Question 1

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The set of all critical numbers of ƒ is

Accepted Answer

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

Question 2

The particular solution of the differential equation $\frac { d y } { d x } = x ^ { 2 } + 2 x + 1$ satisfying the boundary condition when x = 3, y = -1 is

Accepted Answer

A)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 3$ 
B)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x + 1$ 
C)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 22$ 
D)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 21$ 
E)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 23$ 
A)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 3$ 
B)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x + 1$ 
C)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 22$ 
D)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 21$ 
E)  $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { 2 } + x - 23$

Question 3

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ The set of all critical numbers of ƒ is

Accepted Answer

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

Question 4

Suppose two positive integers satisfy the conditions that the sum of the first number and twice the second is 120 and their product is a maximum. Then the two numbers are
​

Accepted Answer

A) 25,70 
B) 27,66 
C) 28,64 
D) 29,62 
E) 30,60 
A) 25,70 
B) 27,66 
C) 28,64 
D) 29,62 
E) 30,60

Question 5

The antiderivative of $f ( x ) = \frac { 2 } { \sqrt { 1 - x ^ { 2 } } } - \sin x$ is

Accepted Answer

A)  $2 \sin ^ { - 1 } x - \cos x + C$ 
B)  $2 \cos ^ { - 1 } x - \cos x + C$ 
C)  $2 \sin ^ { - 1 } x + \cos x + C$ 
D)  $- 2 \sin ^ { - 1 } x + \cos x + C$ 
E)  $- 2 \sin ^ { - 1 } x + \sin x + C$ 
A)  $2 \sin ^ { - 1 } x - \cos x + C$ 
B)  $2 \cos ^ { - 1 } x - \cos x + C$ 
C)  $2 \sin ^ { - 1 } x + \cos x + C$ 
D)  $- 2 \sin ^ { - 1 } x + \cos x + C$ 
E)  $- 2 \sin ^ { - 1 } x + \sin x + C$

Question 6

If the surface area of a sphere is shrinking at the rate of 0.1 sq cm/h when the radius is $\frac { 20 } { \pi } \mathrm { cm } \text {, }$ the rate of change of the radius in cm/s is

Accepted Answer

A)  $- \frac { 1 } { 400 }$ 
B)  $- \frac { 1 } { 800 }$ 
C)  $- \frac { 1 } { 1600 }$ 
D)  $- \frac { 1 } { 3200 }$ 
E)  $- \frac { 1 } { 6400 }$ 
A)  $- \frac { 1 } { 400 }$ 
B)  $- \frac { 1 } { 800 }$ 
C)  $- \frac { 1 } { 1600 }$ 
D)  $- \frac { 1 } { 3200 }$ 
E)  $- \frac { 1 } { 6400 }$

Question 7

The shortest distance from the origin to the line 3x + y = 6 is

Accepted Answer

A)  $\frac { \sqrt { 82 } } { 4 }$ 
B)  $\frac { \sqrt { 86 } } { 5 }$ 
C)  $\frac { \sqrt { 90 } } { 5 }$ 
D)  $\frac { \sqrt { 92 } } { 5 }$ 
E)  $\frac { 1 } { 2 }$ 
A)  $\frac { \sqrt { 82 } } { 4 }$ 
B)  $\frac { \sqrt { 86 } } { 5 }$ 
C)  $\frac { \sqrt { 90 } } { 5 }$ 
D)  $\frac { \sqrt { 92 } } { 5 }$ 
E)  $\frac { 1 } { 2 }$

Question 8

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

Accepted Answer

A) ƒ has an oblique asymptote: y = x + 2. 
B) ƒ has a vertical asymptote: x = 1. 
C) ƒ has a removable discontinuity at x = 1. 
D) ƒ has no horizontal asymptote. 
E) ƒ is discontinuous on $( - \infty , \infty )$ 
A) ƒ has an oblique asymptote: y = x + 2. 
B) ƒ has a vertical asymptote: x = 1. 
C) ƒ has a removable discontinuity at x = 1. 
D) ƒ has no horizontal asymptote. 
E) ƒ is discontinuous on $( - \infty , \infty )$

Question 9

The shortest distance from the point $\left( 2 , \frac { 1 } { 2 } \right)$ to the parabola $y = x ^ { 2 }$ is

Accepted Answer

A)  $\frac { \sqrt { 5 } } { 2 }$ 
B)  $\frac { \sqrt { 6 } } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { 1 } { 2 }$ 
A)  $\frac { \sqrt { 5 } } { 2 }$ 
B)  $\frac { \sqrt { 6 } } { 2 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 2 }$ 
E)  $\frac { 1 } { 2 }$

Question 10

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

Accepted Answer

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

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The set of all critical numbers of ƒ is

The particular solution of the differential equation $\frac { d y } { d x } = x ^ { 2 } + 2 x + 1$ satisfying the boundary condition when x = 3, y = -1 is

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ The set of all critical numbers of ƒ is

Suppose two positive integers satisfy the conditions that the sum of the first number and twice the second is 120 and their product is a maximum. Then the two numbers are

The antiderivative of $f ( x ) = \frac { 2 } { \sqrt { 1 - x ^ { 2 } } } - \sin x$ is

If the surface area of a sphere is shrinking at the rate of 0.1 sq cm/h when the radius is $\frac { 20 } { \pi } \mathrm { cm } \text {, }$ the rate of change of the radius in cm/s is

The shortest distance from the origin to the line 3x + y = 6 is

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

The shortest distance from the point $\left( 2 , \frac { 1 } { 2 } \right)$ to the parabola $y = x ^ { 2 }$ is

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 1$ on $[ - 1,2 ]$ 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

Exam 5: Applications of the Derivative

Let f(x)=x−sin⁡xf ( x ) = x - \sin xf(x)=x−sinx on [0,2π][ 0,2 \pi ][0,2π] The set of all critical numbers of ƒ is

The particular solution of the differential equation dydx=x2+2x+1\frac { d y } { d x } = x ^ { 2 } + 2 x + 1dxdy​=x2+2x+1 satisfying the boundary condition when x = 3, y = -1 is

Let f(x)=x3−x2−x+8f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8f(x)=x3−x2−x+8 The set of all critical numbers of ƒ is

Suppose two positive integers satisfy the conditions that the sum of the first number and twice the second is 120 and their product is a maximum. Then the two numbers are ​

The antiderivative of f(x)=21−x2−sin⁡xf ( x ) = \frac { 2 } { \sqrt { 1 - x ^ { 2 } } } - \sin xf(x)=1−x2​2​−sinx is

If the surface area of a sphere is shrinking at the rate of 0.1 sq cm/h when the radius is 20πcm, \frac { 20 } { \pi } \mathrm { cm } \text {, }π20​cm, the rate of change of the radius in cm/s is

The shortest distance from the origin to the line 3x + y = 6 is

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

The shortest distance from the point (2,12)\left( 2 , \frac { 1 } { 2 } \right)(2,21​) to the parabola y=x2y = x ^ { 2 }y=x2 is

Let f(x)=x4−2x3+1f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 1f(x)=x4−2x3+1 on [−1,2][ - 1,2 ][−1,2] 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

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The set of all critical numbers of ƒ is

The particular solution of the differential equation $\frac { d y } { d x } = x ^ { 2 } + 2 x + 1$ satisfying the boundary condition when x = 3, y = -1 is

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ The set of all critical numbers of ƒ is

Suppose two positive integers satisfy the conditions that the sum of the first number and twice the second is 120 and their product is a maximum. Then the two numbers are

The antiderivative of $f ( x ) = \frac { 2 } { \sqrt { 1 - x ^ { 2 } } } - \sin x$ is

If the surface area of a sphere is shrinking at the rate of 0.1 sq cm/h when the radius is $\frac { 20 } { \pi } \mathrm { cm } \text {, }$ the rate of change of the radius in cm/s is

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

The shortest distance from the point $\left( 2 , \frac { 1 } { 2 } \right)$ to the parabola $y = x ^ { 2 }$ is

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