Free Printable Calculus Review (limits, continuity, derivatives, integration)

1.

what is the derivative of y= sin(2x)?

a)

y^/=cos(2x)

b)

y^/=2sin(2x)

c)

y^/=2sin(x)+cos(2x)

d)

y^/=2cos(2x)

2.

when is the function f(x)=x²+6x+9 increasing?

a)

(3,∞)

b)

(-∞,3)

c)

(-3,-∞)

d)

(-3,∞)

3.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

4.

What is the limit?

a)

Infinity

b)

20

c)

DNE

d)

12

5.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

6.

a)

-7

b)

0

c)

1

d)

DNE

7.

Which of the following best describes the continuity at x = 1?

a)

Continuous

b)

Removable Discontinuity

c)

Infinite Discontinuity

d)

Jump Discontinuity

8.

What is the derivative of f(x) = 2x³- 4x + 5?

a)

f'(x) = 6x² - 4x

b)

f'(x) = 2/3x² - 4x

c)

f'(x) = 3x² - 4

d)

f'(x) = 6x² - 4

9.

What is the 2nd derivative of y = 2x³ - 4x + 6

a)

6x²- 4

b)

12x - 4

c)

12x

d)

6x

10.

if f(x) = x² + 3, then f’’(0) is

a)

0

b)

2

c)

4

d)

Undefined

11.

Does this function have an absolute maximum?

a)

Yes

b)

No

12.

When f'(x) changes from negative to positive, there is(are) ...

a)

a maximum.

b)

a minimum.

c)

no extrema.

13.

When f'(x) changes from positive to negative, there is(are) ...

a)

a maximum.

b)

a minimum.

c)

no extrema.

14.

Find the relative minimum values for f(x)=3x⁴-2x³-9x²

a)

-1

b)

0

c)

1.5

d)

-1 and 1.5

15.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

16.

Identify the critical points of the following function:
g(x)=2x³-3x²

a)

x=-1,1

b)

x=0,0

c)

x=0,-1

d)

x=0,1

17.

Over what intervals is f(x) decreasing?

a)

(-∞, -1) ∪ (1, ∞)

b)

(-∞, -√3) ∪ (0, √3)

c)

(-1, 1)

d)

(-√3, 0) ∪ (√3, ∞)

18.

What will be true at an inflection point? (select the best answer)

a)

f(x)=0

b)

f'(x)=0

c)

f''(x)=0

d)

The function is undefined

19.

Find the derivative of f(x) = x²sinx

a)

f'(x) = 2xsinx - x²cosx

b)

f'(x) = 2xsinx + x²sinx

c)

f'(x) = 2xsinx + x²cosx

d)

f'(x) = 2xcosx

20.

Find the derivative of y = 2x² (3x - 4)

a)

6x³ - 8x²

b)

18x² - 16x

c)

15x²- 6x

d)

6x - 4

21.

Find the derivative. $y=\frac{x^2}{3x-1}$

a)

y' = 9x²- 12

b)

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

c)

$y'=\frac{\left(6x+1\right)^2}{3}$

d)

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

22.

What is the derivative of sin(x)?

a)

-sin(x)

b)

-cos(x)

c)

cos(x)

d)

sin(x)

23.

What is the derivative of cos(x)?

a)

sin(x)

b)

-sin(x)

c)

cos(x)

d)

-cos(x)

24.

Given that $y=\frac{x+2}{x-3}$ , find y'.

a)

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

b)

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

c)

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

d)

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

25.

Evaluate $\frac{\text{d}}{\text{d}x}\left[\log_2x\right]$

a)

$\frac{2}{x}$

b)

$\frac{1}{2x}$

c)

$\frac{1}{\left(\ln2\right)x}$

d)

$\frac{1}{\log_2x}$

26.

Where is the point of infection for the function $y=x^3+6x^2$ ?

a)

0

b)

-4

c)

-2

d)

4

27.

Find dy/dx by Implicit Differentiation
x³ +y³ = 36

a)

6 -x

b)

3x² +3y²

c)

−x²/y²

d)

0

28.

Find dy/dx $xy+y^2=2$

a)

$\frac{-y}{x+2y}$

b)

$\frac{y}{x+2y}$

c)

$\frac{-3y}{x}$

d)

$\frac{-3x}{y}$

29.

Which of the following functions would NOT be continuous?

a)

y = 5x

b)

y = -x² + 6x

c)

y = x³

d)

y = 1/x

30.

Find f'(0) if $f\left(x\right)=\frac{5}{x-1}$ .

a)

-5

b)

5

c)

$\frac{1}{5}$

d)

$-\frac{1}{5}$

31.

Find $f'\left(x\right)$ if $f\left(x\right)=\sqrt[]{2x^2-5x}$

a)

$\frac{1}{\sqrt[]{4x-5}}$

b)

$4x-5$

c)

$\frac{1}{2\ \sqrt[]{2x^2-5x}}$

d)

$\frac{4x-5}{2\ \sqrt[]{2x^2-5x}}$

32.

$\int\left(x^2-2x\right)dx$

a)

$\frac{1}{3}x^3-x^2+C$

b)

$x^{2\ }-2x+C$

c)

$2x-2$

d)

$\frac{1}{3}x^3-x^2$

33.

$\int6x\left(x+2\right)dx$

a)

$6x^2+12x+C$

b)

$x^{2\ }-2x+C$

c)

$3x^2+6x+C$

d)

$2x^3+6x^2+C$

34.

$\int\left(5x^2-7x+6\right)dx$

a)

$\frac{5}{3}x-\frac{7}{2}x+6x+C$

b)

$x+x+x+x+x+C$

c)

$x^3-3x^2+6x+C$

d)

$\frac{5}{3}x^3-\frac{7}{2}x^2+6x+C$

35.

∫_₀^²(3x² - 2x)dx ?

a)

8

b)

4

c)

-8

d)

-4

36.

a)

6

b)

2

c)

9

d)

3

37.

Using the areas of each region given
$\int_a^df\left(x\right)=$

a)

6

b)

20

c)

2

d)

24

38.

a)

Does Not Exist

b)

9

c)

1

d)

0

39.

Find the limit. Hint: Where is the horizontal asymptote?

a)

³/₇

b)

0

c)

$\infty$

d)

$-\infty$

40.

What is the average rate of change of the function $f\left(x\right)\ =\ -\frac{1}{x+2}\ at\ the\ interval\ x\ =\ 0\ and\ x\ =\ 1?$ Input in simplified fraction form.

a)

-5/36

b)

-1/6

c)

1/6

d)

5/36

Calculus Review (limits, continuity, derivatives, integration)

Total questions: 40

Worksheet time: 3hrs 57mins

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