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Precalculus 2e

Review Exercises

Precalculus 2eReview Exercises

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Review Exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.
Figure 1
1.

limx−1+f(x)

2.

limx−1f(x)

3.

limx1f(x)

4.

limx3f(x)

5.

At what values of x is the function discontinuous? What condition of continuity is violated?

6.

Using Table 1, estimate limx0f(x).

x F(x)
−0.12.875
−0.012.92
−0.0012.998
0Undefined
0.0012.9987
0.012.865
0.12.78145
0.152.678
Table 1

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a. If the function has limit as x approaches a, state it. If not, discuss why there is no limit.

7.

f(x)={|x|1,ifx1x3,ifx=1  a=1

8.

f(x)={1x+1,ifx=2(x+1)2,ifx2  a=2

9.

f(x)={x+3,ifx<13x,ifx>1  a=1

Finding Limits: Properties of Limits

For the following exercises, find the limits if limxcf(x)=−3 and limxcg(x)=5.

10.

limxc(f(x)+g(x))

11.

limxcf(x)g(x)

12.

limxc(f(x)g(x))

13.

limx0+f(x),f(x)={3x2+2x+15x+3  x>0x<0

14.

limx0f(x),f(x)={3x2+2x+15x+3  x>0x<0

15.

limx3+(3x[x])

For the following exercises, evaluate the limits using algebraic techniques.

16.

limh0((h+6)236h)

17.

limx25(x2625x5)

18.

limx1(x29xx)

19.

limx4712x+1x4

20.

limx3(13+1x3+x)

Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at x=a. If not, describe the behavior of the graph of the function at x=a.

21.

f(x)=2x4;a=4

22.

f(x)=2(x4)2;a=4

23.

f(x)=xx2x6;a=3

24.

f(x)=6x2+23x+204x225;a=52

25.

f(x)=x39x;a=9

For the following exercises, determine where the given function f(x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

26.

f(x)=x22x15

27.

f(x)=x22x15x5

28.

f(x)=x22xx24x+4

29.

f(x)=x31252x212x+10

30.

f(x)=x21x2x

31.

f(x)=x+2x23x10

32.

f(x)=x+2x3+8

Derivatives

For the following exercises, find the average rate of change f(x+h)f(x)h.

33.

f(x)=3x+2

34.

f(x)=5

35.

f(x)=1x+1

36.

f(x)=ln(x)

37.

f(x)=e2x

For the following exercises, find the derivative of the function.

38.

f(x)=4x6

39.

f(x)=5x23x

40.

Find the equation of the tangent line to the graph of f(x) at the indicated x value.

f(x)=x3+4x; x=2.

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

41.

f(x)=x|x|

42.

Given that the volume of a right circular cone is V=13πr2h and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π

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