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Calculus Volume 1

# Review Exercises

Calculus Volume 1Review Exercises

### 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 .

$lim x → 2 2 x 2 − 3 x − 2 x − 2 lim x → 2 2 x 2 − 3 x − 2 x − 2$

214 .

$lim x → 0 3 x 2 − 2 x + 4 lim x → 0 3 x 2 − 2 x + 4$

215 .

$lim x → 3 x 3 − 2 x 2 − 1 3 x − 2 lim x → 3 x 3 − 2 x 2 − 1 3 x − 2$

216 .

$lim x → π / 2 cot x cos x lim x → π / 2 cot x cos x$

217 .

$lim x → −5 x 2 + 25 x + 5 lim x → −5 x 2 + 25 x + 5$

218 .

$lim x → 2 3 x 2 − 2 x − 8 x 2 − 4 lim x → 2 3 x 2 − 2 x − 8 x 2 − 4$

219 .

$lim x → 1 x 2 − 1 x 3 − 1 lim x → 1 x 2 − 1 x 3 − 1$

220 .

$lim x → 1 x 2 − 1 x − 1 lim x → 1 x 2 − 1 x − 1$

221 .

$lim x → 4 4 − x x − 2 lim x → 4 4 − x x − 2$

222 .

$lim x → 4 1 x − 2 lim x → 4 1 x − 2$

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

223 .

$lim x → 0 x 2 cos ( 2 π x ) = 0 lim x → 0 x 2 cos ( 2 π x ) = 0$

224 .

$lim x → 0 x 3 sin ( π x ) = 0 lim x → 0 x 3 sin ( π x ) = 0$

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 .

$f ( x ) = { x 2 + 1 , x > c 2 x , x ≤ c f ( x ) = { x 2 + 1 , x > c 2 x , x ≤ c$

227 .

$f ( x ) = { x + 1 , x > − 1 x 2 + c , x ≤ − 1 f ( x ) = { x + 1 , x > − 1 x 2 + c , x ≤ − 1$

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

228 .

$lim x → 1 ( 8 x + 16 ) = 24 lim x → 1 ( 8 x + 16 ) = 24$

229 .

$lim x → 0 x 3 = 0 lim x → 0 x 3 = 0$

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 $t=0.01sec.t=0.01sec.$

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