Question 1

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

Accepted Answer

A) 1.16 years 
B) 11.55 years 
C) 1.39 years 
D) 11.90 years 
E) 11.58 years 
A) 1.16 years 
B) 11.55 years 
C) 1.39 years 
D) 11.90 years 
E) 11.58 years B

Question 2

Suppose that the population (in millions) of a Egypt in 2007 is 80.3 and that expected continuous annual rate of change of the population is $0.017$. The exponential growth model for the population by letting $t = 0$ corresponds to 2000 is $P = 71.2910 e ^ { 0.017 t }$. Use the model to predict the population of the country in 2013 . Round your answer to two decimal places.

Accepted Answer

A)  $83.08$ million 
B)  $87.42$ million 
C)  $90.45$ million 
D)  $88.92$ million 
E)  $81.68$ million 
A)  $83.08$ million 
B)  $87.42$ million 
C)  $90.45$ million 
D)  $88.92$ million 
E)  $81.68$ million D

Question 3

$\text { Find the function } y = f ( t ) \text { passing through the point } ( 0,15 ) \text { with the first derivative }$ $\frac { d y } { d t } = \frac { 1 } { 4 } t .$

Accepted Answer

A) $y ( t ) = \frac { t ^ { 2 } } { 8 } + 15$ 
B)  $y ( t ) = 8 t ^ { 2 } + 15$ 
C)  $y ( t ) = 4 t + 15$ 
D)  $y ( t ) = \frac { t ^ { 2 } } { 4 } + 15$ 
E)  $y ( t ) = \frac { t } { 4 } + 15$ 
A) $y ( t ) = \frac { t ^ { 2 } } { 8 } + 15$ 
B)  $y ( t ) = 8 t ^ { 2 } + 15$ 
C)  $y ( t ) = 4 t + 15$ 
D)  $y ( t ) = \frac { t ^ { 2 } } { 4 } + 15$ 
E)  $y ( t ) = \frac { t } { 4 } + 15$ A

Question 4

Solve the first order linear differential equation. $$\frac { d y } { d x } + \frac { 8 } { x } y = 7 x + 2$$

Accepted Answer

A)  $y = \frac { 7 } { 10 } x ^ { 2 } - \frac { 2 } { 9 } x + C x ^ { - 8 }$ 
B)  $y = \frac { 10 } { 7 } x ^ { 2 } + \frac { 9 } { 2 } x + C x ^ { - 8 }$ 
C)  $y = - \frac { 7 } { 10 } x ^ { 2 } + \frac { 2 } { 9 } x + C x ^ { - 8 }$ 
D)  $y = - \frac { 10 } { 7 } x ^ { 2 } + \frac { 9 } { 2 } x + C x ^ { - 8 }$ 
E)  $y = \frac { 7 } { 10 } x ^ { 2 } + \frac { 2 } { 9 } x + C x ^ { - 8 }$ 
A)  $y = \frac { 7 } { 10 } x ^ { 2 } - \frac { 2 } { 9 } x + C x ^ { - 8 }$ 
B)  $y = \frac { 10 } { 7 } x ^ { 2 } + \frac { 9 } { 2 } x + C x ^ { - 8 }$ 
C)  $y = - \frac { 7 } { 10 } x ^ { 2 } + \frac { 2 } { 9 } x + C x ^ { - 8 }$ 
D)  $y = - \frac { 10 } { 7 } x ^ { 2 } + \frac { 9 } { 2 } x + C x ^ { - 8 }$ 
E)  $y = \frac { 7 } { 10 } x ^ { 2 } + \frac { 2 } { 9 } x + C x ^ { - 8 }$

Question 5

$\text { The isotope } { } ^ { 239 } \mathrm { Pu } \text { has a half-life of } 24,100 \text { years. After } 10,000 \text { years, a sample of }$ the isotope is reduced 1.6 grams. What was the initial size of the sample (in grams)? How large was the sample after the first 1,000 years? Round your answers to four decimal places.

Accepted Answer

A) 2.1332 , 2.0727 
B) 2.7731 , 2.6945 
C) 1.2799 , 1.2436 
D) 1.7065 , 1.6582 
E) 1.0666 , 1.0364 
A) 2.1332 , 2.0727 
B) 2.7731 , 2.6945 
C) 1.2799 , 1.2436 
D) 1.7065 , 1.6582 
E) 1.0666 , 1.0364

Question 6

A conservation organization releases 50 foxes into a preserve. After 5 years, there are 85 foxes in the preserve. The preserve has a carrying capacity of 225. Determine the population after 10 years. Discard any fractional part of your answer.

Accepted Answer

A) 126 
B) 118 
C) 139 
D) 131 
E) 205 
A) 126 
B) 118 
C) 139 
D) 131 
E) 205

Question 7

Use integration to find a general solution of the differential equation . $$\frac { d y } { d x } = x \sqrt { x - 15 }$$

Accepted Answer

A)  $y = 2 ( x - 15 ) ^ { 2 } ( 30 - x ) + C$ 
B)  $y = \frac { 2 } { 5 } ( x - 15 ) ^ { 3 } ( 5 + x ) + C$ 
C)  $y = \frac { 1 } { 5 } ( x - 15 ) ^ { 2 / 3 } ( 10 - x ) + C$ 
D)  $y = ( x - 15 ) ^ { 3 / 2 } ( 15 + x ) + C$ 
E)  $y = \frac { 2 } { 5 } ( x - 15 ) ^ { 3 / 2 } ( 10 + x ) + C$ 
A)  $y = 2 ( x - 15 ) ^ { 2 } ( 30 - x ) + C$ 
B)  $y = \frac { 2 } { 5 } ( x - 15 ) ^ { 3 } ( 5 + x ) + C$ 
C)  $y = \frac { 1 } { 5 } ( x - 15 ) ^ { 2 / 3 } ( 10 - x ) + C$ 
D)  $y = ( x - 15 ) ^ { 3 / 2 } ( 15 + x ) + C$ 
E)  $y = \frac { 2 } { 5 } ( x - 15 ) ^ { 3 / 2 } ( 10 + x ) + C$

Question 8

The initial investment in a savings account in which interest is compounded continuously is $\$ 813$. If the time required to double the amount is $9 \frac { 1 } { 4 }$ years, what is the annual rate? Round your answer to two decimal places.

Accepted Answer

A)  $7.49 \%$ 
B)  $7.70 \%$ 
C)  $13.34 \%$ 
D)  $6.29 \%$ 
E)  $8.89 \%$ 
A)  $7.49 \%$ 
B)  $7.70 \%$ 
C)  $13.34 \%$ 
D)  $6.29 \%$ 
E)  $8.89 \%$

Question 9

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. $\frac { d y } { d x } = \frac { - 2 x } { y }$

Accepted Answer

A)  $2 y = 4 x ^ { 2 } + C$ 
B)  $2 y = - 4 x + C$ 
C)  $y ^ { 2 } = - 2 x ^ { 2 } + C$ 
D)  $y ^ { 2 } = - 4 x + C$ 
E)  $2 y ^ { 2 } = 2 x ^ { 2 } + C$ 
A)  $2 y = 4 x ^ { 2 } + C$ 
B)  $2 y = - 4 x + C$ 
C)  $y ^ { 2 } = - 2 x ^ { 2 } + C$ 
D)  $y ^ { 2 } = - 4 x + C$ 
E)  $2 y ^ { 2 } = 2 x ^ { 2 } + C$

Question 10

Solve the differential equation. $$\frac { d y } { d x } = - y - 8$$

Accepted Answer

A)  $y = C e ^ { - x } + 8$ 
B)  $y = C e ^ { x } - 8$ 
C)  $y = C e ^ { x } + 8$ 
D)  $y = C e ^ { - x } - 8$ 
E)  $y = C e ^ { x } + 16$ 
A)  $y = C e ^ { - x } + 8$ 
B)  $y = C e ^ { x } - 8$ 
C)  $y = C e ^ { x } + 8$ 
D)  $y = C e ^ { - x } - 8$ 
E)  $y = C e ^ { x } + 16$

Question 11

$\text { A calf that weighs } 70 \text { pounds at birth gains weight at the rate } \frac { d w } { d t } = k ( 1200 - w ) \text {, }$ where $w$ is weight in pounds and $t$ is time in years. Use a computer algebra system to solve the differential equation for $k = 0.9$.

Accepted Answer

A)  $w = 70 e ^ { - 0.9 t }$ 
B)  $w = 1130 e ^ { - 0.9 t }$ 
C)  $w = 1200 - 1130 e ^ { - 0.9 t }$ 
D)  $w = 1130 - 0.9 e ^ { - t }$ 
E)  $w = 1200 e ^ { t } - 1130 e ^ { - 0.9 t }$ 
A)  $w = 70 e ^ { - 0.9 t }$ 
B)  $w = 1130 e ^ { - 0.9 t }$ 
C)  $w = 1200 - 1130 e ^ { - 0.9 t }$ 
D)  $w = 1130 - 0.9 e ^ { - t }$ 
E)  $w = 1200 e ^ { t } - 1130 e ^ { - 0.9 t }$

Question 12

$\text { A calf that weighs } 75 \text { pounds at birth gains weight at the rate } \frac { d w } { d t } = k ( 1300 - w ) \text {, }$ where $w$ is weight in pounds and $t$ is time in years. If the animal is sold when its weight reaches 750 pounds, find the time of sale using the model $w = 1300 - 1225 e ^ { - 0.8 t }$. Round your answer to two decimal places.

Accepted Answer

A)  $8.17$ years 
B)  $2.88$ years 
C)  $0.92$ year 
D)  1 year 
E)  $0.32$ year 
A)  $8.17$ years 
B)  $2.88$ years 
C)  $0.92$ year 
D)  1 year 
E)  $0.32$ year

Question 13

The isotope${ } ^ { 14 } \mathrm { C }$ has a half-life of $5.715$ years. After 2,000 years, a sample of the isotope is reduced to $1.2$ grams. What was the initial size of the sample (in grams)? How much will remain after 20,000 years (i.e., after another 18000 years)? Round your answers to four decimal places.

Accepted Answer

A)  $1.0706,0.0947$ 
B)  $2.4471,0.2164$ 
C)  $1.5294,0.1352$ 
D)  $2.1412,0.1893$ 
E)  $1.9883,0.1758$ 
A)  $1.0706,0.0947$ 
B)  $2.4471,0.2164$ 
C)  $1.5294,0.1352$ 
D)  $2.1412,0.1893$ 
E)  $1.9883,0.1758$

Question 14

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = 2 x ^ { 2 } + 5 x$

Accepted Answer

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

Question 15

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Accepted Answer

A)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,8 )$ 
B)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,12 )$ 
C)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,6 )$ 
D)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,15 )$ 
E)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 8 } \right) , ( 0,15 )$ 
A)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,8 )$ 
B)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,12 )$ 
C)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,6 )$ 
D)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 15 } \right) , ( 0,15 )$ 
E)  $\frac { d y } { d t } = y \left( y - \frac { 1 } { 8 } \right) , ( 0,15 )$

Question 16

Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where the air resistance is proportional to the velocity. Round numerical answers in your answer to two Places.
(i) Write the velocity as a function of time if the object's velocity after 6 seconds is 3.50 feet per
Second.
(ii) What is the limiting value of the velocity function?

Accepted Answer

A) 
(i) $v ( t ) = 3.5000 \left( 1 - e ^ { - 9.1429 t } \right)$;
(ii) 0 
B)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { 9.1429 t } \right)$;
(ii) 0 
C)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { - 9.1429 t } \right)$;
(ii) $3.5000$ 
D)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { 9.1429 t } \right)$;
(ii) $3.5000$ 
E)  (i) $v ( t ) = 3.5000 \left( 1 + e ^ { 9.1429 t } \right)$;
(ii) limit does not exist 
A) 
(i) $v ( t ) = 3.5000 \left( 1 - e ^ { - 9.1429 t } \right)$;
(ii) 0 
B)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { 9.1429 t } \right)$;
(ii) 0 
C)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { - 9.1429 t } \right)$;
(ii) $3.5000$ 
D)  (i) $v ( t ) = 3.5000 \left( 1 - e ^ { 9.1429 t } \right)$;
(ii) $3.5000$ 
E)  (i) $v ( t ) = 3.5000 \left( 1 + e ^ { 9.1429 t } \right)$;
(ii) limit does not exist

Question 17

$\text { The logistic function } P ( t ) = \frac { 15 } { 1 + 5 e ^ { - 4 t } } \text { models the growth of a population. }$ Determine when the population reaches of the maximum carrying capacity. Round your answer to three decimal places.

Accepted Answer

A) 4.317 
B) 3.000 
C) 0.474 
D) 0.677 
E) 0.301 
A) 4.317 
B) 3.000 
C) 0.474 
D) 0.677 
E) 0.301

Question 18

Match the logistic equation and initial condition with the graph of the solution. $\frac { d y } { d t } = 2 y \left( 1 - \frac { y } { 14 } \right) , y ( 0 ) = 18$

Accepted Answer

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

Question 19

$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$

Accepted Answer

A)  $y = 5 e ^ { - 10 x } + x \sin x$ 
B)  $y = e ^ { - 10 x }$ 
C)  $y = x ^ { 2 } \left( 10 + e ^ { x } \right)$ 
D)  $y = - 10 x \cos x$ 
E)  $y = 9 \ln x$ 
A)  $y = 5 e ^ { - 10 x } + x \sin x$ 
B)  $y = e ^ { - 10 x }$ 
C)  $y = x ^ { 2 } \left( 10 + e ^ { x } \right)$ 
D)  $y = - 10 x \cos x$ 
E)  $y = 9 \ln x$

Question 20

$\text { Sketch the slope field for the differential equation } y ^ {t } = y - 4 x \text { and use the slope field }$ to sketch the solution that passes through the point $( 2,2 )$.

Accepted Answer

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

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

$\text { Find the function } y = f ( t ) \text { passing through the point } ( 0,15 ) \text { with the first derivative }$ $\frac { d y } { d t } = \frac { 1 } { 4 } t .$

Solve the first order linear differential equation. $\frac { d y } { d x } + \frac { 8 } { x } y = 7 x + 2$

A conservation organization releases 50 foxes into a preserve. After 5 years, there are 85 foxes in the preserve. The preserve has a carrying capacity of 225. Determine the population after 10 years. Discard any fractional part of your answer.

Use integration to find a general solution of the differential equation . $\frac { d y } { d x } = x \sqrt { x - 15 }$

The initial investment in a savings account in which interest is compounded continuously is $\$ 813$ . If the time required to double the amount is $9 \frac { 1 } { 4 }$ years, what is the annual rate? Round your answer to two decimal places.

Solve the differential equation. $\frac { d y } { d x } = - y - 8$

$\text { A calf that weighs } 70 \text { pounds at birth gains weight at the rate } \frac { d w } { d t } = k ( 1200 - w ) \text {, }$ where $w$ is weight in pounds and $t$ is time in years. Use a computer algebra system to solve the differential equation for $k = 0.9$ .

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = 2 x ^ { 2 } + 5 x$

Match the logistic differential equation and initial condition with the graph of its solution shown below.

$\text { The logistic function } P ( t ) = \frac { 15 } { 1 + 5 e ^ { - 4 t } } \text { models the growth of a population. }$ Determine when the population reaches of the maximum carrying capacity. Round your answer to three decimal places.

Match the logistic equation and initial condition with the graph of the solution. $\frac { d y } { d t } = 2 y \left( 1 - \frac { y } { 14 } \right) , y ( 0 ) = 18$

$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$

$\text { Sketch the slope field for the differential equation } y ^ {t } = y - 4 x \text { and use the slope field }$ to sketch the solution that passes through the point $( 2,2 )$ .

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

Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6% compounded continuously. Round your answer to two decimal places.

Solve the first order linear differential equation. dydx+8xy=7x+2\frac { d y } { d x } + \frac { 8 } { x } y = 7 x + 2dxdy​+x8​y=7x+2

A conservation organization releases 50 foxes into a preserve. After 5 years, there are 85 foxes in the preserve. The preserve has a carrying capacity of 225. Determine the population after 10 years. Discard any fractional part of your answer.

Use integration to find a general solution of the differential equation . dydx=xx−15\frac { d y } { d x } = x \sqrt { x - 15 }dxdy​=xx−15​

The initial investment in a savings account in which interest is compounded continuously is $813\$ 813$813 . If the time required to double the amount is 9149 \frac { 1 } { 4 }941​ years, what is the annual rate? Round your answer to two decimal places.

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. dydx=−2xy\frac { d y } { d x } = \frac { - 2 x } { y }dxdy​=y−2x​

Solve the differential equation. dydx=−y−8\frac { d y } { d x } = - y - 8dxdy​=−y−8

Use integration to find a general solution of the differential equation. dydx=2x2+5x\frac { d y } { d x } = 2 x ^ { 2 } + 5 xdxdy​=2x2+5x

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Match the logistic equation and initial condition with the graph of the solution. dydt=2y(1−y14),y(0)=18\frac { d y } { d t } = 2 y \left( 1 - \frac { y } { 14 } \right) , y ( 0 ) = 18dtdy​=2y(1−14y​),y(0)=18

Which of the following is a solution of the differential equation y(4)−10,000y=0 ? \text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? } Which of the following is a solution of the differential equation y(4)−10,000y=0 ?

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

Solve the first order linear differential equation. $\frac { d y } { d x } + \frac { 8 } { x } y = 7 x + 2$

Use integration to find a general solution of the differential equation . $\frac { d y } { d x } = x \sqrt { x - 15 }$

The initial investment in a savings account in which interest is compounded continuously is $\$ 813$ . If the time required to double the amount is $9 \frac { 1 } { 4 }$ years, what is the annual rate? Round your answer to two decimal places.

Solve the differential equation. $\frac { d y } { d x } = - y - 8$

Use integration to find a general solution of the differential equation. $\frac { d y } { d x } = 2 x ^ { 2 } + 5 x$

Match the logistic equation and initial condition with the graph of the solution. $\frac { d y } { d t } = 2 y \left( 1 - \frac { y } { 14 } \right) , y ( 0 ) = 18$

$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$