Free Printable Unit 2a: Derivative Review AP calculus

1.

Select all x-values where the graph is not differentiable

a)

-4

b)

-3

c)

-2

d)

0

e)

2

2.

a)

1/2

b)

-1/2

c)

0

d)

3/2

3.

At which x-value is f continuous but not differentiable?

a)

a

b)

b

c)

c

d)

d

4.

Find y' if y = cos⁵(4x³)

a)

-5cos⁴(4x³)sin(12x²)

b)

-60x²sin⁴(4x³)

c)

-5cos⁴(4x³)sin(4x³)

d)

-60x²cos⁴(4x³)sin(4x³)

5.

Set up the derivative of y=(3x⁴- 7)*(5x²+ 1)

a)

(12x³)*(10x)

b)

(3x⁴ - 7)*(10x) + (12x³)*(5x² + 1)

c)

(3x⁴ - 7)*(10x) - (12x³)*(5x² + 1)

d)

(3x⁴ - 7)/(10x) + (12x³)/(5x² + 1)

6.

Find the derivative of:

a)

a

b)

b

c)

c

d)

d

7.

Find the derivative of
$f\left(x\right)=x^4\sin\left(x\right)$

a)

$x^4\cos\left(x\right)-4x^3\sin\left(x\right)$

b)

$x^4\cos\left(x\right)+4x^3\sin\left(x\right)$

c)

$4x^3\cos\left(x\right)$

d)

$-4x^3\cos\left(x\right)$

8.

Find the derivative of
$g\left(x\right)=\frac{3x-2}{x^2+2}$

a)

$\frac{3}{2x}$

b)

$\frac{3\left(x^2+2\right)}{\left(x^2+2\right)^2}$

c)

$\frac{-3x^2+4x+6}{\left(x^2+2\right)^2}$

d)

$\frac{9x^2-4x+6}{\left(x^2+2\right)^2}$

9.

The position of an object is given as a function of time by x = 3t² + 5t³- 2t
What is the acceleration of the object at time t = 2 s?

a)

64 m/s/s

b)

60 m/s/s

c)

66 m/s/s

d)

70 m/s/s

10.

s(t) = t² - 20
Find the average velocity from t = 3 to t = 5.

a)

2

b)

4

c)

6

d)

8

11.

Which of the following can be used to determine when a particle is at rest?

a)

x(t)=0

b)

v(t)=0

c)

a(t)=0

12.

a)

2 only

b)

2 and 4

c)

0 and 2 only

d)

0,1 and 2

13.

Find the value for that makes f(x) continuous.

a)

0

b)

1

c)

1.5

d)

2

e)

No such value exists

14.

What is the limit?

a)

DNE

b)

2/3

c)

1/4

d)

Infinity

15.

what is the average rate of change of the function f(x)=x²-6x+6 from 2 to 6?

a)

4

b)

2

c)

3

d)

23/4

16.

a)

A

b)

B

c)

C

d)

D

e)

E

17.

If f(x) = 4tan(x/2), then f'(x) =

a)

4sec²(x/2)

b)

-4csc²(x/2)

c)

2sec²(x/2)

d)

-2csc²(x/2)

18.

Find the derivative f(x) = tanxcosx

a)

f'(x) = sec²xcosx - tanxsinx

b)

f'(x) = sec²xcosx + tanxsinx

c)

f'(x) = sec²xsinx

d)

f'(x) = sec²xcosx - tanxcosx

19.

What is the derivative of sec(x)?

a)

sec(x)tan(x)

b)

csc(x)cot(x)

c)

-sec(x)tan(x)

d)

-csc(x)cot(x)

20.

What is the derivative of tan(x)?

a)

-sec²(x)

b)

-csc²(x)

c)

sec²(x)

d)

csc²(x)

21.

What is the derivative of cot(x)?

a)

sec²(x)

b)

-sec²(x)

c)

csc²(x)

d)

-csc²(x)

22.

What is the 2nd derivative of csc(x)?

a)

-csc(x)cot(x)

b)

$\csc x\cot^2x+\csc^3x$

c)

$-\csc^2x\cot x+\cot^3x$

d)

-cot²(x)

23.

What is the slope of the line normal to the curve y = x² + x at x = 1?

a)

-1

b)

-1/2

c)

-1/3

d)

-1/4

24.

What is the equation of the line tangent to f(x)= 4x²+2x-1 at x=0?

a)

y+1=2(x+1)

b)

y=2x-1

c)

y=2x+2

d)

y= -x-1

25.

Write the equation of the normal line of: $f\left(x\right)=2x^2-\frac{1}{x}$ at x = 1

a)

$y=-\frac{1}{5}x+\frac{4}{5}$

b)

$y=5x-6$

c)

$y-1=5\left(x-1\right)$

d)

$y-1=-\frac{1}{5}\left(x-1\right)$

26.

Find where the function has a horizontal tangent line.

y=2x^2-8x+3

a)

x = -2

b)

x = 1

c)

x = 2

d)

x = 1/2

27.

Differentiate:
f(x) = x⁷ (5 + 8x)³

a)

f '(x) = x² (5 + 8x)⁶ (35 + 80x)

b)

f '(x) = x⁶ (5 + 8x)² (35 + 80x)

c)

f '(x) = 8x⁷ (5 + 8x)² (35 + 80x)

d)

f '(x) = x⁶ (5 + 8x)³ (35 + 80x)

28.

Which option shows the derivative?

a)

b)

c)

d)

29.

a)

1

b)

1.5

c)

0

d)

Does not exist

30.

Evaluate the limit:

a)

0

b)

1

c)

-9/4

d)

∞

31.

$\lim_{h\rightarrow0}\left(\frac{5\left(x+h\right)^2-5x^2}{h}\right)$

a)

$5x^2$

b)

$10x$

c)

10

d)

DNE

32.

Evaluate the limit

a)

a

b)

b

c)

c

d)

d

33.

Evaluate the limit

a)

a

b)

b

c)

c

d)

d

34.

Find the limit as x approaches 1.

a)

0

b)

3

c)

DNE

d)

∞

35.

You are given a table containing some values of differentiable functions [eval(f,x)], [eval(g,x)] and their derivatives. Use the table data and the rules of differentiation to solve each problem.
If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ Find $h'\left(2\right)$

a)

$\frac{3}{2}$

b)

$2$

c)

$\frac{5}{6}$

d)

$-3$

36.

You are given a table containing some values of differentiable functions [eval(f,x)], [eval(g,x)] and their derivatives. Use the table data and the rules of differentiation to solve each problem.
If $h\left(x\right)=f\left(x\right).g\left(x\right)$ Find $h'\left(3\right)$

a)

$0$

b)

$2$

c)

$\frac{5}{6}$

d)

$-3$

37.

$If\ y\ =\sec\sqrt{x\ }\ then\ \frac{dy}{dx}=?$

a)

$\frac{\tan^2\sqrt{x}}{\sqrt{x}}$

b)

$\frac{\sec\sqrt{x}.\tan\sqrt{x}}{2\sqrt{x}}$

c)

$\frac{\tan\sqrt{x}}{2\sqrt{x}}$

d)

$\sec\sqrt{x}.\tan\sqrt{x}.2\sqrt{x}$

38.

Find $y'$ if $y=\tan\left(3x^2+2\right)$ .

a)

$y'=\sec^2\left(3x^2+2\right)$

b)

$y'=\sec^2\left(3x^2+2\right)\cdot6x$

c)

$y'=\sec^2x\cdot\left(3x^2+2\right)\cdot6x$

d)

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

39.

a)

A

b)

B

c)

C

d)

D

40.

a)

1/5

b)

1

c)

5

d)

Does not exist

41.

a)

1

b)

Does not exist

c)

0

d)

-1

42.

The function g is continuous on the interval [0, 5] and has select values in the table. The equation g(x) = 1 must have at least 2 solutions in the interval [0, 5] if k =

a)

6

b)

2

c)

0

d)

1

43.

Let f be a continuous function for which f(-2)=1 and f(5)=-3. The Intermediate Value Theorem guarantees that

a)

f(c)=2 for at least one c between -3 and 1

b)

f(c)=0 for at least one c between -2 and 5

c)

f(c)=0 for at least one c between -3 and 1

d)

f(c)=2 for at least one c between -2 and 5

44.

a)

I only

b)

II only

c)

I, II, III

d)

None

45.

a)

A

b)

B

c)

C

d)

D

46.

a)

A

b)

B

c)

C

d)

E

47.

a)

0

b)

1

c)

2

d)

DNE

48.

a)

-4

b)

-1

c)

0

d)

DNE

49.

a)

2

b)

4

c)

32

d)

DNE

50.

Let f be a continuous function for which f(-2)=1 and f(5)=-3. The Intermediate Value Theorem guarantees that

a)

f(c)=2 for at least one c between -3 and 1

b)

f(c)=0 for at least one c between -2 and 5

c)

f(c)=0 for at least one c between -3 and 1

d)

f(c)=2 for at least one c between -2 and 5

51.

a)

I only

b)

II only

c)

I, II, III

d)

None

52.

a)

IVT

b)

EVT

c)

MVT

53.

Find the value that makes the function continuous

a)

c=1/3

b)

c=3

c)

c=-3

d)

c=-1/3

54.

Is the above function differentiable for all values of x? Why?

a)

Yes, the slope of the tangent line can be calculated for all values of x.

b)

Yes, the function is continuous for all values of x so the function is differentiable.

c)

No, the function has a sharp turn in the first quadrant so if is not differentiable.

d)

No, the function is discontinuous so it is not differentiable.

55.

For what value(s) of x is the function continuous but not differentiable?

a)

x = -1

b)

x = 0

c)

x = 2

d)

x = 3; x = -2

56.

Is the function continuous, differentiable, both, or neither?

a)

continuous

b)

differentiable

c)

both

d)

neither

57.

Is the function continuous, differentiable, both, or neither?

a)

continuous

b)

differentiable

c)

both

d)

neither

58.

Find the values of a and b that make the function f differentiable.

(a)

59.

Find the values of a and b that make the function f differentiable.

(a)

60.

The function shown

a)

is continuous at x = 1

b)

is differentiable at x = 1

c)

has a limit that exists at x = 1

d)

exists at x = 1

61.

The function shown

a)

is continuous at x = 2

b)

is differentiable at x = 2

c)

has a limit that exists at x = 2

d)

exists at x = 2

62.

The function shown

a)

is continuous at x = b

b)

is differentiable at x = b

c)

has a limit that exists at x = b

d)

exists at x = b

63.

The function shown

a)

is continuous at x = c

b)

is differentiable at x = c

c)

has a limit that exists at x = c

d)

exists at x = c

64.

a)

A

b)

B

c)

C

d)

D

65.

a)

A or C

b)

B

c)

D

d)

E

66.

a)

b)

c)

d)

67.

$find\ g'\left(x\right)\ for\ g\left(x\right)=\left(h\left(x\right)\right)^3$

a)

$g'\left(x\right)=3\left(h'\left(x\right)\right)^2$

b)

$g'\left(x\right)=3h'\left(x\right)^2$

c)

$g'\left(x\right)=3\left(h\left(x\right)\right)^2\cdot h'\left(x\right)$

d)

$g'\left(x\right)=3h\left(x\right)^2\cdot h'\left(x\right)$

68.

a)

A

b)

B

c)

C

d)

D

69.

a)

A

b)

B

c)

C

d)

D

70.

a)

A

b)

B

c)

C

d)

D

71.

a)

A

b)

B

c)

C

d)

D

72.

If $y=a\left(b\left(x\right)\right)$ , find $y'\left(3\right)$

a)

$-25$

b)

$-20$

c)

$9$

d)

$15$

Unit 2a: Derivative Review AP calculus

Total questions: 72

Worksheet time: 4hrs 38mins

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