Question 1

Let $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1$ Then ƒ has a relative maximum at x =

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Determine which of the following is not true for the graph of $f ( x ) = 2 x ^ { 3 } - 6 x + 4$

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 = 0. 
E) is concave upward on $( - \infty , 1 )$ 
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 = 0. 
E) is concave upward on $( - \infty , 1 )$

Question 6

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 12 - x ^ { 2 }$ The largest area of the rectangle is

Accepted Answer

A) 42 
B) 36 
C) 32 
D) 16 
E) 12 
A) 42 
B) 36 
C) 32 
D) 16 
E) 12

Question 7

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } x \cot x$

Accepted Answer

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

Question 8

Let $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$ Then ƒ has a relative maximum at x =

Accepted Answer

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

Question 9

The largest set on which the function $f ( x ) = 5 ^ { - x }$ is decreasing is

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , - 5 )$ 
C)  $( - 5 , \infty )$ 
D)  $( - 5,5 )$ 
E)  $( 5 , \infty )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , - 5 )$ 
C)  $( - 5 , \infty )$ 
D)  $( - 5,5 )$ 
E)  $( 5 , \infty )$

Question 10

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

Accepted Answer

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

Question 11

Determine the value of the limit $\lim _ { x \rightarrow \infty } x ^ { 2 } 2 ^ { - x } .$ ?

Accepted Answer

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

Question 12

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 2$ 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 13

The antiderivative of $f ( x ) = ( 3 + 2 x ) ^ { 2 }$ is

Accepted Answer

A)  $\frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } + 9 x + C$ 
B)  $\frac { 4 x ^ { 3 } } { 3 } - 6 x ^ { 2 } + 9 x + C$ 
C)  $\frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } - 9 x + C$ 
D)  $\frac { 4 x ^ { 3 } } { 3 } - 6 x ^ { 2 } - 9 x + C$ 
E)  $- \frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } + 9 x + C$ 
A)  $\frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } + 9 x + C$ 
B)  $\frac { 4 x ^ { 3 } } { 3 } - 6 x ^ { 2 } + 9 x + C$ 
C)  $\frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } - 9 x + C$ 
D)  $\frac { 4 x ^ { 3 } } { 3 } - 6 x ^ { 2 } - 9 x + C$ 
E)  $- \frac { 4 x ^ { 3 } } { 3 } + 6 x ^ { 2 } + 9 x + C$

Question 14

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 5$ The set of all critical numbers of ƒ is

Accepted Answer

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

Question 15

If A = xy and 2x + 3y = 12, the maximum value of A is
​

Accepted Answer

A) 6 
B) 8 
C) 10 
D) 12 
E) 14 
A) 6 
B) 8 
C) 10 
D) 12 
E) 14

Question 16

The antiderivative of $f ( x ) = \sinh x + \cos x - \frac { 1 } { 1 + x ^ { 2 } }$ is

Accepted Answer

A)  $\cosh x + \sin x - \tan ^ { - 1 } x + C$ 
B)  $\cosh x - \sin x + \tan ^ { - 1 } x + C$ 
C)  $\cosh x - \sin x - \tan ^ { - 1 } x + C$ 
D)  $- \cosh x + \sin x - \tan ^ { - 1 } x + C$ 
E)  $\cosh x + \sin x + \tan ^ { - 1 } x + C$ 
A)  $\cosh x + \sin x - \tan ^ { - 1 } x + C$ 
B)  $\cosh x - \sin x + \tan ^ { - 1 } x + C$ 
C)  $\cosh x - \sin x - \tan ^ { - 1 } x + C$ 
D)  $- \cosh x + \sin x - \tan ^ { - 1 } x + C$ 
E)  $\cosh x + \sin x + \tan ^ { - 1 } x + C$

Question 17

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The absolute minimum and maximum of ƒ are located respectively at x =

Accepted Answer

A)  $0 , \frac { \pi } { 2 }$ 
B)  $0 , \pi$ 
C)  $\frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 2 } , 2 \pi$ 
E)  $0,2 \pi$ 
A)  $0 , \frac { \pi } { 2 }$ 
B)  $0 , \pi$ 
C)  $\frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 2 } , 2 \pi$ 
E)  $0,2 \pi$

Question 18

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

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 upward on (-1,1). 
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 upward on (-1,1).

Question 19

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 27 - x ^ { 2 }$ The largest area of the rectangle is

Accepted Answer

A) 100 
B) 105 
C) 108 
D) 125 
E) 128 
A) 100 
B) 105 
C) 108 
D) 125 
E) 128

Question 20

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x$ Then ƒ has a point of inflection at x =

Accepted Answer

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

Let $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1$ Then ƒ has a relative maximum at x =

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

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

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

Determine which of the following is not true for the graph of $f ( x ) = 2 x ^ { 3 } - 6 x + 4$

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 12 - x ^ { 2 }$ The largest area of the rectangle is

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } x \cot x$

Let $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$ Then ƒ has a relative maximum at x =

The largest set on which the function $f ( x ) = 5 ^ { - x }$ is decreasing is

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

Determine the value of the limit $\lim _ { x \rightarrow \infty } x ^ { 2 } 2 ^ { - x } .$ ?

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

The antiderivative of $f ( x ) = ( 3 + 2 x ) ^ { 2 }$ is

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 5$ The set of all critical numbers of ƒ is

If A = xy and 2x + 3y = 12, the maximum value of A is

The antiderivative of $f ( x ) = \sinh x + \cos x - \frac { 1 } { 1 + x ^ { 2 } }$ is

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The absolute minimum and maximum of ƒ are located respectively at x =

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

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 27 - x ^ { 2 }$ The largest area of the rectangle is

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x$ Then ƒ has a point of inflection 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

Exam 5: Applications of the Derivative

Let f(x)=−x3−6x2−9x+1f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1f(x)=−x3−6x2−9x+1 Then ƒ has a relative maximum at x =

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 relative maximum at x =

Determine the value of the limit lim⁡x→∞x22−x\lim _ { x \rightarrow \infty } x ^ { 2 } 2 ^ { - x }limx→∞​x22−x ?

Determine the value of the limit lim⁡x→∞(1+12x)x2\lim _ { x \rightarrow \infty } \left( 1 + \frac { 1 } { 2 x } \right) ^ { x ^ { 2 } }limx→∞​(1+2x1​)x2

Determine which of the following is not true for the graph of f(x)=2x3−6x+4f ( x ) = 2 x ^ { 3 } - 6 x + 4f(x)=2x3−6x+4

A rectangle has its base on the x-axis and its upper two vertices on the parabola y=12−x2y = 12 - x ^ { 2 }y=12−x2 The largest area of the rectangle is

Determine the value of the limit lim⁡x→0+tan⁡−1xcot⁡x\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } x \cot xlimx→0+​tan−1xcotx

Let f(x)=x3−9x2+15xf ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 xf(x)=x3−9x2+15x Then ƒ has a relative maximum at x =

The largest set on which the function f(x)=5−xf ( x ) = 5 ^ { - x }f(x)=5−x is decreasing is

Let f(x)=x3−x+2f ( x ) = x ^ { 3 } - x + 2f(x)=x3−x+2 on [−1,1][ - 1,1 ][−1,1] Then the set of all c in (-1,1) guaranteed by Rolle's Theorem is

Determine the value of the limit lim⁡x→∞x22−x.\lim _ { x \rightarrow \infty } x ^ { 2 } 2 ^ { - x } .limx→∞​x22−x. ?

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

The antiderivative of f(x)=(3+2x)2f ( x ) = ( 3 + 2 x ) ^ { 2 }f(x)=(3+2x)2 is

Let f(x)=2x3−15x2+36x+5f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 5f(x)=2x3−15x2+36x+5 The set of all critical numbers of ƒ is

If A = xy and 2x + 3y = 12, the maximum value of A is ​

The antiderivative of f(x)=sinh⁡x+cos⁡x−11+x2f ( x ) = \sinh x + \cos x - \frac { 1 } { 1 + x ^ { 2 } }f(x)=sinhx+cosx−1+x21​ is

Let f(x)=x−sin⁡xf ( x ) = x - \sin xf(x)=x−sinx on [0,2π][ 0,2 \pi ][0,2π] 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)=x2−4x2−1f ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } - 1 }f(x)=x2−1x2−4​

A rectangle has its base on the x-axis and its upper two vertices on the parabola y=27−x2y = 27 - x ^ { 2 }y=27−x2 The largest area of the rectangle is

Let f(x)=12x3−3x2+6xf ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 xf(x)=21​x3−3x2+6x Then ƒ has a point of inflection 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

Let $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1$ Then ƒ has a relative maximum at x =

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

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

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

Determine which of the following is not true for the graph of $f ( x ) = 2 x ^ { 3 } - 6 x + 4$

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 12 - x ^ { 2 }$ The largest area of the rectangle is

Determine the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } \tan ^ { - 1 } x \cot x$

Let $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$ Then ƒ has a relative maximum at x =

The largest set on which the function $f ( x ) = 5 ^ { - x }$ is decreasing is

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

Determine the value of the limit $\lim _ { x \rightarrow \infty } x ^ { 2 } 2 ^ { - x } .$ ?

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

The antiderivative of $f ( x ) = ( 3 + 2 x ) ^ { 2 }$ is

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 5$ The set of all critical numbers of ƒ is

If A = xy and 2x + 3y = 12, the maximum value of A is

The antiderivative of $f ( x ) = \sinh x + \cos x - \frac { 1 } { 1 + x ^ { 2 } }$ is

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The absolute minimum and maximum of ƒ are located respectively at x =

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

A rectangle has its base on the x-axis and its upper two vertices on the parabola $y = 27 - x ^ { 2 }$ The largest area of the rectangle is

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x$ Then ƒ has a point of inflection at x =