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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 limxaf(x)limxaf(x) exists.

209.

You can use the quotient rule to evaluate limx0sinxx.limx0sinxx.

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 limxaf(x)limxaf(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. limx1f(x)limx1f(x)
  3. limx0+f(x)limx0+f(x)
  4. limx2f(x)limx2f(x)
A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).

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)=x2+xexf(x)=x2+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)=t22t+4,x(t)=t22t+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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