Question 1

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

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative maximum at x = 1. 
C) ƒ has a relative minimum at x = 1. 
D) ƒ has a point of inflection at $x = \frac { 1 } { 3 }$ 
E) ƒ is concave upward on $\left( \frac { 1 } { 3 } , \infty \right)$ 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has a relative maximum at x = 1. 
C) ƒ has a relative minimum at x = 1. 
D) ƒ has a point of inflection at $x = \frac { 1 } { 3 }$ 
E) ƒ is concave upward on $\left( \frac { 1 } { 3 } , \infty \right)$

Question 2

Let $f ( x ) = x ^ { 2 } + 2 x - 1$ on $[ 0,1 ]$ . Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Accepted Answer

A)  $\left\{ \frac { 1 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 3 } \right\}$ 
C)  $\left\{ \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ \frac { 2 } { 3 } \right\}$ 
E)  $\left\{ \frac { 3 } { 4 } \right\}$ 
A)  $\left\{ \frac { 1 } { 4 } \right\}$ 
B)  $\left\{ \frac { 1 } { 3 } \right\}$ 
C)  $\left\{ \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ \frac { 2 } { 3 } \right\}$ 
E)  $\left\{ \frac { 3 } { 4 } \right\}$

Question 3

Let $s ^ { 2 } = 200 + x ^ { 2 }$ If $\frac { d x } { d t } = - 6$ when $x = 5 \text {, }$ then $\frac { d s } { d t }$ is

Accepted Answer

A) -6 
B) -4 
C) -2 
D) 2 
E) 4 
A) -6 
B) -4 
C) -2 
D) 2 
E) 4

Question 4

If each edge of a cube is increasing at the rate of 3 cm/s when the length of an edge is 2 cm long, then the rate of increase of its volume is

Accepted Answer

A)  $6 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
B)  $12 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
C)  $24 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
D)  $36 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
E)  $72 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
A)  $6 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
B)  $12 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
C)  $24 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
D)  $36 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ 
E)  $72 \mathrm {~cm} ^ { 3 } / \mathrm { s }$

Question 5

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

Accepted Answer

A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has a vertical asymptote: x = -1. 
D) ƒ has a horizontal asymptote: y = 1. 
E) ƒ is concave downward on $( - \infty , - 1 )$ 
A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has a vertical asymptote: x = -1. 
D) ƒ has a horizontal asymptote: y = 1. 
E) ƒ is concave downward on $( - \infty , - 1 )$

Question 6

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln x } { \ln ( x + 3 ) }$ ?

Accepted Answer

A)  $- \infty$ 
B) -1 
C) 1 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B) -1 
C) 1 
D) 0 
E)  $\infty$

Question 7

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( 1 + e ^ { x } \right) } { 3 x }$ ?

Accepted Answer

A)  $- \infty$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $\frac { 1 } { 3 }$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- \frac { 1 } { 3 }$ 
C)  $\frac { 1 } { 3 }$ 
D) 0 
E)  $\infty$

Question 8

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 ƒ increasing?

Accepted Answer

A)  $( - \infty , 0 )$ 
B)  $( - 3,3 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , - 3 ) \cup ( 3 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $( - \infty , 0 )$ 
B)  $( - 3,3 )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , - 3 ) \cup ( 3 , \infty )$ 
E)  $( - \infty , \infty )$

Question 9

Let $f ( x ) = \left\{ \begin{array} { l l } 
x ^ { 2 } & - 2 \leq x < 1 \
x ^ { 3 } & 1 \leq x \leq 2
\end{array} \right.$ The set of all critical numbers of ƒ is

Accepted Answer

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

Question 10

Let $f ( x ) = - 2 x ^ { 3 } + 6 x ^ { 2 } - 3$ Then ƒ has a point of inflection at x =

Accepted Answer

A)  $- \frac { 1 } { 3 }$ 
B) 1 
C)  $\frac { 1 } { 3 }$ 
D) -1 
E) 2 
A)  $- \frac { 1 } { 3 }$ 
B) 1 
C)  $\frac { 1 } { 3 }$ 
D) -1 
E) 2

Question 11

Let $f ( x ) = x ^ { 3 } + 1$ on $[ 0,2 ]$ Then the set of all c in (0,2) guaranteed by the Mean Value Theorem is

Accepted Answer

A)  $\left\{ \frac { \sqrt { 3 } } { 4 } \right\}$ 
B)  $\left\{ \frac { 3 } { 4 } \right\}$ 
C)  $\left\{ \frac { 2 \sqrt { 3 } } { 3 } \right\}$ 
D)  $\left\{ \frac { 5 } { 4 } \right\}$ 
E)  $\{ 1 \}$ 
A)  $\left\{ \frac { \sqrt { 3 } } { 4 } \right\}$ 
B)  $\left\{ \frac { 3 } { 4 } \right\}$ 
C)  $\left\{ \frac { 2 \sqrt { 3 } } { 3 } \right\}$ 
D)  $\left\{ \frac { 5 } { 4 } \right\}$ 
E)  $\{ 1 \}$

Question 12

Let $f ( x ) = x ^ { \frac { 2 } { 3 } }$ on $[ 0,1 ]$ Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Accepted Answer

A)  $\left\{ \frac { 2 } { 9 } \right\}$ 
B)  $\left\{ \frac { 1 } { 3 } \right\}$ 
C)  $\left\{ \frac { 4 } { 9 } \right\}$ 
D)  $\left\{ \frac { 8 } { 27 } \right\}$ 
E)  $\left\{ \frac { 11 } { 27 } \right\}$ 
A)  $\left\{ \frac { 2 } { 9 } \right\}$ 
B)  $\left\{ \frac { 1 } { 3 } \right\}$ 
C)  $\left\{ \frac { 4 } { 9 } \right\}$ 
D)  $\left\{ \frac { 8 } { 27 } \right\}$ 
E)  $\left\{ \frac { 11 } { 27 } \right\}$

Question 13

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

Accepted Answer

A)  $- \infty$ 
B)  $- \ln 2$ 
C)  $\ln 2$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- \ln 2$ 
C)  $\ln 2$ 
D) 0 
E)  $\infty$

Question 14

The particular solution of the differential equation $\frac { d y } { d x } = 9 x ^ { 2 } - 4 x + 5$ satisfying the boundary condition $y ( - 1 ) = 0$ is

Accepted Answer

A)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 10$ 
B)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 10$. 
C)  $y = - 3 x ^ { 3 } - 2 x ^ { 2 } - 5 x + 10$ 
D)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 10$ 
E)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x - 10$ 
A)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 10$ 
B)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 10$. 
C)  $y = - 3 x ^ { 3 } - 2 x ^ { 2 } - 5 x + 10$ 
D)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 10$ 
E)  $y = 3 x ^ { 3 } - 2 x ^ { 2 } + 5 x - 10$

Question 15

Let y = ƒ(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 horizontal 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 16

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

Accepted Answer

A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has no vertical asymptote. 
C) ƒ has a horizontal asymptote: y = 0. 
D) ƒ has no point of inflection. 
E) ƒ has a relative maximum and a relative minimum. 
A) ƒ is continuous on $( - \infty , \infty )$ 
B) ƒ has no vertical asymptote. 
C) ƒ has a horizontal asymptote: y = 0. 
D) ƒ has no point of inflection. 
E) ƒ has a relative maximum and a relative minimum.

Question 17

Let $x y = 1$ If $\frac { d x } { d t } = 12$ when $x = 2$ and $y = \frac { 1 } { 2 } ,$ then $\frac { d y } { d t }$ is

Accepted Answer

A) -8 
B) -6 
C) -4 
D) -3 
E) -2 
A) -8 
B) -6 
C) -4 
D) -3 
E) -2

Question 18

Determine which of the following is not true for the graph of $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 1 }$

Accepted Answer

A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has a vertical asymptote: x = 1. 
D) ƒ has no horizontal asymptote. 
E) ƒ is concave downward on $( - \infty , \infty )$ 
A) ƒ has no relative maximum. 
B) ƒ has no relative minimum. 
C) ƒ has a vertical asymptote: x = 1. 
D) ƒ has no horizontal asymptote. 
E) ƒ is concave downward on $( - \infty , \infty )$

Question 19

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

Accepted Answer

A)  $- \infty$ 
B)  $- \ln 2$ 
C)  $\ln 2$ 
D) 0 
E)  $\infty$ 
A)  $- \infty$ 
B)  $- \ln 2$ 
C)  $\ln 2$ 
D) 0 
E)  $\infty$

Question 20

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ on $[ - 3,0 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

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

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

Let $f ( x ) = x ^ { 2 } + 2 x - 1$ on $[ 0,1 ]$ . Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Let $s ^ { 2 } = 200 + x ^ { 2 }$ If $\frac { d x } { d t } = - 6$ when $x = 5 \text {, }$ then $\frac { d s } { d t }$ is

If each edge of a cube is increasing at the rate of 3 cm/s when the length of an edge is 2 cm long, then the rate of increase of its volume is

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

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln x } { \ln ( x + 3 ) }$ ?

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( 1 + e ^ { x } \right) } { 3 x }$ ?

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 ƒ increasing?

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ The set of all critical numbers of ƒ is

Let $f ( x ) = - 2 x ^ { 3 } + 6 x ^ { 2 } - 3$ Then ƒ has a point of inflection at x =

Let $f ( x ) = x ^ { 3 } + 1$ on $[ 0,2 ]$ Then the set of all c in (0,2) guaranteed by the Mean Value Theorem is

Let $f ( x ) = x ^ { \frac { 2 } { 3 } }$ on $[ 0,1 ]$ Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

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

The particular solution of the differential equation $\frac { d y } { d x } = 9 x ^ { 2 } - 4 x + 5$ satisfying the boundary condition $y ( - 1 ) = 0$ is

Let y = ƒ(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 horizontal tangent?

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

Let $x y = 1$ If $\frac { d x } { d t } = 12$ when $x = 2$ and $y = \frac { 1 } { 2 } ,$ then $\frac { d y } { d t }$ is

Determine which of the following is not true for the graph of $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 1 }$

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

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ on $[ - 3,0 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

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

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Exam 5: Applications of the Derivative

Determine which of the following is not true for the graph of f(x)=x3−x2−x+4f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 4f(x)=x3−x2−x+4

Let f(x)=x2+2x−1f ( x ) = x ^ { 2 } + 2 x - 1f(x)=x2+2x−1 on [0,1][ 0,1 ][0,1] . Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Let s2=200+x2s ^ { 2 } = 200 + x ^ { 2 }s2=200+x2 If dxdt=−6\frac { d x } { d t } = - 6dtdx​=−6 when x=5, x = 5 \text {, }x=5, then dsdt\frac { d s } { d t }dtds​ is

If each edge of a cube is increasing at the rate of 3 cm/s when the length of an edge is 2 cm long, then the rate of increase of its volume is

Determine which of the following is not true for the graph of f(x)=xx+1f ( x ) = \frac { x } { x + 1 }f(x)=x+1x​

Determine the value of the limit lim⁡x→∞ln⁡xln⁡(x+3)\lim _ { x \rightarrow \infty } \frac { \ln x } { \ln ( x + 3 ) }limx→∞​ln(x+3)lnx​ ?

Determine the value of the limit lim⁡x→∞ln⁡(1+ex)3x\lim _ { x \rightarrow \infty } \frac { \ln \left( 1 + e ^ { x } \right) } { 3 x }limx→∞​3xln(1+ex)​ ?

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 ƒ increasing?

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​ The set of all critical numbers of ƒ is

Let f(x)=−2x3+6x2−3f ( x ) = - 2 x ^ { 3 } + 6 x ^ { 2 } - 3f(x)=−2x3+6x2−3 Then ƒ has a point of inflection at x =

Let f(x)=x3+1f ( x ) = x ^ { 3 } + 1f(x)=x3+1 on [0,2][ 0,2 ][0,2] Then the set of all c in (0,2) guaranteed by the Mean Value Theorem is

Let f(x)=x23f ( x ) = x ^ { \frac { 2 } { 3 } }f(x)=x32​ on [0,1][ 0,1 ][0,1] Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Determine the value of the limit lim⁡x→∞ln⁡(2xx−1)\lim _ { x \rightarrow \infty } \ln \left( \frac { 2 x } { x - 1 } \right)limx→∞​ln(x−12x​) ?

The particular solution of the differential equation dydx=9x2−4x+5\frac { d y } { d x } = 9 x ^ { 2 } - 4 x + 5dxdy​=9x2−4x+5 satisfying the boundary condition y(−1)=0y ( - 1 ) = 0y(−1)=0 is

Let y = ƒ(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 horizontal tangent? ​

Determine which of the following is not true for the graph of f(x)=4xx2+4f ( x ) = \frac { 4 x } { x ^ { 2 } + 4 }f(x)=x2+44x​

Let xy=1x y = 1xy=1 If dxdt=12\frac { d x } { d t } = 12dtdx​=12 when x=2x = 2x=2 and y=12,y = \frac { 1 } { 2 } ,y=21​, then dydt\frac { d y } { d t }dtdy​ is

Determine which of the following is not true for the graph of f(x)=x2−9x−1f ( x ) = \frac { x ^ { 2 } - 9 } { x - 1 }f(x)=x−1x2−9​

Determine the value of the limit lim⁡x→02x−1x2x\lim _ { x \rightarrow 0 } \frac { 2 ^ { x } - 1 } { x 2 ^ { x } }limx→0​x2x2x−1​ ?

Let f(x)=x3−12x+1f ( x ) = x ^ { 3 } - 12 x + 1f(x)=x3−12x+1 on [−3,0][ - 3,0 ][−3,0] The absolute minimum and maximum of ƒ are located respectively at x =

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

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

Let $f ( x ) = x ^ { 2 } + 2 x - 1$ on $[ 0,1 ]$ . Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

Let $s ^ { 2 } = 200 + x ^ { 2 }$ If $\frac { d x } { d t } = - 6$ when $x = 5 \text {, }$ then $\frac { d s } { d t }$ is

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

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln x } { \ln ( x + 3 ) }$ ?

Determine the value of the limit $\lim _ { x \rightarrow \infty } \frac { \ln \left( 1 + e ^ { x } \right) } { 3 x }$ ?

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 ƒ increasing?

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ The set of all critical numbers of ƒ is

Let $f ( x ) = - 2 x ^ { 3 } + 6 x ^ { 2 } - 3$ Then ƒ has a point of inflection at x =

Let $f ( x ) = x ^ { 3 } + 1$ on $[ 0,2 ]$ Then the set of all c in (0,2) guaranteed by the Mean Value Theorem is

Let $f ( x ) = x ^ { \frac { 2 } { 3 } }$ on $[ 0,1 ]$ Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

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

The particular solution of the differential equation $\frac { d y } { d x } = 9 x ^ { 2 } - 4 x + 5$ satisfying the boundary condition $y ( - 1 ) = 0$ is

Let y = ƒ(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 horizontal tangent?

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

Let $x y = 1$ If $\frac { d x } { d t } = 12$ when $x = 2$ and $y = \frac { 1 } { 2 } ,$ then $\frac { d y } { d t }$ is

Determine which of the following is not true for the graph of $f ( x ) = \frac { x ^ { 2 } - 9 } { x - 1 }$

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

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ on $[ - 3,0 ]$ The absolute minimum and maximum of ƒ are located respectively at x =