Precalculus

# Practice Test

PrecalculusPractice Test

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

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.

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.

$lim h→0 ( h 2 +25 −5 h 2 ) lim h→0 ( h 2 +25 −5 h 2 )$

11.

$lim h→0 ( 1 h − 1 h 2 +h ) lim h→0 ( 1 h − 1 h 2 +h )$

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.

$f(x)= x 3 −4 x 2 −9x+36 x 3 −3 x 2 +2x−6 f(x)= x 3 −4 x 2 −9x+36 x 3 −3 x 2 +2x−6$

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.

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.

$f(x)=| x−2 |−| x+2 | f(x)=| x−2 |−| x+2 |$

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.

$s(2)−s(1) 2−1 s(2)−s(1) 2−1$

24.

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

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

25.

$lim x→0 sin(x) 3x lim x→0 sin(x) 3x$

26.

$lim x→0 tan 2 (x) 2x lim x→0 tan 2 (x) 2x$

27.

$lim x→0 sin(x)(1−cos(x)) 2 x 2 lim x→0 sin(x)(1−cos(x)) 2 x 2$

28.

Evaluate the limit by hand.

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

$f(x)={ 4x−7 x≠1 x 2 −4 x=1 f(x)={ 4x−7 x≠1 x 2 −4 x=1$

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

Figure 3
29.

Find the average rate of change of the function from

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:

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

33.

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

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:$lim h→0 f(x+h)−f(x) h lim h→0 f(x+h)−f(x) h$

35.

$f(x)=2x−8 f(x)=2x−8$

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.

$f(x)= 3 x−1 f(x)= 3 x−1$

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.

$f(x)= x−1 f(x)= x−1$