Free Printable Mathematics

1.

We want to construct a box whose base length is 3 times the base width. If the box must have a volume of 50 ft³, determine the dimensions that will minimize the amount of material used.

a)

w=2.027ft, h=4.055ft, l=6.082ft

b)

w=1.488ft, h=3.347ft, l=6.694ft

c)

w=2.231ft, h=3.347ft, l=6.694ft

d)

w=0.485ft, h=2.111ft, l=4.222ft

2.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12

3.

(3 min) The area of a circular region is increasing at a rate of 96 π square meters per second. When the area of the region is 64 π square meters, how fast, in meters per second, is the radius of the region increasing?

a)

6

b)

8

c)

16

d)

4√3

4.

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

a)

(-1,0)

b)

(0,1)

c)

(-∞,-1) and (1,∞)

d)

(0,1) and (-1,0)

5.

Find the point(s) of inflection for f(x) = 2(x)^1/5 + 3

a)

None

b)

x = 0

c)

x = 0, 2

d)

x = -2, 0, 2

6.

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

a)

100ft x 200ft

b)

102ft x 196 ft

c)

50 ft x 300 ft

d)

50 ft x 175 ft

7.

If y=2x-8, what is the minimum value of the product xy?

a)

-16

b)

-8

c)

-4

d)

2

8.

Which of the following is true?

a)

the derivative is a way to show rate of change, that is - the amount by which a function is changing at one given point

b)

dx/dy is the derivative

9.

a)

cosx

b)

-cosx

c)

-sinx cosx

d)

sinx cosx

10.

a)

sinx

b)

-sinx

c)

-cscx cotx

d)

sec²x

11.

Find f(2).

a)

1

b)

-1

c)

5

d)

DNE

12.

Find the limit of the function as x approaches 2.

a)

1

b)

-1

c)

5

d)

DNE

13.

If a function has a derivative that is negative, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The concavity of the function is up

d)

The concavity of the function is down

14.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

15.

The image displays the ______

a)

Limit definition of the first derivative

b)

The bane of my existance

16.

What is an antiderivative?

a)

The opposite of a derivative

b)

The same as a derivative

c)

A second derivative

d)

It always represents velocity.

17.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

18.

Differentiate y= 12x^-2

a)

24x^-1

b)

-24x^-3

c)

-24x^-1

d)

6x^-3

19.

If f'(x) = 0 what does that imply about the x value?

a)

It is a critical point, it is a possible max, min, or point of inflection.

b)

That the limit does not exist.

20.

When looking for critical points we did....

a)

took the limit of the function. 2. Graphed the critical points.

b)

Found f'(x). 2. Set f'(x) = 0 and solved for x. 3. Created a sign diagram. 4. Took the limit

c)

Found f'(x). 2. Set f'(x) = 0 and solved for x. 3. Created a sign diagram. 4. Checked out intervals.

21.

When applying calculus. The second derivative helps find...

a)

the distance traveled by an object.

b)

The velocity of a particle at any given point

c)

acceleration of an object at any given time

22.

Rate of change is another way of saying...

a)

The gradient

b)

The slope

c)

dy/dx

23.

Integration is the inverse of differentiation but it applications it can be used to....

a)

find the area under a curve

b)

calculate the force of an object

c)

Find the altitude of an objects perimeter

24.

Which of the following is true.

a)

The derivative is the same as volume

b)

optimization is the same as integration.

c)

The derivative is often written using "dy over dx" (meaning the difference in y divided by the difference in x). The d's are not

variable, and therefore cannot be cancelled out.

25.

Is differentiating the same as taking the derivative?

a)

yes

b)

no

c)

banana on my rice

26.

FInd the derivative:

f(x) = (-2/3)x³ - (1/2)x² + 9x

a)

-2x² - x + 9

b)

-2x² + x + 9

c)

2x² - x + 9

d)

2x² - x - 9

27.

Find the second derivative of the function:
f (x) = 2x - 5x⁶

a)

f ''(x)= 2 - 30x

b)

f ''(x) = 2-30x⁵

c)

f ''(x) = -30x⁵

d)

f ''(x) = -150x⁴

28.

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)

29.

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

a)

6x³ - 8x²

b)

18x² - 16x

c)

15x²- 6x

d)

6x - 4

30.

Differentiate f(x) =(2/ x⁵) - 5.

a)

x⁵-3

b)

-10x⁶

c)

x⁵-5

d)

-10x^-6

31.

Find the derivative of the given equation
f(x) = x⁴ + 4x³- 2x²

a)

x³ + x²- x

b)

4x³ + 12x²+ 4x

c)

4x + 12x - 4x

d)

4x³ + 12x²- 4x

32.

Find the derivative.

a)

x⁴cosx - 4x³sinx

b)

x⁴cosx + 4x³sinx

c)

4x³cosx

d)

-4x³cosx

33.

.

a)

(3x²-2)cos(x³-2x)

b)

-(3x²-2)cos(x³-2x)

c)

cos(2x²-2)

d)

sin(2x²-2)

34.

Find the derivative of g(x)=(3x-2)/(x²+2)

a)

9x²+2

b)

3(x²+2)/(x²+2)²

c)

(-3x²+4x +6)/(x²+2)²

d)

(-3x²+10)/(x²+2)²

35.

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

36.

The velocity of an object is given as v = 2t + 3t³. what is the acceleration of the object at t = 2 s?

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

37.

If the position function for a particle is s(t) = -t² - t, what is the instantaneous velocity function for the particle?

a)

v(t) = -2

b)

v(t) = -2t - 1

c)

v(t) = t³

d)

v(t) = -t

38.

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

39.

Find the derivative
f(t) = (t² + 2t)⁵

a)

f'(t) = 5(2t+2)⁴

b)

f'(t) = 5(t² + 2t)⁴

c)

f'(t) = 5(t² + 2t)⁴(2t)

d)

f'(t) = 5(t² + 2t)⁴(2t + 2)

40.

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

41.

f(x)= 9/∛x
f'(x) =

a)

9x^-1/3

b)

-3x^-4/3

c)

-9x^2/3

d)

-3x^2/3

42.

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

43.

Given x³-3x - 4, the numbers 1 and -3 are

a)

terms

b)

coefficients

c)

variables

d)

constants

44.

State another name for additive inverse

a)

Negative

b)

Opposite

c)

Reciprocal

d)

Constant

45.

Letters in an expression

a)

variables

b)

coefficients

c)

exponents

d)

constant

46.

Given 3/7, 7 is called

a)

numerator

b)

coefficient

c)

expression

d)

denominator

47.

A collection of letters and real numbers combined using addition, subtraction, multiplication, division, and exponentiation is

a)

terms

b)

expression

c)

coefficients

d)

constants

48.

Given x² - 3x + 4, x² and -3x are ______ terms.

a)

variable

b)

reciprocal

c)

denominator

d)

constant

49.

Another name for multiplicative inverse

a)

denominator

b)

expression

c)

reciprocal

d)

opposite

50.

1/3 is the ___________ inverse of 3.

a)

expression

b)

multiplicative

c)

variable

d)

additive

51.

Real numbers in an expression are called ___________

a)

constants

b)

terms

c)

coefficient

d)

numerator

52.

9 is the __________ inverse of -9

a)

multiplicative

b)

constant

c)

additive

d)

reciprocal

53.

x² -3x - 7, 7 is the ______ term

a)

reciprocal

b)

term

c)

variable

d)

constant

54.

Parts of an expression that are separated by addition symbols

a)

variable

b)

denominator

c)

numerator

d)

terms

55.

Given 2/7, 2 is called the

a)

denominator

b)

term

c)

numerator

d)

expression

56.

What is the derivative of cot(x)?

a)

sec²(x)

b)

-sec²(x)

c)

csc²(x)

d)

-csc²(x)

57.

What is the derivative of cos(x)?

a)

sin(x)

b)

-sin(x)

c)

cos(x)

d)

-cos(x)

58.

Which of the following is true?

a)

the derivative is a way to show rate of change, that is - the amount by which a function is changing at one given point

b)

dx/dy is the derivative

59.

Rate of change is another way of saying...

a)

The gradient

b)

The slope

c)

dy/dx

60.

a)

b)

c)

d)

61.

a)

A

b)

B

c)

C

d)

D

62.

Find f’(x) when f(x) = sin(4x+5)

a)

3cos(3x)

b)

4sin(4x+5)

c)

4cos(4x+5)

d)

4sin(4x)

63.

What is the acceleration function for the particle whose position is defined by s(t) = 3t * lnt

a)

a(t) = ln(x) +3

b)

a(t) = 3/x

c)

a(t)= 3ln(x)+1

d)

a(x)= 3ln(x) + 3

64.

$\int u^ndu=$

a)

$\ln u+C$

b)

$u^{n+1}+C$

c)

$\frac{u^{n+1}}{n+1}+C$

d)

$u^{n-1}+C$

65.

$\int u^ndu=$

a)

$\ln u+C$

b)

$u^{n+1}+C$

c)

$\frac{u^{n+1}}{n+1}+C$

d)

$u^{n-1}+C$

66.

$\frac{\text{d}}{\text{d}x}\left(\ln\left|x\right|\right)=$

a)

$e^x$

b)

$-x^{-1}$

c)

$-x^{-2}$

d)

$\frac{1}{\left|x\right|}$

67.

As x gets close to 1, then $\frac{x^2-1}{x-1}$ gets close to

a)

1.5

b)

1.9

c)

1.99

d)

2

68.

Evaluate the limit of the given expression.

a)

1

b)

2

c)

3

d)

4

69.

Evaluate the limit.

a)

5

b)

6

c)

7

d)

8

70.

Evaluate the limit.

a)

8

b)

4

c)

32

d)

48

71.

Which of the following is INCORRECT?

a)

If a sequence of values of the variable x approaches c as a limit , then a sequence of values of the function f(x) = x will also approach c as a limit.

b)

The limit of a sum is equal to the sum of the limits.

c)

The limit of a product is equal to the product of the limits.

d)

The limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is 0.

72.

Evaluate the limit using the given graph

a)

-2

b)

4

c)

1/2

d)

6

73.

Evaluate the limit.

a)

1

b)

-1

c)

2

d)

4

74.

Evaluate.

a)

-3

b)

-4

c)

-5

d)

-6

75.

In this discontinuity, the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.

(a)

76.

It is a discontinuity at which the limit of the function does not exist

(a)

77.

If the limit of a rational function produces 0/0 form, the following should be done EXCEPT

a)

Factor the numerator and denominator

b)

Divide out the common factor(s)

c)

Re-evaluate the limit

d)

Add its variables

78.

If the limit of a rational function produces 0/0 form, the following should be done EXCEPT

a)

Factor the numerator and denominator

b)

Divide out the common factor(s)

c)

Re-evaluate the limit

d)

Add its variables

79.

Evaluate.

a)

0

b)

1

c)

2

d)

3

80.

Evaluate.

a)

1

b)

2

c)

3

d)

4

81.

a)

2

b)

3

c)

4

d)

The limit doesn't exist

82.

a)

-5

b)

-3

c)

6

d)

The limit doesn't exist

83.

What is the derivative of cot(x)?

a)

sec²(x)

b)

-sec²(x)

c)

csc²(x)

d)

-csc²(x)

84.

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

85.

Differentiate y= x³ + 2x

a)

3x³+2x

b)

3x²+2

c)

3x+2

d)

3x

86.

Find the derivative of the given equation
f(x) = x⁴ + 4x³- 2x²

a)

x³ + x²- x

b)

4x³ + 12x²+ 4x

c)

4x + 12x - 4x

d)

4x³ + 12x²- 4x

87.

Find the first order partial derivative with respect to y
$f(x,y)=x^3y^2+3xe^y$
.

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

$f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$

c)

$f_y(x,y)=2x^3y+3xe^y$

d)

$f_y(x,y)=6x^2+3e^y$

88.

Find $\frac{\partial f}{\partial x}$ of $f\left(x,y\right)=\sin\left(\frac{x}{1+y}\right)$

a)

$\cos\left(\frac{x}{1+y}\right)$

b)

$\cos\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

c)

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

d)

$\sin\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

89.

Differentiate y= x³ + 2x

a)

3x³+2x

b)

3x²+2

c)

3x+2

d)

3x

90.

Find the derivative of the given equation
f(x) = x⁴ + 4x³- 2x²

a)

x³ + x²- x

b)

4x³ + 12x²+ 4x

c)

4x + 12x - 4x

d)

4x³ + 12x²- 4x

91.

Find the first order partial derivative with respect to y
$f(x,y)=x^3y^2+3xe^y$
.

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

$f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$

c)

$f_y(x,y)=2x^3y+3xe^y$

d)

$f_y(x,y)=6x^2+3e^y$

92.

Find $\frac{\partial f}{\partial x}$ of $f\left(x,y\right)=\sin\left(\frac{x}{1+y}\right)$

a)

$\cos\left(\frac{x}{1+y}\right)$

b)

$\cos\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

c)

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

d)

$\sin\left(\frac{x}{1+y}\right)\left(\frac{1}{1+y}\right)$

93.

$find\ \frac{\partial f}{\partial y},\ given\ f\left(x,y\right)=e^x+2xy^2-4y\ at\ \left(0,3\right)$

a)

$8$

b)

$-4$

c)

$-12$

94.

Suppose at the point (1,2, f(1,2)), $f_{xx}\left(1,2\right)=-5$ , $f_{yy}\left(1,2\right)=4$ , and $f_{xy}\left(1,2\right)=-2$ . Then at that point there is a:

a)

Saddle point

b)

Local Max

c)

Local Min

d)

Cannot be determined

95.

Find $\frac{\partial z}{\partial t}$ when s = 1 and t = 2.

a)

-47

b)

-43

c)

69

d)

108

96.

When finding local extrema, suppose the D>0 and $f_{x x} < 0$ at a point. Then that point is:

a)

A local max

b)

A local min

c)

A saddle point

d)

Cannot be determined.

97.

When finding local extrema, suppose the D>0 and $f_{x x} < 0$ at a point. Then that point is:

a)

A local max

b)

A local min

c)

A saddle point

d)

Cannot be determined.

98.

a)

b)

c)

d)

99.

a)

b)

c)

d)

100.

a)

b)

c)

d)

101.

The function f is continuous on the closed interval [0,6] and has values given in the table above. The trapezoidal approximation found with 3 subintervals is 52. What is the value of k?

a)

2

b)

6

c)

7

d)

10

102.

A particle moves along an axis so that at any time t>0, its velocity is given by v(t)=4-6t². If the particle is at position p=7 at t=1, what is the position of the particle at t=2?

a)

-10

b)

-5

c)

-3

d)

3

103.

a)

b)

c)

d)

104.

What is the area of the region enclosed by the graphs of f(x)=x-2x² and g(x)=-5x?

a)

7/3

b)

16/3

c)

20/3

d)

9

105.

The velocity of a particle moving along an axis is given by v(t)=2-t² for t>0, what is the average velocity of the particle from t=1 to t=3?

a)

-4

b)

-3

c)

8/3

d)

- 7/3

106.

a)

10

b)

20

c)

23

d)

35

107.

a)

b)

c)

d)

Mathematics

Total questions: 107

Worksheet time: 3hrs 19mins

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