Question 1

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

Accepted Answer

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

Question 2

Suppose that, when a stone is thrown upwards with a speed of 120 ft/sec, its height, h feet, after t seconds, is given by $h ( t ) = 120 t - 16 t ^ { 2 }$ The maximum height of the stone is

Accepted Answer

A) 200 ft 
B) 225 ft 
C) 240 ft 
D) 250 ft 
E) 280 ft 
A) 200 ft 
B) 225 ft 
C) 240 ft 
D) 250 ft 
E) 280 ft

Question 3

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2$ Then ƒ has a point of inflection at x =

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B) -1 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E)  $\frac { 5 } { 2 }$ 
A)  $- \frac { 1 } { 2 }$ 
B) -1 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E)  $\frac { 5 } { 2 }$

Question 4

Let $f ( x ) = x ^ { 4 } + x ^ { 2 }$ 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 5

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

Accepted Answer

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

Question 6

A page of print contains 24 square inches of printed region, a margin of 1.5 inches at the top and bottom, and a margin of 1 inch at the sides. The dimensions of the smallest page that will fill these requirements are
​

Accepted Answer

A) 5 in by 10 in 
B) 6 in by 9 in 
C) 7 in by 8 in 
D) 5.5 in by 9.5 in 
E) 6.5 in by 8.5 in 
A) 5 in by 10 in 
B) 6 in by 9 in 
C) 7 in by 8 in 
D) 5.5 in by 9.5 in 
E) 6.5 in by 8.5 in

Question 7

The shortest distance from the point (2, 0) to the curve $y ^ { 2 } - x ^ { 2 } = 1$ is

Accepted Answer

A)  $\sqrt { 2 }$ 
B)  $\sqrt { 3 }$ 
C) 2 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 6 }$ 
A)  $\sqrt { 2 }$ 
B)  $\sqrt { 3 }$ 
C) 2 
D)  $\sqrt { 5 }$ 
E)  $\sqrt { 6 }$

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 f concave up?

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 $f ( x ) = x ^ { 4 } - 3$ on $[ - 2,2 ]$ Then the set of all c in $( - 2,2 )$ guaranteed by Rolle's Theorem is

Accepted Answer

A) $\{ - 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 3 } \right\}$ 
C)  $\left\{ \frac { \sqrt { 3 } } { 3 } \right\}$ 
D)  $\{ 0 \}$ 
E)  $\{ 1 \}$ 
A) $\{ - 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 3 } \right\}$ 
C)  $\left\{ \frac { \sqrt { 3 } } { 3 } \right\}$ 
D)  $\{ 0 \}$ 
E)  $\{ 1 \}$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $\left\{ \frac { 3 } { 4 } \right\}$ 
B)  $\{ 1 \}$ 
C)  $\left\{ \frac { 5 } { 4 } \right\}$ 
D)  $\left\{ \frac { 3 } { 2 } \right\}$ 
E)  $\left\{ \frac { 7 } { 4 } \right\}$ 
A)  $\left\{ \frac { 3 } { 4 } \right\}$ 
B)  $\{ 1 \}$ 
C)  $\left\{ \frac { 5 } { 4 } \right\}$ 
D)  $\left\{ \frac { 3 } { 2 } \right\}$ 
E)  $\left\{ \frac { 7 } { 4 } \right\}$

Question 12

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

Accepted Answer

A)  $\{ - 2,0 \}$ 
B)  $\{ 2,3 \}$ 
C)  $\{ 0 , - 3 \}$ 
D)  $\{ - 2,0,3 \}$ 
E)  $\{ - 2,0 \}$. 
A)  $\{ - 2,0 \}$ 
B)  $\{ 2,3 \}$ 
C)  $\{ 0 , - 3 \}$ 
D)  $\{ - 2,0,3 \}$ 
E)  $\{ - 2,0 \}$.

Question 13

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

Accepted Answer

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

Question 14

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ The set of all critical numbers of ƒ is

Accepted Answer

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

Question 15

The particular solution of the differential equation $\frac { d v } { d t } = 8 t + \csc ^ { 2 } t$ satisfying the boundary condition $v \left( \frac { \pi } { 2 } \right) = - 7$ is

Accepted Answer

A)  $v = 4 x ^ { 2 } - \cot t - 7 + \pi ^ { 2 }$ 
B)  $v = 4 x ^ { 2 } + \cot t + 7 + \pi ^ { 2 }$ 
C)  $v = 4 x ^ { 2 } + \cot t - 7 - \pi ^ { 2 }$ 
D)  $v = 4 x ^ { 2 } - \cot t - 7 - \pi ^ { 2 }$ 
E)  $v = 4 x ^ { 2 } - \cot t + 7 + \pi ^ { 2 }$ 
A)  $v = 4 x ^ { 2 } - \cot t - 7 + \pi ^ { 2 }$ 
B)  $v = 4 x ^ { 2 } + \cot t + 7 + \pi ^ { 2 }$ 
C)  $v = 4 x ^ { 2 } + \cot t - 7 - \pi ^ { 2 }$ 
D)  $v = 4 x ^ { 2 } - \cot t - 7 - \pi ^ { 2 }$ 
E)  $v = 4 x ^ { 2 } - \cot t + 7 + \pi ^ { 2 }$

Question 16

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 minimum?

Accepted Answer

A) x = 0 
B) x = -2 
C) x = -2, x = 1, and x = 0 
D) x = 1 
E) None 
A) x = 0 
B) x = -2 
C) x = -2, x = 1, and x = 0 
D) x = 1 
E) None

Question 17

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

Accepted Answer

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

Question 18

A rectangular sheet of cardboard is 24 inches by 9 inches. Equal squares are cut out at the corners and the flaps are turned up to form an open box. The volume of the box, in cubic inches, is
​

Accepted Answer

A) 180 
B) 200 
C) 220 
D) 240 
E) 260 
A) 180 
B) 200 
C) 220 
D) 240 
E) 260

Question 19

Let $x ^ { 2 } + y ^ { 2 } = 25$ If $\frac { d y } { d t } = - 4$ when $x = 4$ and $y = 3 \text {, }$ then $\frac { d x } { d t }$ is

Accepted Answer

A) 5 
B) 3 
C)  $\frac { 3 } { 2 }$ 
D) 1 
E)  $\frac { 1 } { 2 }$ 
A) 5 
B) 3 
C)  $\frac { 3 } { 2 }$ 
D) 1 
E)  $\frac { 1 } { 2 }$

Question 20

Suppose that, when a stone is thrown upwards with a speed of 60 miles an hour, its height, h feet, after t seconds, is given by $h ( t ) = 88 t - 16 t ^ { 2 }$ The maximum height of the stone is

Accepted Answer

A) 100 ft 
B) 120 ft 
C) 121 ft 
D) 125 ft 
E) 200 ft 
A) 100 ft 
B) 120 ft 
C) 121 ft 
D) 125 ft 
E) 200 ft

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

Suppose that, when a stone is thrown upwards with a speed of 120 ft/sec, its height, h feet, after t seconds, is given by $h ( t ) = 120 t - 16 t ^ { 2 }$ The maximum height of the stone is

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2$ Then ƒ has a point of inflection at x =

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

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

A page of print contains 24 square inches of printed region, a margin of 1.5 inches at the top and bottom, and a margin of 1 inch at the sides. The dimensions of the smallest page that will fill these requirements are

The shortest distance from the point (2, 0) to the curve $y ^ { 2 } - x ^ { 2 } = 1$ 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 up?

Let $f ( x ) = x ^ { 4 } - 3$ on $[ - 2,2 ]$ Then the set of all c in $( - 2,2 )$ guaranteed by Rolle's Theorem is

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

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

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

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ The set of all critical numbers of ƒ is

The particular solution of the differential equation $\frac { d v } { d t } = 8 t + \csc ^ { 2 } t$ satisfying the boundary condition $v \left( \frac { \pi } { 2 } \right) = - 7$ 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 minimum?

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

A rectangular sheet of cardboard is 24 inches by 9 inches. Equal squares are cut out at the corners and the flaps are turned up to form an open box. The volume of the box, in cubic inches, is

Let $x ^ { 2 } + y ^ { 2 } = 25$ If $\frac { d y } { d t } = - 4$ when $x = 4$ and $y = 3 \text {, }$ then $\frac { d x } { d t }$ is

Suppose that, when a stone is thrown upwards with a speed of 60 miles an hour, its height, h feet, after t seconds, is given by $h ( t ) = 88 t - 16 t ^ { 2 }$ The maximum height of the stone 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

Determine the value of the limit lim⁡x→0+ln⁡xln⁡(x2+2x)\lim _ { x \rightarrow 0 ^ { + } } \frac { \ln x } { \ln \left( x ^ { 2 } + 2 x \right) }limx→0+​ln(x2+2x)lnx​ ?

Suppose that, when a stone is thrown upwards with a speed of 120 ft/sec, its height, h feet, after t seconds, is given by h(t)=120t−16t2h ( t ) = 120 t - 16 t ^ { 2 }h(t)=120t−16t2 The maximum height of the stone is

Let f(x)=2x3−15x2−36x−2f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2f(x)=2x3−15x2−36x−2 Then ƒ has a point of inflection at x =

Let f(x)=x4+x2f ( x ) = x ^ { 4 } + x ^ { 2 }f(x)=x4+x2 on [−2,2][ - 2,2 ][−2,2] Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

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

A page of print contains 24 square inches of printed region, a margin of 1.5 inches at the top and bottom, and a margin of 1 inch at the sides. The dimensions of the smallest page that will fill these requirements are ​

The shortest distance from the point (2, 0) to the curve y2−x2=1y ^ { 2 } - x ^ { 2 } = 1y2−x2=1 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 up?

Let f(x)=x4−3f ( x ) = x ^ { 4 } - 3f(x)=x4−3 on [−2,2][ - 2,2 ][−2,2] Then the set of all c in (−2,2)( - 2,2 )(−2,2) guaranteed by Rolle's Theorem is

Determine the value of the limit lim⁡x→∞ln⁡x−ln⁡(x+1)\lim _ { x \rightarrow \infty } \ln x - \ln ( x + 1 )limx→∞​lnx−ln(x+1) ?

Let f(x)=x+1xff ( x ) = x + \frac { 1 } { x } ff(x)=x+x1​f on [12,2].\left[ \frac { 1 } { 2 } , 2 \right] .[21​,2]. Then the set of all c in (12,2)\left( \frac { 1 } { 2 } , 2 \right)(21​,2) guaranteed by the Mean Value Theorem is

Let f(x)=14x4−13x3−3x2+1f ( x ) = \frac { 1 } { 4 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } - 3 x ^ { 2 } + 1f(x)=41​x4−31​x3−3x2+1 The set of all critical numbers of ƒ is

Determine the value of the limit lim⁡x→0(1x−csc⁡x)\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x } - \csc x \right)limx→0​(x1​−cscx) ?

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 The set of all critical numbers of ƒ is

The particular solution of the differential equation dvdt=8t+csc⁡2t\frac { d v } { d t } = 8 t + \csc ^ { 2 } tdtdv​=8t+csc2t satisfying the boundary condition v(π2)=−7v \left( \frac { \pi } { 2 } \right) = - 7v(2π​)=−7 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 minimum?

Determine the value of the limit lim⁡x→0(1sin⁡x−1x)\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right)limx→0​(sinx1​−x1​)

A rectangular sheet of cardboard is 24 inches by 9 inches. Equal squares are cut out at the corners and the flaps are turned up to form an open box. The volume of the box, in cubic inches, is ​

Let x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25 If dydt=−4\frac { d y } { d t } = - 4dtdy​=−4 when x=4x = 4x=4 and y=3, y = 3 \text {, }y=3, then dxdt\frac { d x } { d t }dtdx​ is

Suppose that, when a stone is thrown upwards with a speed of 60 miles an hour, its height, h feet, after t seconds, is given by h(t)=88t−16t2h ( t ) = 88 t - 16 t ^ { 2 }h(t)=88t−16t2 The maximum height of the stone 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

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

Suppose that, when a stone is thrown upwards with a speed of 120 ft/sec, its height, h feet, after t seconds, is given by $h ( t ) = 120 t - 16 t ^ { 2 }$ The maximum height of the stone is

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2$ Then ƒ has a point of inflection at x =

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

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

A page of print contains 24 square inches of printed region, a margin of 1.5 inches at the top and bottom, and a margin of 1 inch at the sides. The dimensions of the smallest page that will fill these requirements are

The shortest distance from the point (2, 0) to the curve $y ^ { 2 } - x ^ { 2 } = 1$ 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 up?

Let $f ( x ) = x ^ { 4 } - 3$ on $[ - 2,2 ]$ Then the set of all c in $( - 2,2 )$ guaranteed by Rolle's Theorem is

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

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

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

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

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ The set of all critical numbers of ƒ is

The particular solution of the differential equation $\frac { d v } { d t } = 8 t + \csc ^ { 2 } t$ satisfying the boundary condition $v \left( \frac { \pi } { 2 } \right) = - 7$ 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 minimum?

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

A rectangular sheet of cardboard is 24 inches by 9 inches. Equal squares are cut out at the corners and the flaps are turned up to form an open box. The volume of the box, in cubic inches, is

Let $x ^ { 2 } + y ^ { 2 } = 25$ If $\frac { d y } { d t } = - 4$ when $x = 4$ and $y = 3 \text {, }$ then $\frac { d x } { d t }$ is

Suppose that, when a stone is thrown upwards with a speed of 60 miles an hour, its height, h feet, after t seconds, is given by $h ( t ) = 88 t - 16 t ^ { 2 }$ The maximum height of the stone is