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Precalculus

Review Exercises

PrecalculusReview Exercises

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.

lim x −1 + f(x) lim x −1 + f(x)

2.

lim x −1 f(x) lim x −1 f(x)

3.

lim x1 f(x) lim x1 f(x)

4.

lim x3 f(x) lim x3 f(x)

5.

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

6.

Using Table 1, estimate lim x0 f(x). lim x0 f(x).

xx F(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 x approaches a. a. If the function has limit as x x approaches a, a, state it. If not, discuss why there is no limit.

7.

f(x)={ | x |1, if x1 x 3 , if x=1   a=1 f(x)={ | x |1, if x1 x 3 , if x=1   a=1

8.

f(x)={ 1 x+1 , if x=2 (x+1) 2 , if x2   a=2 f(x)={ 1 x+1 , if x=2 (x+1) 2 , if x2   a=2

9.

f(x)={ x+3 , if x<1 x 3 , if x>1   a=1 f(x)={ x+3 , if x<1 x 3 , if x>1   a=1

Finding Limits: Properties of Limits

For the following exercises, find the limits if lim xc f( x )=−3 lim xc f( x )=−3 and lim xc g( x )=5. lim xc g( x )=5.

10.

lim xc ( f(x)+g(x) ) lim xc ( f(x)+g(x) )

11.

lim xc f(x) g(x) lim xc f(x) g(x)

12.

lim xc ( f(x)g(x) ) lim xc ( f(x)g(x) )

13.

lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0 lim x 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

14.

lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0 lim x 0 f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

15.

lim x 3 + ( 3x[x] ) lim x 3 + ( 3x[x] )

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

16.

lim h0 ( ( h+6 ) 2 36 h ) lim h0 ( ( h+6 ) 2 36 h )

17.

lim x25 ( x 2 625 x 5 ) lim x25 ( x 2 625 x 5 )

18.

lim x1 ( x 2 9x x ) lim x1 ( x 2 9x x )

19.

lim x4 7 12x+1 x4 lim x4 7 12x+1 x4

20.

lim x3 ( 1 3 + 1 x 3+x ) lim x3 ( 1 3 + 1 x 3+x )

Continuity

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

21.

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

22.

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

23.

f(x)= x x 2 x6 ;a=3 f(x)= x x 2 x6 ;a=3

24.

f(x)= 6 x 2 +23x+20 4 x 2 25 ;a= 5 2 f(x)= 6 x 2 +23x+20 4 x 2 25 ;a= 5 2

25.

f(x)= x 3 9x ;a=9 f(x)= x 3 9x ;a=9

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

26.

f(x)= x 2 2x15 f(x)= x 2 2x15

27.

f(x)= x 2 2x15 x5 f(x)= x 2 2x15 x5

28.

f(x)= x 2 2x x 2 4x+4 f(x)= x 2 2x x 2 4x+4

29.

f(x)= x 3 125 2 x 2 12x+10 f(x)= x 3 125 2 x 2 12x+10

30.

f(x)= x 2 1 x 2x f(x)= x 2 1 x 2x

31.

f(x)= x+2 x 2 3x10 f(x)= x+2 x 2 3x10

32.

f(x)= x+2 x 3 +8 f(x)= x+2 x 3 +8

Derivatives

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

33.

f(x)=3x+2 f(x)=3x+2

34.

f(x)=5 f(x)=5

35.

f(x)= 1 x+1 f(x)= 1 x+1

36.

f(x)=ln(x) f(x)=ln(x)

37.

f(x)= e 2x f(x)= e 2x

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

38.

f(x)=4x6 f(x)=4x6

39.

f(x)=5 x 2 3x f(x)=5 x 2 3x

40.

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

f(x)= x 3 +4x f(x)= x 3 +4x ; x=2. 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 | f(x)= x | x |

42.

Given that the volume of a right circular cone is V= 1 3 π r 2 h V= 1 3 π r 2 h 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 π

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