Question 1

Write and solve the differential equation that models the following verbal statement: The rate of change of $Y$ with respect to $s$ is proportional to 
$50 - s$.

Accepted Answer

A)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { - 1 } , Y = - k \ln ( 50 - s ) ^ { 2 } + C$ 
B)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { - 1 } , Y = - k ( 50 - s ) + C$ 
C)  $\frac { d Y } { d s } = k ( 50 - s ) , Y = - \frac { k } { 2 } ( 50 - s ) ^ { 2 } + C$ 
D)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { 3 } , Y = - \frac { k } { 4 } ( 50 - s ) ^ { 4 } + C$ 
E)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { 2 } , Y = - \frac { k } { 3 } ( 50 - s ) ^ { 3 } + C$ 
A)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { - 1 } , Y = - k \ln ( 50 - s ) ^ { 2 } + C$ 
B)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { - 1 } , Y = - k ( 50 - s ) + C$ 
C)  $\frac { d Y } { d s } = k ( 50 - s ) , Y = - \frac { k } { 2 } ( 50 - s ) ^ { 2 } + C$ 
D)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { 3 } , Y = - \frac { k } { 4 } ( 50 - s ) ^ { 4 } + C$ 
E)  $\frac { d Y } { d s } = k ( 50 - s ) ^ { 2 } , Y = - \frac { k } { 3 } ( 50 - s ) ^ { 3 } + C$

Question 2

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = x \sqrt { x + 2 }$

Accepted Answer

A)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } - \frac { 4 } { 3 } ( x + 2 ) ^ { 5 / 2 } + C$ 
B)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } - ( x + 2 ) ^ { 3 / 2 } + C$ 
C)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 5 / 2 } - \frac { 4 } { 3 } ( x + 2 ) ^ { 3 / 2 } + C$ 
D)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } + \frac { 4 } { 3 } ( x + 2 ) ^ { 5 / 2 } + C$ 
E) $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 2 } - \frac { 4 } { 3 } ( x + 2 ) + C$ 
A)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } - \frac { 4 } { 3 } ( x + 2 ) ^ { 5 / 2 } + C$ 
B)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } - ( x + 2 ) ^ { 3 / 2 } + C$ 
C)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 5 / 2 } - \frac { 4 } { 3 } ( x + 2 ) ^ { 3 / 2 } + C$ 
D)  $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 3 / 2 } + \frac { 4 } { 3 } ( x + 2 ) ^ { 5 / 2 } + C$ 
E) $y = \frac { 2 } { 5 } ( x + 2 ) ^ { 2 } - \frac { 4 } { 3 } ( x + 2 ) + C$

Question 3

Use integration to find a general solution of the differential equation. $$\frac { d y } { d x } = x ^ { 10 } e ^ { x ^ { 11 } }$$

Accepted Answer

A)  $y = \frac { 1 } { 10 } e ^ { x ^ { 11 } } + C$ 
B)  $y = \frac { 1 } { 11 } e ^ { x ^ { 11 } } + C$ 
C)  $y = \frac { x ^ { 11 } } { 11 } e ^ { x ^ { 11 } } + C$ 
D)  $y = 11 e ^ { x ^ { 11 } } + C$ 
E)  $y = 10 e ^ { x ^ { 11 } } + C$ 
A)  $y = \frac { 1 } { 10 } e ^ { x ^ { 11 } } + C$ 
B)  $y = \frac { 1 } { 11 } e ^ { x ^ { 11 } } + C$ 
C)  $y = \frac { x ^ { 11 } } { 11 } e ^ { x ^ { 11 } } + C$ 
D)  $y = 11 e ^ { x ^ { 11 } } + C$ 
E)  $y = 10 e ^ { x ^ { 11 } } + C$

Question 4

A 100-gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time $t = 0$, distilled water is added to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as $t \rightarrow \infty$.

Accepted Answer

A)  10 
B)  26 
C)  25 
D)  0 
E)  1 
A)  10 
B)  26 
C)  25 
D)  0 
E)  1

Question 5

Select from the choices below the slope field for the differential equation. $$\frac { d y } { d x } = e ^ { - 2 x }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E)  none of the above 
A) 
  
B) 
  
C) 
  
D) 
  
E)  none of the above

Question 6

A container of hot liquid is placed in a freezer that is kept at a constant temperature of $20 ^ { \circ } \mathrm { F }$. The initial temperature of the liquid is $170 ^ { \circ } \mathrm { F }$. After 3 minutes, the liquid's temperature is $62 ^ { \circ } \mathrm { F }$. How much longer will it take for its temperature to decrease to $31 ^ { \circ } \mathrm { F }$ ? Round your answer to two decimal places.

Accepted Answer

A)  $1.89$ minutes 
B)  $2.84$ minutes 
C)  $3.16$ minutes 
D)  $1.26$ minutes 
E)  $3.47$ minutes 
A)  $1.89$ minutes 
B)  $2.84$ minutes 
C)  $3.16$ minutes 
D)  $1.26$ minutes 
E)  $3.47$ minutes

Question 7

$\text { Find the orthogonal trajectories of the family } 3 x ^ { 2 } + 6 y ^ { 2 } = C \text {. }$

Accepted Answer

A)  $6 y ^ { 2 } - 3 x ^ { 2 } = k$ 
B)  $| y | ^ { 3 } = k | x | ^ { 6 }$ 
C)  $| y | ^ { 6 } = k | x | ^ { - 6 }$ 
D)  $6 y ^ { 2 } + 3 x ^ { 2 } = k$ 
E)  $| y | ^ { 3 } = k | x | ^ { - 6 }$ 
A)  $6 y ^ { 2 } - 3 x ^ { 2 } = k$ 
B)  $| y | ^ { 3 } = k | x | ^ { 6 }$ 
C)  $| y | ^ { 6 } = k | x | ^ { - 6 }$ 
D)  $6 y ^ { 2 } + 3 x ^ { 2 } = k$ 
E)  $| y | ^ { 3 } = k | x | ^ { - 6 }$

Question 8

Select from the choices below the slope field for the differential equation. $$\frac { d y } { d x } = \cos ( 5 x )$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E)  none of the above 
A) 
  
B) 
  
C) 
  
D) 
  
E)  none of the above

Question 9

$\text { Solve the first-order linear differential equation } \frac { d y } { d x } = ( y - 1 ) \sin ( 2 x ) \text {. }$

Accepted Answer

A)  $y = 1 + C e ^ { \sin ( 2 x ) }$ 
B)  $y = 1 + C e ^ -{ \frac { \cos ( 2 x ) } { 2 } }$ 
C)  $y = 1 + C e ^ { \frac { \sin ( 2 x ) } { 2 } }$ 
D)  $y = 1 + C e ^- { \frac { \sin ( 2 x ) } { 2 } }$ 
E)  $y = 1 + C e ^ { \cos ( 2 x ) }$ 
A)  $y = 1 + C e ^ { \sin ( 2 x ) }$ 
B)  $y = 1 + C e ^ -{ \frac { \cos ( 2 x ) } { 2 } }$ 
C)  $y = 1 + C e ^ { \frac { \sin ( 2 x ) } { 2 } }$ 
D)  $y = 1 + C e ^- { \frac { \sin ( 2 x ) } { 2 } }$ 
E)  $y = 1 + C e ^ { \cos ( 2 x ) }$

Question 10

Solve the first order linear differential equation. $y ^ { \prime } - 2 x y = e ^ { x ^ { 2 } }$

Accepted Answer

A)  $y = e ^ { x ^ { 2 } } ( 2 x + C )$ 
B)  $y = e ^ { x ^ { 2 } } ( x + C )$ 
C)  $y = e ^ { x ^ { 2 } } ( 3 x + C )$ 
D)  $y = e ^ { x ^ { - 2 } } ( x + C )$ 
E)  $y = e ^ { x ^ { - 2 } } \left( x + C ^ { x } \right)$ 
A)  $y = e ^ { x ^ { 2 } } ( 2 x + C )$ 
B)  $y = e ^ { x ^ { 2 } } ( x + C )$ 
C)  $y = e ^ { x ^ { 2 } } ( 3 x + C )$ 
D)  $y = e ^ { x ^ { - 2 } } ( x + C )$ 
E)  $y = e ^ { x ^ { - 2 } } \left( x + C ^ { x } \right)$

Question 11

$\text { The logistic function } P ( t ) = \frac { 20 } { 1 + 0.5 e ^ { - 0.2 t } } \text { models the growth of a population. }$ Identify the maximum carrying capacity.

Accepted Answer

A) 20 
B) 1.7 
C) 0.5 
D) 0.2 
E) 2.2 
A) 20 
B) 1.7 
C) 0.5 
D) 0.2 
E) 2.2

Question 12

$\text { The logistic function } P ( t ) = \frac { 10 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. }$ Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.

Accepted Answer

A) 0.549 
B) 3.333 
C) 1.151 
D) 5.000 
E) 1.000 
A) 0.549 
B) 3.333 
C) 1.151 
D) 5.000 
E) 1.000

Question 13

$\text { Identify the graph of the logistic function } y = \frac { 20 } { 1 + e ^ { - t } } \text {. }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 14

Find the principal that must be invested at the rate 8%, compounded monthly, so that $1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.

Accepted Answer

A) $250,000.00 
B) $18,560.39 
C) $717,324.37 
D) $333,333.33 
E) $21,321.23 
A) $250,000.00 
B) $18,560.39 
C) $717,324.37 
D) $333,333.33 
E) $21,321.23

Question 15

Solve the differential equation. $$y ^ { t } = x ( 1 + y )$$

Accepted Answer

A)  $\ln | 1 + y | = x ^ { 2 } + C$ 
B)  $\3 \ln | 1 + y | = x ^ { 3 } + C$ 
C)  $4 \ln | 1 + y | = x ^ { 4 } + C$ 
D)  $\ln | 1 + y | = x + C$ 
E)  $2 \ln | 1 + y | = x ^ { 2 } + C$ 
A)  $\ln | 1 + y | = x ^ { 2 } + C$ 
B)  $\3 \ln | 1 + y | = x ^ { 3 } + C$ 
C)  $4 \ln | 1 + y | = x ^ { 4 } + C$ 
D)  $\ln | 1 + y | = x + C$ 
E)  $2 \ln | 1 + y | = x ^ { 2 } + C$

Question 16

$\text { Use the differential equation } \frac { d y } { d x } = \frac { 2 x } { y } \text { and its slope field to find the slope at the }$ $\text { point } ( 4,8 )$

Accepted Answer

A) 32 
B) 1 
C) 8 
D) 64 
E) 16 
A) 32 
B) 1 
C) 8 
D) 64 
E) 16

Question 17

A conservation organization releases coyotes into a preserve. After years, 40 4 There are70 coyotes in the preserve. The preserve has a carrying capacity of 175. Write a logistic Function that models the population of coyotes in the preserve.

Accepted Answer

A) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.202733 t } }$$ 
B) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.702733 t } }$$ 
C) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.402733 t } }$$ 
D) 
$$y = \frac { 175 } { 1 + 4.875 e ^ { - 0.352733 t } }$$ 
E) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 1.002733 t } }$$ 
A) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.202733 t } }$$ 
B) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.702733 t } }$$ 
C) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 0.402733 t } }$$ 
D) 
$$y = \frac { 175 } { 1 + 4.875 e ^ { - 0.352733 t } }$$ 
E) 
$$y = \frac { 175 } { 1 + 3.375 e ^ { - 1.002733 t } }$$

Question 18

$\text { Each of the following graphs is from a logistic function } y = \frac { 20 } { 1 + b e ^ { - t } } \text {. Which one has }$ the largest value of b?

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 19

$\text { Each of the following graphs is from a logistic function } y = \frac { 14 } { 1 + b e ^ { - t } } \text {. Which one has }$ the smallest value of b?

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 20

$\text { Solve the Bernoulli differential equation } x y ^ { t} + y = x y ^ { 12 } \text {. }$

Accepted Answer

A)  $y ^ { - 11 } = 10 x + C x ^ { 11 }$ 
B)  $y ^ { - 11 } = 11 x - C x ^ { 11 }$ 
C)  $y ^ { - 11 } = 9 x + C x ^ { 11 }$ 
D)  $y ^ { - 11 } = \frac { 10 } { 11 } x + C x ^ { 10 }$ 
E)  $y ^ { - 11 } = \frac { 11 } { 10 } x + C x ^ { 11 }$ 
A)  $y ^ { - 11 } = 10 x + C x ^ { 11 }$ 
B)  $y ^ { - 11 } = 11 x - C x ^ { 11 }$ 
C)  $y ^ { - 11 } = 9 x + C x ^ { 11 }$ 
D)  $y ^ { - 11 } = \frac { 10 } { 11 } x + C x ^ { 10 }$ 
E)  $y ^ { - 11 } = \frac { 11 } { 10 } x + C x ^ { 11 }$

Write and solve the differential equation that models the following verbal statement: The rate of change of $Y$ with respect to $s$ is proportional to $50 - s$ .

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = x \sqrt { x + 2 }$

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = x ^ { 10 } e ^ { x ^ { 11 } }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = e ^ { - 2 x }$

$\text { Find the orthogonal trajectories of the family } 3 x ^ { 2 } + 6 y ^ { 2 } = C \text {. }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = \cos ( 5 x )$

$\text { Solve the first-order linear differential equation } \frac { d y } { d x } = ( y - 1 ) \sin ( 2 x ) \text {. }$

Solve the first order linear differential equation. $y ^ { \prime } - 2 x y = e ^ { x ^ { 2 } }$

$\text { The logistic function } P ( t ) = \frac { 20 } { 1 + 0.5 e ^ { - 0.2 t } } \text { models the growth of a population. }$ Identify the maximum carrying capacity.

$\text { The logistic function } P ( t ) = \frac { 10 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. }$ Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.

$\text { Identify the graph of the logistic function } y = \frac { 20 } { 1 + e ^ { - t } } \text {. }$

Find the principal that must be invested at the rate 8%, compounded monthly, so that $1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.

Solve the differential equation. $y ^ { t } = x ( 1 + y )$

A conservation organization releases coyotes into a preserve. After years, 40 4 There are70 coyotes in the preserve. The preserve has a carrying capacity of 175. Write a logistic Function that models the population of coyotes in the preserve.

$\text { Each of the following graphs is from a logistic function } y = \frac { 20 } { 1 + b e ^ { - t } } \text {. Which one has }$ the largest value of b?

$\text { Each of the following graphs is from a logistic function } y = \frac { 14 } { 1 + b e ^ { - t } } \text {. Which one has }$ the smallest value of b?

$\text { Solve the Bernoulli differential equation } x y ^ { t} + y = x y ^ { 12 } \text {. }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 6: Slope Fields and Eulers Method

Write and solve the differential equation that models the following verbal statement: The rate of change of YYY with respect to sss is proportional to 50−s50 - s50−s .

Use integration to find a general solution of the differential equation. dydx=xx+2\frac { d y } { d x } = x \sqrt { x + 2 }dxdy​=xx+2​

Use integration to find a general solution of the differential equation. dydx=x10ex11\frac { d y } { d x } = x ^ { 10 } e ^ { x ^ { 11 } }dxdy​=x10ex11

Select from the choices below the slope field for the differential equation. dydx=e−2x\frac { d y } { d x } = e ^ { - 2 x }dxdy​=e−2x

Find the orthogonal trajectories of the family 3x2+6y2=C. \text { Find the orthogonal trajectories of the family } 3 x ^ { 2 } + 6 y ^ { 2 } = C \text {. } Find the orthogonal trajectories of the family 3x2+6y2=C.

Select from the choices below the slope field for the differential equation. dydx=cos⁡(5x)\frac { d y } { d x } = \cos ( 5 x )dxdy​=cos(5x)

Solve the first-order linear differential equation dydx=(y−1)sin⁡(2x). \text { Solve the first-order linear differential equation } \frac { d y } { d x } = ( y - 1 ) \sin ( 2 x ) \text {. } Solve the first-order linear differential equation dxdy​=(y−1)sin(2x).

Solve the first order linear differential equation. y′−2xy=ex2y ^ { \prime } - 2 x y = e ^ { x ^ { 2 } }y′−2xy=ex2

Identify the graph of the logistic function y=201+e−t. \text { Identify the graph of the logistic function } y = \frac { 20 } { 1 + e ^ { - t } } \text {. } Identify the graph of the logistic function y=1+e−t20​.

Find the principal that must be invested at the rate 8%, compounded monthly, so that $1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.

Solve the differential equation. yt=x(1+y)y ^ { t } = x ( 1 + y )yt=x(1+y)

A conservation organization releases coyotes into a preserve. After years, 40 4 There are70 coyotes in the preserve. The preserve has a carrying capacity of 175. Write a logistic Function that models the population of coyotes in the preserve.

Solve the Bernoulli differential equation xyt+y=xy12. \text { Solve the Bernoulli differential equation } x y ^ { t} + y = x y ^ { 12 } \text {. } Solve the Bernoulli differential equation xyt+y=xy12.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Write and solve the differential equation that models the following verbal statement: The rate of change of $Y$ with respect to $s$ is proportional to $50 - s$ .

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = x \sqrt { x + 2 }$

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = x ^ { 10 } e ^ { x ^ { 11 } }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = e ^ { - 2 x }$

$\text { Find the orthogonal trajectories of the family } 3 x ^ { 2 } + 6 y ^ { 2 } = C \text {. }$

Select from the choices below the slope field for the differential equation. $\frac { d y } { d x } = \cos ( 5 x )$

$\text { Solve the first-order linear differential equation } \frac { d y } { d x } = ( y - 1 ) \sin ( 2 x ) \text {. }$

Solve the first order linear differential equation. $y ^ { \prime } - 2 x y = e ^ { x ^ { 2 } }$

$\text { Identify the graph of the logistic function } y = \frac { 20 } { 1 + e ^ { - t } } \text {. }$

Solve the differential equation. $y ^ { t } = x ( 1 + y )$

$\text { Solve the Bernoulli differential equation } x y ^ { t} + y = x y ^ { 12 } \text {. }$