Free Printable Final Exam Reviewer

1.

When a function gets closer and closer to 7 as the variable gets closer to 4 the limit of the function?

a)

equals 4

b)

equals 7

c)

does not exist

d)

cannot be determined

2.

Evaluate $\lim_{x\rightarrow-4}\ \frac{4x+16}{x^2-16}$ using table of values

a)

$\infty$

b)

$-\infty$

c)

0.5

d)

-0.5

3.

Evaluate $\lim_{x\rightarrow2^+}\ g\left(x\right)$ given the graph .

a)

2

b)

-1

c)

0

d)

1

4.

Evaluate $\lim_{x\rightarrow-1^+}\ f\left(x\right)\$ given the graph.

a)

$\infty$

b)

- $\infty$

c)

-2

d)

-1

5.

Given , $f\left(x\right)\ =\ x^2-1\ if\ x\ >\ 2,\ \ and\ x\ -\ 4\ \ if\ x\ \le\ 2\ \ ,\ \lim_{x\rightarrow2^-}f\left(x\right)$ Given the piecewise function, evaluate the given limit.

a)

3

b)

2

c)

-2

d)

-3

6.

Given the piecewise function $f\left(x\right)=\ x^2-1\ ,\ if\ x\ >\ 3\ ,\ \ x\ -\ 4\ ,\ \ if\ x\ \ge3\$ evalaute f(3)

a)

-1

b)

8

c)

0

d)

DNE

7.

Evaluate $\lim_{x\rightarrow3}\left(x^2+7x-5\right)$

a)

25

b)

19

c)

14

d)

-25

8.

Evaluate $\lim_{x\rightarrow1}\sqrt{\frac{8x+1}{x+3}}$

a)

$\frac{\sqrt{9}}{2}$

b)

$\frac{9}{2}$

c)

$\sqrt{\frac{9}{4}}$

d)

$\frac{3}{2}$

9.

Evaluate $\lim_{x\rightarrow-3}\ \frac{x^2+x-6}{x^2+4x+3}$

a)

$\frac{10}{2}$

b)

$\frac{5}{2}$

c)

$\frac{-5}{2}$

d)

$\frac{-3}{2}$

10.

Evaluate $\lim_{x\rightarrow1}\ \frac{x-1}{\sqrt{x+3}-2}$

a)

indeterminate

b)

undefined

c)

4

d)

-4

11.

Evaluate $\lim_{x\rightarrow-\infty}\ \frac{5y^3-2y^2+y\ -\ 7}{4y^2-5y+2}$

a)

$-\infty$

b)

$\infty$

c)

$\frac{5}{4}$

d)

0

12.

Evaluate $\lim_{x\rightarrow9}\ \frac{9-x}{3-\sqrt{x}}$

a)

indeterminate

b)

6

c)

12

d)

9

13.

Evaluate $\lim_{x\rightarrow2}\ \ \frac{x^2-5x+6}{x-2}$

a)

5

b)

undefined

c)

-1

d)

indeterminate

14.

Evaluate $\lim_{x\rightarrow2}\ \frac{x^3-8}{x^4-16}$

a)

12

b)

undefined

c)

indeterminate

d)

1

15.

If $f\left(x\right)=\frac{x^2-16}{x-4}$ , why does f(4) does not exist?

a)

4 is not the domain of the function

b)

its indeterminate in form

c)

the vertical asymptote is x = 4

d)

the function is undefined

16.

$\lim_{x\rightarrow\infty}\ \left(\frac{1}{4}\right)^x$

a)

$-\infty$

b)

$\infty$

c)

1

d)

0

17.

Evaluate $\lim_{x\rightarrow0}\ \ \frac{\sin\ 2x}{x}$ using table of values

a)

0.959

b)

0.998

c)

1.999

d)

1

18.

Evaluate $\lim_{x\rightarrow\infty}\ e^{3x-5}$

a)

0

b)

3x-5

c)

$\infty$

d)

$e^{3x-5}$

19.

Evaluate $\lim_{x\rightarrow\ 3}\ \left[\log_{\frac{3}{2}}\ \frac{7x+6}{x^2-1}\right]$

(a)

20.

Evaluate $\lim_{x\rightarrow2\pi}\ \tan\frac{x}{2}$

(a)

21.

Evaluate $\lim_{x\rightarrow0}\ \frac{\sin\ \frac{1}{2}x}{\sin\ \pi x}$

(a)

22.

Evaluate $\lim_{x\rightarrow0}\ \frac{\sin\ \frac{3}{4}x}{x}$

(a)

23.

Evaluate $\lim_{x\rightarrow0}\ \frac{\sin\ 5x}{\sin\ 7x}$

(a)

24.

Evaluate $\lim_{x\rightarrow0}\ \frac{1-\cos\ 9x}{7x}$

(a)

25.

Evaluate $\lim_{x\rightarrow-\infty}\ \frac{e^{4x}+2}{e^x+e^{2x}}$ . Show your solution

a)

$\infty$

b)

$-\infty$

c)

0

d)

1

26.

The function $f\left(x\right)=\frac{x-3}{x^2-9}$ is continuous at _____

a)

5

b)

3

c)

-3

27.

Find the derivative of $f\left(x\right)=\ 4x^3-8x^2+4x\$ using differentiation rule.

a)

$f'\left(x\right)=12x^2-16x+4$

b)

$f'\left(x\right)=4x^2-8x$

c)

$f\left(x\right)=12x^2-16x+4$

d)

$f\left(x\right)=12x^2-16x$

28.

What is the equation of the tangent line to the curve $y=x^2-2x-3$ at (-1,0). Express in general form

(a)

29.

What is the slope of the tangent line to the graph of the function $f\left(x\right)=2x^3-3x^2$ at a given point

a)

$f'\left(x\right)=6x^2$

b)

$f'\left(x\right)\ =\ 2x^2-3x$

c)

$f'\left(x\right)=6x^2-6$

d)

$f'\left(x\right)\ =\ 6x^2-6x$

30.

What is the slope of the tangent curve $y=3x^2+6x+2$ at (-2,1)?

(a)

31.

Identify the type of discontinuity $f\left(x\right)\ =\ \frac{x^2+7x+10}{x^2+4x+5}$

a)

removable ata x = 5, infinite at x = -1

b)

infinite at x = -1

c)

removable at x = -5 , infinite at x = 1

d)

infinite at x = 2

32.

Identify the type of discontinuity $f\left(x\right)=\frac{x^2-1}{x-4}$

a)

infinite at x = 4

b)

removable at x = 4

c)

the function is continuous

d)

jump at x = 4

33.

Name the discontinuity at x = 2 given the graph

a)

infinite

b)

jump

c)

removable

d)

continuous at x = 2

34.

What will make the function $f\left(x\right)=\ \frac{3x^2+8}{x^2-4}$ continuous?

a)

2

b)

3

c)

-2

d)

-3

35.

If a function f has a domain $\left(-\infty,\infty\right)$ , then it is ____

a)

increasing

b)

continuous

c)

extraneous

d)

undefined

36.

Assume f(x) is continuous over the interval [-2,3] . The function f has at least how many zeros?

a)

1

b)

2

c)

3

d)

4

37.

Consider the function $f\left(x\right)=\frac{1}{x-3}-5$ ,Can you conclude that there must be a zero between f(2) and f(3.1)?

a)

Yes, because f(2) is negative and f(3.1) is positive.

b)

No, because f(2) is negative and f(3.1) is also negative.

c)

Yes, because f(2) is positive and f(3.1) is negative.

d)

No, because there is a discontinuity at x = 3.

38.

Which of the intervals is NOT continuous of the function $f\left(x\right)=\frac{x^2+x-20}{x-4}$

a)

$\left(\left(0,\infty\right)\right)$

b)

$\left[0,\infty\right]$

c)

(0,2)

d)

(1,2)

39.

Which of the following is false to determine that there is a solution to the given function $f\left(x\right)=x^3-x-1\ between\ x\ =\ 1\ and\ x\ =\ 2$

a)

There is more than one solution on the interval [1,2].

b)

The function is continuous because it is a polynomial function

c)

It is defined on the closed interval [1,2]

d)

If f(x)=0, then -1< 0 < 5

40.

Which of the following is false to prove that the function $f\left(x\right)=\ x^{4\ }+x^{3\ }-3x^{2\ }+2x\ -4\$ has a zero on the closed interval [-2,2]?

a)

if f(x)= 0 , then -12 < 0 < 12

b)

f(-2)=12, f(12)=12

c)

The function has a zero on [-2,2]

d)

The function has more than one zero on the closed interval

41.

Find the third derivative of the function $f\left(x\right)=\ x^{5\ }-x^4+2x^3-3x^2+5x+9$

a)

$f'''\ =\ 20x^3-12x^2+12x-6$

b)

$f'''=60x^2-24x+12$

c)

$f'''\ =\ 5x^4-4x^3+6x^2-6x+5$

d)

$f'''=120x\ -\ 24$

42.

Find the first derivative of $y=\frac{1}{x^3}+2x$

a)

$y'=\frac{-3}{x^3}+2x$

b)

$y'=\frac{-3}{x^2}+2$

c)

$y'\ =\ \frac{3}{x^4}+2x$

d)

$y'=\frac{-3}{x^4}+2$

43.

Find the first derivative of $f\left(x\right)=\frac{1}{4}x^5-\frac{1}{3}x^3+2$

a)

$f'\left(x\right)=\frac{5}{4x^4}-x^2$

b)

$f'\left(x\right)=\frac{5}{4}x^4-x^2$

c)

$f'\left(x\right)=\frac{5}{4}x^4-3x^2$

d)

$f'\left(x\right)=\frac{5}{4}x^4-x$

44.

Find the derivative of $y\ =\ \left(x^3+2x\right)\left(2x-1\right)$ .

a)

$\frac{\text{d}y}{\text{d}x}=2x^3+4x\ -\ 2$

b)

$\frac{\text{d}y}{\text{d}x}=8x^3-3x^2+8x-2$

c)

$\frac{\text{d}y}{\text{d}x}=8x^3+3x^2+8x\ +2$

d)

$\frac{\text{d}y}{\text{d}x}=8x^3-3x^2+8x$

45.

Find $\frac{\text{d}y}{\text{d}x}$ given $y=\frac{x^2+1}{x^3-1}$

a)

$\frac{dy}{dx}=\frac{x\left(x^3+3x+2\right)}{\left(x^3-1\right)^2}$

b)

$\frac{\text{d}y}{\text{d}x}=\frac{2x^4-2x-3x^4-3x^2}{\left(x^3-1\right)^2}$

46.

A particle moves in one direction along a line so that after t seconds its distance is given by $s\left(t\right)=6t^4-6t^2$ meters from the origin. Find the instantaneous velocity at t = 3 seconds