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Precalculus

Practice Test

PrecalculusPractice Test

Practice Test

For the following exercises, use the graph of f f in Figure 1.

Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.
Figure 1
1.

f(1) f(1)

2.

lim x −1 + f(x) lim x −1 + f(x)

3.

lim x −1 f(x) lim x −1 f(x)

4.

lim x−1 f(x) lim x−1 f(x)

5.

lim x−2 f(x) lim x−2 f(x)

6.

At what values of x x is f f discontinuous? What property of continuity is violated?

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 a limit as x x approaches a, a, state it. If not, discuss why there is no limit

7.

f(x)={ 1 x 3, if x2 x 3 +1,if x>2   a=2 f(x)={ 1 x 3, if x2 x 3 +1,if x>2   a=2

8.

f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1 f(x)={ x 3 +1, if x<1 3 x 2 1, if x=1 x+3 +4, if x>1 a=1

For the following exercises, evaluate each limit using algebraic techniques.

9.

lim x−5 ( 1 5 + 1 x 10+2x ) lim x−5 ( 1 5 + 1 x 10+2x )

10.

lim h0 ( h 2 +25 5 h 2 ) lim h0 ( h 2 +25 5 h 2 )

11.

lim h0 ( 1 h 1 h 2 +h ) lim h0 ( 1 h 1 h 2 +h )

For the following exercises, determine whether or not the given function f f is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12.

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

13.

f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6 f(x)= x 3 4 x 2 9x+36 x 3 3 x 2 +2x6

For the following exercises, use the definition of a derivative to find the derivative of the given function at x=a. x=a.

14.

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

15.

f(x)= 3 x f(x)= 3 x

16.

f(x)=2 x 2 +9x f(x)=2 x 2 +9x

17.

For the graph in Figure 2, determine where the function is continuous/discontinuous and differentiable/not differentiable.

Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.
Figure 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.

18.

f(x)=| x2 || x+2 | f(x)=| x2 || x+2 |

19.

f(x)= 2 1+ e 2 x f(x)= 2 1+ e 2 x

For the following exercises, explain the notation in words when the height of a projectile in feet, s, s, is a function of time t t in seconds after launch and is given by the function s(t). s(t).

20.

s(0) s(0)

21.

s(2) s(2)

22.

s'(2) s'(2)

23.

s(2)s(1) 21 s(2)s(1) 21

24.

s(t)=0 s(t)=0

For the following exercises, use technology to evaluate the limit.

25.

lim x0 sin(x) 3x lim x0 sin(x) 3x

26.

lim x0 tan 2 (x) 2x lim x0 tan 2 (x) 2x

27.

lim x0 sin(x)(1cos(x)) 2 x 2 lim x0 sin(x)(1cos(x)) 2 x 2

28.

Evaluate the limit by hand.

lim x1 f(x), where  f(x)={ 4x7 x1 x 2 4 x=1 lim x1 f(x), where  f(x)={ 4x7 x1 x 2 4 x=1

At what value(s) of x x is the function below discontinuous?

f(x)={ 4x7x1 x 2 4x=1 f(x)={ 4x7x1 x 2 4x=1

For the following exercises, consider the function whose graph appears in Figure 3.

Graph of a positive parabola.
Figure 3
29.

Find the average rate of change of the function from x=1 to x=3. x=1 to x=3.

30.

Find all values of x x at which f'(x)=0. f'(x)=0.

31.

Find all values of x x at which f'(x) f'(x) does not exist.

32.

Find an equation of the tangent line to the graph of f f the indicated point: f(x)=3 x 2 2x6,  x=2 f(x)=3 x 2 2x6,  x=2

For the following exercises, use the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 .

33.

Graph the function f(x)=x ( 1x ) 2 5 f(x)=x ( 1x ) 2 5 by entering f(x)=x ( ( 1x ) 2 ) 1 5 f(x)=x ( ( 1x ) 2 ) 1 5 and then by entering f(x)=x ( ( 1x ) 1 5 ) 2 f(x)=x ( ( 1x ) 1 5 ) 2 .

34.

Explore the behavior of the graph of f(x) f(x) around x=1 x=1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x=1. x=1.

For the following exercises, find the derivative of each of the functions using the definition: lim h0 f(x+h)f(x) h lim h0 f(x+h)f(x) h

35.

f(x)=2x8 f(x)=2x8

36.

f(x)=4 x 2 7 f(x)=4 x 2 7

37.

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

38.

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

39.

f(x)= 3 x1 f(x)= 3 x1

40.

f(x)= x 3 +1 f(x)= x 3 +1

41.

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

42.

f(x)= x1 f(x)= x1

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