Question 1

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 } { 2 } x ^ { 2 } + 1$ The set of all critical numbers of ƒ is

Accepted Answer

A)  $\{ - \sqrt { 3 } , \sqrt { 3 } \}$ 
B)  $\{ - \sqrt { 3 } , 0 \}$ 
C)  $\{ 0 , \sqrt { 3 } \}$ 
D)  $\{ 0 \}$ 
E)  $\{ - \sqrt { 3 } , 0 , \sqrt { 3 } \}$ 
A)  $\{ - \sqrt { 3 } , \sqrt { 3 } \}$ 
B)  $\{ - \sqrt { 3 } , 0 \}$ 
C)  $\{ 0 , \sqrt { 3 } \}$ 
D)  $\{ 0 \}$ 
E)  $\{ - \sqrt { 3 } , 0 , \sqrt { 3 } \}$

Question 2

Determine the value of the limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)$ ?

Accepted Answer

A)  $- \infty$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- \frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E)  $\infty$

Question 3

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of ƒ have a vertical tangent?

Accepted Answer

A) x = 0 only 
B) x = -3 only 
C) x = 3 only 
D) x = -3 and x = 3 
E) None 
A) x = 0 only 
B) x = -3 only 
C) x = 3 only 
D) x = -3 and x = 3 
E) None

Question 4

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A) 1,3 
B) 3,1 
C) 1,2 
D) 2,3 
E) 3,2 
A) 1,3 
B) 3,1 
C) 1,2 
D) 2,3 
E) 3,2

Question 5

Determine which of the following is not true for the graph of $f ( x ) = \frac { 4 } { x ^ { 2 } + 1 }$

Accepted Answer

A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has no vertical asymptote. 
D) ƒ has a horizontal asymptote: y = 0. 
E) ƒ is concave downward on $\left( - \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) \text {. }$ 
A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has no vertical asymptote. 
D) ƒ has a horizontal asymptote: y = 0. 
E) ƒ is concave downward on $\left( - \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) \text {. }$

Question 6

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a relative minimum at x =

Accepted Answer

A)  -2 
B) -1 
C) 0 
D) 1 
E) 2 
A)  -2 
B) -1 
C) 0 
D) 1 
E) 2

Question 7

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 4 } { 3 } } + 4 x ^ { \frac { 1 } { 3 } } + 2$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has no relative maximum. 
C) ƒ has a relative minimum at x = -1. 
D) ƒ has a point of inflection at x = 2. 
E) ƒ is concave downward on $( - \infty , 0 )$ 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has no relative maximum. 
C) ƒ has a relative minimum at x = -1. 
D) ƒ has a point of inflection at x = 2. 
E) ƒ is concave downward on $( - \infty , 0 )$

Question 8

Let $f ( x ) = \left\{ \begin{array} { l l } 
x ^ { 2 } & - 2 \leq x < 1 \
x ^ { 3 } & 1 \leq x \leq 2
\end{array} \right.$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A)  $- 2,0$ 
B)  $0 , - 2$ 
C)  $0,2$ 
D)  $- 2,2$ 
E)  $- 3 , - 2$ 
A)  $- 2,0$ 
B)  $0 , - 2$ 
C)  $0,2$ 
D)  $- 2,2$ 
E)  $- 3 , - 2$

Question 9

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒhas a relative maximum at x = -1. 
C) ƒ has a relative minimum at x = 5. 
D) ƒ has a point of inflection at x = 3. 
E) ƒ is concave downward on $( - \infty , 3 )$ 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒhas a relative maximum at x = -1. 
C) ƒ has a relative minimum at x = 5. 
D) ƒ has a point of inflection at x = 3. 
E) ƒ is concave downward on $( - \infty , 3 )$

Question 10

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ The set of all critical numbers of ƒ is

Accepted Answer

A)  $\{ - 2 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ - 2,2 \}$ 
D)  $\{ 0 , - 2 \}$ 
E)  $\{ 0,2 \}$ 
A)  $\{ - 2 \}$ 
B)  $\{ 2 \}$ 
C)  $\{ - 2,2 \}$ 
D)  $\{ 0 , - 2 \}$ 
E)  $\{ 0,2 \}$

Question 11

A company is selling tables at a price of $400 each. The total cost required to produce x tables is $C ( x ) = 2 x ^ { 2 } + 80 x + 6000$ The maximum profit of this company, in dollars, is

Accepted Answer

A) 6,800 
B) 7,000 
C) 7,200 
D) 7,500 
E) 8,000 
A) 6,800 
B) 7,000 
C) 7,200 
D) 7,500 
E) 8,000

Question 12

Let $f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } - 3 x - \frac { 1 } { 5 }$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A) 0,3 
B) 3,0 
C) 1,3 
D) 3,1 
E) 2,3 
A) 0,3 
B) 3,0 
C) 1,3 
D) 3,1 
E) 2,3

Question 13

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 1$ on $[ 0,2 ]$ . The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A) 0,2 
B) 0,1 
C)  $\sqrt { 3 } , 0$ 
D)  $\sqrt { 3 } , 2$ 
E) 1,2 
A) 0,2 
B) 0,1 
C)  $\sqrt { 3 } , 0$ 
D)  $\sqrt { 3 } , 2$ 
E) 1,2

Question 14

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

Accepted Answer

A)  $\{ - 1 \}$ 
B)  $\{ 0 \}$ 
C)  $\{ 1 \}$ 
D)  $\{ 0,1 \}$ 
E)  $\{ - 1,0,1 \}$ 
A)  $\{ - 1 \}$ 
B)  $\{ 0 \}$ 
C)  $\{ 1 \}$ 
D)  $\{ 0,1 \}$ 
E)  $\{ - 1,0,1 \}$

Question 15

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a local maximum?

Accepted Answer

A) x = 0 only 
B) x = -3 only 
C) x = 3 only 
D) x = -3 and x = 3 
E) None 
A) x = 0 only 
B) x = -3 only 
C) x = 3 only 
D) x = -3 and x = 3 
E) None

Question 16

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 1 } { \ln x } } .$

Accepted Answer

A)  $- \infty$ 
B)  $- e$ 
C)  $e$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- e$ 
C)  $e$ 
D) 0 
E)  $\infty$

Question 17

Determine the value of the limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } } { \sin x - x }$ ?

Accepted Answer

A)  $- \infty$ 
B)  $- 6$ 
C) 6 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- 6$ 
C) 6 
D) 0 
E)  $\infty$

Question 18

Let $A = \pi r ^ { 2 }$ If $\frac { d r } { d t } = 1.5$ when then $r = 6,$ is

Accepted Answer

A)  $18 \pi$ 
B)  $16 \pi$ 
C)  $12 \pi$ 
D)  $8 \pi$ 
E)  $4 \pi$ 
A)  $18 \pi$ 
B)  $16 \pi$ 
C)  $12 \pi$ 
D)  $8 \pi$ 
E)  $4 \pi$

Question 19

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
On what interval(s) is the graph of f concave down?

Accepted Answer

A)  $( - \infty , 1 )$ 
B)  $( - \infty , - 2 ) \cup ( 0 , \infty )$ 
C) (-2, 0) 
D)  $( 1 , \infty )$ 
E) (-2, 1) 
A)  $( - \infty , 1 )$ 
B)  $( - \infty , - 2 ) \cup ( 0 , \infty )$ 
C) (-2, 0) 
D)  $( 1 , \infty )$ 
E) (-2, 1)

Question 20

Let $\sqrt { x } + \sqrt { y } = 5$ If $\frac { d x } { d t } = - \frac { 3 } { 4 }$ when $x = 1 ,$ then $\frac { d y } { d t }$ is

Accepted Answer

A) 4 
B) 3 
C) 2 
D) -2 
E) -3 
A) 4 
B) 3 
C) 2 
D) -2 
E) -3

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 } { 2 } x ^ { 2 } + 1$ The set of all critical numbers of ƒ is

Determine the value of the limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)$ ?

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of ƒ have a vertical tangent?

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = \frac { 4 } { x ^ { 2 } + 1 }$

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a relative minimum at x =

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 4 } { 3 } } + 4 x ^ { \frac { 1 } { 3 } } + 2$

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ The set of all critical numbers of ƒ is

A company is selling tables at a price of $400 each. The total cost required to produce x tables is $C ( x ) = 2 x ^ { 2 } + 80 x + 6000$ The maximum profit of this company, in dollars, is

Let $f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } - 3 x - \frac { 1 } { 5 }$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 1$ on $[ 0,2 ]$ . The absolute minimum and maximum of ƒ are located respectively at x =

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local maximum?

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 1 } { \ln x } } .$

Determine the value of the limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } } { \sin x - x }$ ?

Let $A = \pi r ^ { 2 }$ If $\frac { d r } { d t } = 1.5$ when then $r = 6,$ is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: On what interval(s) is the graph of f concave down?

Let $\sqrt { x } + \sqrt { y } = 5$ If $\frac { d x } { d t } = - \frac { 3 } { 4 }$ when $x = 1 ,$ then $\frac { d y } { d t }$ is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Exam 5: Applications of the Derivative

Let f(x)=x44−32x2+1f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 } { 2 } x ^ { 2 } + 1f(x)=4x4​−23​x2+1 The set of all critical numbers of ƒ is

Determine the value of the limit lim⁡x→0(1x2−1x2cos⁡x)\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)limx→0​(x21​−x2cosx1​) ?

Let y=f(x)y = f ( x )y=f(x) be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of ƒ have a vertical tangent?

Let f(x)=12x3−3x2+6x−4f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4f(x)=21​x3−3x2+6x−4 on [1,3][ 1,3 ][1,3] The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of f(x)=4x2+1f ( x ) = \frac { 4 } { x ^ { 2 } + 1 }f(x)=x2+14​

Let f(x)=x4−4x3+16xf ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 xf(x)=x4−4x3+16x Then ƒ has a relative minimum at x =

Determine which of the following is not true for the graph of f(x)=x43+4x13+2f ( x ) = x ^ { \frac { 4 } { 3 } } + 4 x ^ { \frac { 1 } { 3 } } + 2f(x)=x34​+4x31​+2

Let f(x)={x2−2≤x<1x31≤x≤2f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.f(x)={x2x3​−2≤x<11≤x≤2​ on [−2,2][ - 2,2 ][−2,2] The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of f(x)=x3−9x2+15xf ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 xf(x)=x3−9x2+15x

Let f(x)=x3−12x+1f ( x ) = x ^ { 3 } - 12 x + 1f(x)=x3−12x+1 The set of all critical numbers of ƒ is

A company is selling tables at a price of $400 each. The total cost required to produce x tables is C(x)=2x2+80x+6000C ( x ) = 2 x ^ { 2 } + 80 x + 6000C(x)=2x2+80x+6000 The maximum profit of this company, in dollars, is

Let f(x)=15x3−95x2−3x−15f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } - 3 x - \frac { 1 } { 5 }f(x)=51​x3−59​x2−3x−51​ on [1,3][ 1,3 ][1,3] The absolute minimum and maximum of ƒ are located respectively at x =

Let f(x)=x44−3x22+1f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 1f(x)=4x4​−23x2​+1 on [0,2][ 0,2 ][0,2] . The absolute minimum and maximum of ƒ are located respectively at x =

Let f(x)=x4−2x2+1f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1f(x)=x4−2x2+1 on [−2,2][ - 2,2 ][−2,2] Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

Let y=f(x)y = f ( x )y=f(x) be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local maximum?

Determine the value of the limit lim⁡x→0+x1ln⁡x.\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 1 } { \ln x } } .limx→0+​xlnx1​.

Determine the value of the limit lim⁡x→0x3sin⁡x−x\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } } { \sin x - x }limx→0​sinx−xx3​ ?

Let A=πr2A = \pi r ^ { 2 }A=πr2 If drdt=1.5\frac { d r } { d t } = 1.5dtdr​=1.5 when then r=6,r = 6,r=6, is

Let y=f(x)y = f ( x )y=f(x) be a differentiable function for which the graph of its derivative, ƒ ', is given below: On what interval(s) is the graph of f concave down?

Let x+y=5\sqrt { x } + \sqrt { y } = 5x​+y​=5 If dxdt=−34\frac { d x } { d t } = - \frac { 3 } { 4 }dtdx​=−43​ when x=1,x = 1 ,x=1, then dydt\frac { d y } { d t }dtdy​ is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 } { 2 } x ^ { 2 } + 1$ The set of all critical numbers of ƒ is

Determine the value of the limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 2 } \cos x } \right)$ ?

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of ƒ have a vertical tangent?

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = \frac { 4 } { x ^ { 2 } + 1 }$

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a relative minimum at x =

Determine which of the following is not true for the graph of $f ( x ) = x ^ { \frac { 4 } { 3 } } + 4 x ^ { \frac { 1 } { 3 } } + 2$

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Determine which of the following is not true for the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ The set of all critical numbers of ƒ is

A company is selling tables at a price of $400 each. The total cost required to produce x tables is $C ( x ) = 2 x ^ { 2 } + 80 x + 6000$ The maximum profit of this company, in dollars, is

Let $f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } - 3 x - \frac { 1 } { 5 }$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 1$ on $[ 0,2 ]$ . The absolute minimum and maximum of ƒ are located respectively at x =

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: At what x-value(s), if any, does the graph of f have a local maximum?

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \frac { 1 } { \ln x } } .$

Determine the value of the limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 3 } } { \sin x - x }$ ?

Let $A = \pi r ^ { 2 }$ If $\frac { d r } { d t } = 1.5$ when then $r = 6,$ is

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below: On what interval(s) is the graph of f concave down?

Let $\sqrt { x } + \sqrt { y } = 5$ If $\frac { d x } { d t } = - \frac { 3 } { 4 }$ when $x = 1 ,$ then $\frac { d y } { d t }$ is