text{ln}(x)}(4t+e^t)dt$

12.  If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

13.  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$. Find the average cost per gigabyte of RAM for 1980 to 2010.

14.  Find the average cost of 1GB RAM for 2005 to 2010.

15.  The velocity of a bullet from a rifle can be approximated by $v(t)=6400t^2-6505t+2686$, 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: $0\le t\le 0.5$. What is the total distance the bullet travels in 0.5 sec?

16.  What is the average velocity of the bullet for the first half-second?