Free Printable MCV4U Exam Review Day 1

1.

At what point on the graph of $y=\frac{1}{2}x^2$ is the tangent line parallel to the line $y=\frac{1}{2}x-5$ ?

a)

(0.5, 0.5)

b)

(0.5, 0.125)

c)

(1, 0.25)

d)

(1, 0.5)

e)

(2, 2)

2.

The slope of $f\left(x\right)=x^3+12x-6$ at x = 4 is:

a)

48

b)

-60

c)

60

d)

96

e)

DNE

3.

Let $f\left(x\right)=\frac{3x-2}{2x+1}$ . Find f'(x).

a)

$\frac{3}{2}$

b)

$-\frac{7}{\left(2x+1\right)^2}$

c)

$\frac{12x+1}{\left(2x+1\right)^2}$

d)

$\frac{12x-1}{\left(2x+1\right)^2}$

e)

$\frac{7}{\left(2x+1\right)^2}$

4.

Let $f\left(x\right)=\tan^2x$ . Find $f'\left(\frac{\pi}{4}\right)$ .

a)

4

b)

2

c)

1

d)

$2\sqrt{2}$

e)

Not Possible (right now)... requires chain rule

5.

Find the derivative of the given equation $x^3+x^2+12$

a)

$3x^2+2x$

b)

$5x$

c)

$3x^3+2x^2$

d)

$x^2+x$

6.

Find the first step of the derivative of y = 2x² (3x - 4) using the product rule.

a)

6x³ - 8x²

b)

2x² (3) + (3x - 4)(4x)

c)

(3x)(2x²)+ (4x² )(3x - 4)

d)

4x² (3)

7.

Find the tangent line when x = 2 on the curve $y=\frac{12}{x^2}$

a)

$y=3x-3$

b)

$y=-12x+25$

c)

$y=-3x+9$

d)

$y=12x-21$

8.

Find the derivative of
$f\left(x\right)=-\frac{2}{3}x^3-\frac{1}{2}x^2+9x$

a)

$-2x^2-x+9$

b)

$-2x^3-x^2+9x$

c)

$-\frac{2}{12}x^4-\frac{1}{6}x^3+\frac{9}{\ 2}x^2$

d)

$-2x^4-x^3+9x^2$

9.

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)

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

10.

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

a)

$f'\left(x\right)=e^x$

b)

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

c)

$f'\left(x\right)=e^x-xe^x$

d)

$f'\left(x\right)=e^x+xe^x$

11.

$\frac{d}{dx}\left[\sin\left(x^3-2x\right)\right]=$

a)

$\left(3x^2-2\right)\cos\left(x^3-2x\right)$

b)

$-\left(3x^2-2\right)\cos\left(x^3-2x\right)$

c)

$\cos\left(3x^2-2\right)$

d)

$\sin\left(3x^2-2\right)$

12.

Find the derivative of f(x) = (x⁶ + 4)⁵

a)

f '(x) = 5x⁵(x⁴ + 4)⁴

b)

f '(x) = 6x⁵(x⁶ + 4)⁴

c)

f '(x) = 30x⁵(x⁶ + 4)⁴

d)

f '(x) = 30x⁶(x⁶ + 4)

13.

Find the slope of the tangent line to the graph of f at x=4 given that

f\left(x\right)=-x^2+4\sqrt{x}

a)

-8

b)

-12

c)

-9

d)

-7

14.

The equation of the tangent line of

y=x\sqrt{3+x^2}

at the point (1, 2) is

a)

$y=\frac{3}{2}x-\frac{1}{2}$

b)

$y=\frac{1}{2}x+\frac{1}{2}$

c)

$y=\frac{5}{2}x+\frac{1}{2}$

d)

$y=\frac{5}{2}x-\frac{1}{2}$

15.

Find dy/dx if y = cos⁵x

a)

5cos⁴(x)

b)

-5cos⁴(x)sin⁴(x)

c)

5cos⁴(x)sin(x)

d)

-5cos⁴(x)sin(x)

16.

Find g'(x) if g(x) = sin^1/2(4x)

a)

2cos^-1/2(4x)

b)

4cos^1/2(4x)

c)

2sin^-1/2(4x)cos(4x)

d)

1/2sin^-1/2(4x)cos(4x)

17.

If f(x) = sin³(2x), then f'(x) =

a)

6sin²(2x)cos(2x)

b)

6sin(2x)

c)

3sin²(2x)cos(2x)

d)

3cos(2x)

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.

The exact value of $f'\left(-2\right)$ if $f\left(x\right)=2e^{-3x}$ is:

a)

$A.\ \ \ \ \ -6e^6$

b)

$B.\ \ \ \ \ 6e^6$

c)

$C.\ \ \ \ -6e^{-6}$

d)

$D.\ \ \ \ \ 6e^{-6}$

e)

$E.\ \ \ \ \ -e^6$

20.

Find the derivative.
f(x) = -8x^-3 + 5x - e^x

a)

f'(x) = -24x^-4 + 5 - e^x

b)

f'(x) = 24x^-4 + 5 - e^x

c)

f'(x) = 24x^-2 + 5 - e^x

d)

f'(x) = 24x^-4 + 5 - e^x-1

21.

Find the derivative:

y=4e^x²

a)

A. y'=8xe^x

b)

B. y'=8xe^x²

c)

C. y'=8xe^2x

d)

D. y'=8xe^4x²

22.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

23.

Differentiate $y=\ e^{\sin\left(2x\right)}+\ e^{\cos\left(2x\right)}$

a)

$e^{\sin\left(2x\right)}\cos\left(2x\right)-e^{\cos\left(2x\right)}\sin\left(2x\right)$

b)

$2e^{\sin\left(2x\right)}\cos\left(2x\right)-e^{\cos\left(2x\right)}\sin\left(2x\right)$

c)

$2e^{\sin\left(2x\right)}\cos\left(2x\right)-2e^{\cos\left(2x\right)}\sin\left(2x\right)$

d)

$-2e^{\sin\left(2x\right)}\cos\left(2x\right)+2e^{\cos\left(2x\right)}\sin\left(2x\right)$

24.

$y=\frac{4e^{3x}-e^x}{e^x}$

Find the derivative of

a)

$8e^{2x}$

b)

$4e^{2x}$

c)

$4e^{2x}-1$

d)

$8e^{2x}-1$

25.

Which of the following is an example of exponential decay?

a)

y = 3(5)^x

b)

y = .5(5)^x

c)

y = 2(.5)^x

d)

y = 8(1)^x

26.

Which of the following is an example of exponential growth?

a)

y = 8(.5)^x

b)

y = .8(.95)^x

c)

y = 8(5)^x

d)

y = .5(.5)^x

27.

Consider the given function that represents the balance in a bank account.

$b\left(x\right)=850\left(1.025\right)^x$

What is the initial balance in the account?

(a)

28.

Penicillin decays exponentially in the human body. Suppose you receive a 300-milligram dose of penicillin to combat strep throat. About 180 milligrams will remain active in your body after 1 day. Which equation represents the amount of penicillin remaining in your body?

a)

$y=300\left(0.6\right)^x$

b)

$y=300\left(1.6\right)^x$

c)

$y=300\left(1.67\right)^x$

d)

$y=300\left(0.4\right)^x$

29.

What is the slope of the curve y = 2sin(x) at x = π/3

a)

-√3

b)

√3

c)

-1

d)

1

30.

When we "take the derivative" of a function what are we finding?

a)

What's a derivative?

b)

The rate at which our struggles in Calculus are increasing.

c)

The slope of the secant line

d)

The slope of the tangent line

31.

Solve for x in the exponential equation.

$3^x-5=12$

Round your answer to the nearest hundreth.

(a)

32.

Evaluate:

a)

23

b)

114/7

c)

does not exist

d)

infinity

33.

a)

does not exist

b)

1

c)

0.5

d)

infinity

34.

a)

0.5

b)

1

c)

does not exist

d)

negative infinity

35.

a)

false

b)

true

c)

not enough information

36.

a)

true

b)

false

c)

not enough information

37.

a)

3

b)

2

c)

does not exist

d)

not enough information

38.

a)

does not exist

b)

not enough information

c)

3

d)

2

39.

a)

f(x) is not continuous

b)

f(x) is not differentiable

c)

f(x) is neither continuous nor differentiable

d)

not enough information

40.

a)

f'(3) = 0

b)

f'(3) > 0

c)

f'(3) < 0

d)

f'(3) does not exist

41.

a)

f is concave down at x = 3

b)

f is concave up at x = 3

c)

f has a point of inflection at x = 3

d)

none of these are true

42.

A tangent line touches a function at exactly one point.

a)

true

b)

false

c)

not enough information

43.

If a function has a vertical tangent at x = a, then f'(a) does not exist.

a)

true

b)

false

c)

not enough information

44.

If the function is not differentiable at x = a, then it is not continuous at x = a.

a)

never true (i.e. false)

b)

sometimes true

c)

always true

d)

not enough information

45.

If f(x) = sin(x), then f'(x) =

a)

sin(x)

b)

cos(x)

c)

-cos(x)

d)

-sin(x)

46.

If f(x) = cos(x), then f'(x) =

a)

sin(x)

b)

cos(x)

c)

-cos(x)

d)

-sin(x)

47.

If f(x) = e^x, then f'(x) =

a)

e^x

b)

ln(x)

c)

1/x

48.

If f(x) = 3^x, then f'(x) =

a)

3^x

b)

3^x ln(3)

c)

e^x

d)

1/(x ln3)

49.

If y = tan(x), then y' =

a)

sec^2 x

b)

cot x

c)

csc x tan x

d)

- cot^2 x

50.

If f(x) = e ^sin(x)then f'(x) =

a)

e^(sin(x))

b)

e^(cos(x))

c)

e^(sin(x)) cos x

d)

e^(sin(x)) ln(cos(x))

51.

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

a)

There is a local max at x=3

b)

There is a local min at x = 3

c)

There is an inflection point at x = 3

d)

There is an x-intercept at x = 3

52.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

53.

If a function's FIRST derivative is negative at a certain point, what does that tell you?

a)

The function is increasing at that point

b)

The function is decreasing at that point

c)

The concavity of the function is up at that point

d)

The concavity of the function is down at that point

54.

What are the intervals of the graph increasing for f(x) = 2x⁴- 4x² + 1

a)

(-1,0)

b)

(0,1)

c)

(-∞,-1) and (0,1)

d)

(1,∞) and (-1,0)

55.

What is the maximum value of f(x) = x³ - 3x² - 1 on the interval [-3, 2]?

a)

0

b)

-1

c)

2

d)

5

56.

What is a point of inflection?

a)

When a function goes from increasing to decreasing.

b)

When the concavity changes.

c)

When the derivative changes.

d)

When the function crosses the x-axis.

57.

How do you find the critical numbers of a function?

a)

Critical numbers only occur when f'(x) = 0.

b)

Critical numbers only occur when f'(x) is undefined.

c)

Critical numbers occur when f'(x) = 0 or where f'(x) is undefined.

d)

Critical numbers occur as x-intercepts on the graph.

58.

Use the sign chart for f'(x). There is(are) ...

a)

a local maximum at x = -2.

b)

a local minimum at

x = -2 and a local maximum at x = 4.

c)

a local maximum at

x = -2 and a local minimum at x = 4.

d)

no extrema.

59.

If (a,b) is a local maximum, then what will be true about f''(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

60.

Where is the point of infection for the function y = x³ + 6x²?

a)

0

b)

-4

c)

-2

d)

4

61.

On what interval(s) is the function y=x³ + 6x²

concave down?

a)

(-∞,-4)

b)

(-∞,-2)

c)

(-2,∞)

d)

(0,∞)

62.

Over what intervals is f(x) decreasing?

a)

(-∞, -1) ∪ (1, ∞)

b)

(1, ∞)

c)

(-1, 1)

d)

(-√3, 0) ∪ (√3, ∞)

63.

Find the Relative Maximum and Minimum. Round to nearest tenth.

a)

A

b)

B

c)

C

d)

D

64.

If (a,b) is a local minimum, and f'(a) exists, then what is true about f'(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

65.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

66.

What is mostly likely to be true at an inflection point?

a)

f(x)=0

b)

f'(x)=0

c)

f''(x)=0

d)

f'''(x)=0

67.

Find the intervals of concavity for

f(x) = x² + 2x + 1.

a)

concave up: (-∞,∞)

b)

concave down: (-∞,∞)

c)

concave up: (2, ∞)

concave down: (-∞,2)

d)

concave up: (-∞,2)

concave down: (2, ∞)

68.

Find the critical points of f(x) = 2x⁴- 4x² + 1

a)

x= 0

b)

x = -1, 1

c)

x = -1, 0, 1

d)

no critical points

69.

Which of the following could be used to take the derivative for the function f(x)=6x³(4x)⁶ ??

(Select all that apply)

a)

Power Rule

b)

Sum and Difference

c)

Product

d)

Quotient

70.

Which of the following rules could NOT be used to take the derivative of the function

f(x)= 3x²-10x+7

(Select all that apply)

a)

Power Rule

b)

Sum and Difference

c)

Product

d)

Quotient

71.

What type of graph would be the derivative for a cubic function?

a)

Quartic

b)

Cubic

c)

Parabola

d)

Linear

72.

What type of graph would be the derivative for a quartic function?

a)

Quartic

b)

Cubic

c)

Parabola

d)

Linear

73.

If the tangent line is horizontal at x=c, which of the following must be true?

a)

c is a point of inflection

b)

c must be a maximum

c)

c must be a critical point

d)

c must be a minimum

74.

If x=c is an inflection point, what must be true? (Select all that apply)

a)

f'(c)=0

b)

the concavity changes

c)

f''(c)=0

d)

the function changes from increasing to decreasing

75.

True/False: if the limit of a function exists at x=c, then f(c) exists.

a)

True

b)

False

76.

True/False: if the limit of a function at x=c is 5, then f(c)=5

a)

True

b)

False

77.

True/False: If a function is continuous at x=c, then $\lim_{x\rightarrow c}f\left(x\right)=f\left(c\right)$ .

a)

True

b)

False

78.

True/False: The velocity of a function is the derivative of the position function.

a)

True

b)

False

79.

True/False: The slope of the tangent line at a point is the same as the instantaneous rate of change of the function at that point.

a)

True

b)

False

80.

True/False: If c is a critical value and f''(c)<0, then c is a relative minimum.

a)

True

b)

False

81.

What is the limit?

a)

5/2

b)

-2/3

c)

Infinity

d)

17/3

82.

a)

1/2

b)

0

c)

Positive Infinity

d)

Negative Infinity

83.

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

a)

a maximum.

b)

a minimum.

c)

no extrema.

84.

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

a)

a maximum.

b)

a minimum.

c)

no extrema.

85.

If (a,b) is a local minimum, then what will be true about f'(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

86.

Evaluate:

a)

23

b)

114/7

c)

does not exist

d)

infinity

87.

a)

f(x) is not continuous

b)

f(x) is not differentiable

c)

f(x) is neither continuous nor differentiable

d)

not enough information

88.

If a function has a vertical tangent at x = a, then f'(a) does not exist.

a)

true

b)

false

c)

not enough information

89.

If the function is not differentiable at x = a, then it is not continuous at x = a.

a)

never true (i.e. false)

b)

sometimes true

c)

always true

d)

not enough information

90.

The concavity of a function is described by its (a) derivative

91.

The (a) numbers can be found by setting f'(x) = 0 and solving for x.

92.

If a function's FIRST derivative is negative at a certain point, what does that tell you?

a)

The function is increasing at that point

b)

The concavity of the function is up at that point

c)

The concavity of the function is down at that point

d)

The function is decreasing at that point

93.

In the diagram given, which points lie on the curve f(x) where f'(x) < 0?

a)

A, C, E, G

b)

B, F

c)

B, D, F

d)

C, E

94.

What is the horizontal asymptote of the function in the image?

a)

y = 5/4

b)

y = -7/2

c)

y = 1

d)

y = 2

95.

When is a function concave up?

a)

f"(c)<0

b)

f'(c)<0

c)

f"(c)>0

d)

f'(c)>0

96.

When is a function concave down?

a)

f"(c)<0

b)

f"(c)>0

c)

f'(c)<0

d)

f'(c)>0

97.

What is a point of inflection?

a)

When a function goes from increasing to decreasing

b)

When a function goes from concave up to concave down

98.

How do you find the critical numbers of a function?

a)

f'(x) = 0

b)

f(x) = 0

c)

f"(x) = 0

d)

f'(x) = 0 and where f'(x) is undefined

99.

Find the vertical asymptotes and holes of the function in the image

a)

Holes: None;

VA: x = 1, -3

b)

Holes: x = -3;

VA: x = 1

c)

Holes: x = -3, 1

VA: None

d)

Holes: x = 1

VA: x = -3

100.

Use the sign chart image for f'(x). There is(are) ...

a)

a local maximum at

x = -2 and a local minimum at x = 4.

b)

a local minimum at

x = -2 and a local maximum at x = 4.

c)

no extrema.

d)

a local maximum at x = -2.

101.

Find the critical points of f(x) = 2x⁴- 4x² + 1

a)

x= 0

b)

x = -1, 0, 1

c)

x = -1, 1

d)

no critical points

102.

Find the intervals of concavity for

f(x) = x² + 2x + 1.

a)

concave up: (2, ∞)

concave down: (-∞,2)

b)

concave up: (-∞,2)

concave down: (2, ∞)

c)

concave down: (-∞,∞)

d)

concave up: (-∞,∞)

103.

Evaluate the limit (if it exists): $\lim_{x\rightarrow4}\ \frac{x+1}{x-4}$

a)

-1

b)

$\infty$

c)

DNE

d)

0

104.

Evaluate the limit (if it exists): $\lim_{x\rightarrow\infty}\ x^3-17x^2+3x+6$

a)

6

b)

$\infty$

c)

DNE

d)

$-\infty$

105.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

slope of the secant line

106.

Which rule would you need to use?

a)

Power Rule

b)

Product Rule

c)

Quotient Rule

d)

Chain Rule

107.

Find the derivative of f(x)=(x³-2x)²

a)

6x⁵ - 12x³+8x

b)

6x⁵ - 16x³+8x

c)

x⁶-4x⁴+4x²

d)

6x⁵ - 16x^3-8x

108.

When do you use the chain rule?

a)

anytime you want

b)

when there is a function in a function

c)

where there are multiple layers to a lasagna (yum)

d)

when there is division

109.

Find the slope of the tangent line to f(x) = -3x²-6x at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

110.

Determine the type(s) and location(s) of all discontinuities

a)

jump at x = -3, infinite at x = 2, removable at x = 2.

b)

jump at x = -3, removable at x = 2, removable at x = 4.

c)

removable at x = -3, removable at x = 2, removable at x = 4.

d)

jump at x = -3, removable at x = 2, jump at x = 4.

111.

Determine the type(s) and location(s) of discontinuities

a)

infinite at x = ∞.

b)

jump at x = -∞, removable at x = 2.

c)

the graph is continuous (no discontinuities)

d)

infinite at x = ∞, removable at x = 0

112.

At what point on the graph of $y=\frac{1}{2}x^2$ is the tangent line parallel to the line $y=\frac{1}{2}x-5$ ?

a)

(0.5, 0.5)

b)

(0.5, 0.125)

c)

(1, 0.25)

d)

(1, 0.5)

e)

(2, 2)

113.

Find the tangent line when x = 2 on the curve $y=\frac{12}{x^2}$

a)

$y=3x-3$

b)

$y=-12x+25$

c)

$y=-3x+9$

d)

$y=12x-21$

114.

Find the slope of the tangent line to the graph of f at x=4 given that

f\left(x\right)=-x^2+4\sqrt{x}

a)

-8

b)

-12

c)

-9

d)

-7

115.

The equation of the tangent line of

y=x\sqrt{3+x^2}

at the point (1, 2) is

a)

$y=\frac{3}{2}x-\frac{1}{2}$

b)

$y=\frac{1}{2}x+\frac{1}{2}$

c)

$y=\frac{5}{2}x+\frac{1}{2}$

d)

$y=\frac{5}{2}x-\frac{1}{2}$

116.

the smells: Peppermint and rosemary were found to support an attentive mood and may improve test scores.

a)

crazy but true

b)

insane

117.

Chewing gum increases blood flow to the brain. Increased heart rate improves oxygen delivery to the brain which can enhance our cognitive powers.

a)

so then- chewing gum could improve my test scores

b)

this is crazy - gum is not allowed in a test site

c)

worth a try - but I promise not to dispose of my old gum inappropriately

MCV4U Exam Review Day 1

Total questions: 117

Worksheet time: 7hrs 35mins

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