Calculus Volume 1

# Chapter Review Exercises

Calculus Volume 1Chapter Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

208.

A function has to be continuous at $x=ax=a$ if the $limx→af(x)limx→af(x)$ exists.

209.

You can use the quotient rule to evaluate $limx→0sinxx.limx→0sinxx.$

210.

If there is a vertical asymptote at $x=ax=a$ for the function $f(x),f(x),$ then f is undefined at the point $x=a.x=a.$

211.

If $limx→af(x)limx→af(x)$ does not exist, then f is undefined at the point $x=a.x=a.$

212.

Using the graph, find each limit or explain why the limit does not exist.

1. $limx→−1f(x)limx→−1f(x)$
2. $limx→1f(x)limx→1f(x)$
3. $limx→0+f(x)limx→0+f(x)$
4. $limx→2f(x)limx→2f(x)$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

213.

$limx→22x2−3x−2x−2limx→22x2−3x−2x−2$

214.

$limx→03x2−2x+4limx→03x2−2x+4$

215.

$limx→3x3−2x2−13x−2limx→3x3−2x2−13x−2$

216.

$limx→π/2cotxcosxlimx→π/2cotxcosx$

217.

$limx→−5x2+25x+5limx→−5x2+25x+5$

218.

$limx→23x2−2x−8x2−4limx→23x2−2x−8x2−4$

219.

$limx→1x2−1x3−1limx→1x2−1x3−1$

220.

$limx→1x2−1x−1limx→1x2−1x−1$

221.

$limx→44−xx−2limx→44−xx−2$

222.

$limx→41x−2limx→41x−2$

In the following exercises, use the squeeze theorem to prove the limit.

223.

$limx→0x2cos(2πx)=0limx→0x2cos(2πx)=0$

224.

$limx→0x3sin(πx)=0limx→0x3sin(πx)=0$

225.

Determine the domain such that the function $f(x)=x−2+xexf(x)=x−2+xex$ is continuous over its domain.

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

226.

$f(x)={x2+1,x>c2x,x≤cf(x)={x2+1,x>c2x,x≤c$

227.

$f(x)={x+1,x>−1x2+c,x≤−1f(x)={x+1,x>−1x2+c,x≤−1$

In the following exercises, use the precise definition of limit to prove the limit.

228.

$limx→1(8x+16)=24limx→1(8x+16)=24$

229.

$limx→0x3=0limx→0x3=0$

230.

A ball is thrown into the air and the vertical position is given by $x(t)=−4.9t2+25t+5.x(t)=−4.9t2+25t+5.$ Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

231.

A particle moving along a line has a displacement according to the function $x(t)=t2−2t+4,x(t)=t2−2t+4,$ where x is measured in meters and t is measured in seconds. Find the average velocity over the time period $t=[0,2].t=[0,2].$

232.

From the previous exercises, estimate the instantaneous velocity at $t=2t=2$ by checking the average velocity within $t=0.01sec.t=0.01sec.$