Question 1

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ with P(0) = 10. Find the population after 500 years.

Accepted Answer

A) 50 
B) 94 
C) 70 
D) 500 
E) 80
F)100
G)35
H)30 
A) 50 
B) 94 
C) 70 
D) 500 
E) 80
F)100
G)35
H)30

Question 2

A thermometer is taken outside from a room where the temperature is 72 $^\circ$ F. Outdoors, the temperature is 48 $^\circ$ F. After one minute, the thermometer reads 55 $^\circ$ F. After how many minutes does the thermometer read 50 $^\circ$ F?

Accepted Answer

A) 2.107 
B) 1.107 
C) 3.100 
D) 1.503 
E) 2.017
F)1.017
G)3.010
H)1.013 
A) 2.107 
B) 1.107 
C) 3.100 
D) 1.503 
E) 2.017
F)1.017
G)3.010
H)1.013

Question 3

A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after $t$ hours is:

Accepted Answer

A)  $200 \left( 9 ^ { - t } \right)$ 
B)  $200 \left( 9 ^ { t } \right)$ 
C)  $200 \left( 3 ^ { - t } \right)$ 
D)  $200 \left( 3 ^ { t } \right)$ 
E)  $200 \left( e ^ { - t } \right)$ 
F) $200 \left( e ^ { t } \right)$ 
G) $200 \left( e ^ { - 3 t } \right)$ 
H) $200 \left( e ^ { 3 t } \right)$ 
A)  $200 \left( 9 ^ { - t } \right)$ 
B)  $200 \left( 9 ^ { t } \right)$ 
C)  $200 \left( 3 ^ { - t } \right)$ 
D)  $200 \left( 3 ^ { t } \right)$ 
E)  $200 \left( e ^ { - t } \right)$ 
F) $200 \left( e ^ { t } \right)$ 
G) $200 \left( e ^ { - 3 t } \right)$ 
H) $200 \left( e ^ { 3 t } \right)$

Question 4

Find the solution of the initial-value problem $y ^ { \prime } = \frac { \ln x } { x y }$ , $y ( 1 ) = 2$ .

Accepted Answer

A)  $y = \frac { 1 + x } { 1 + \ln x }$ 
B)  $y = \frac { 8 x } { ( 1 + x ) ^ { 2 } }$ 
C)  $y = 2 + 2 \ln x$ 
D)  $y = \sqrt { 4 + ( \ln x ) ^ { 2 } }$ 
E)  $y = x \ln x + 2 x$ 
F) $y = x \left( 1 + x ^ { 2 } \right)$ 
G) $y = x + \sqrt { 1 + \ln x }$ 
H) $y = \sqrt { x } ( 1 + x )$ 
A)  $y = \frac { 1 + x } { 1 + \ln x }$ 
B)  $y = \frac { 8 x } { ( 1 + x ) ^ { 2 } }$ 
C)  $y = 2 + 2 \ln x$ 
D)  $y = \sqrt { 4 + ( \ln x ) ^ { 2 } }$ 
E)  $y = x \ln x + 2 x$ 
F) $y = x \left( 1 + x ^ { 2 } \right)$ 
G) $y = x + \sqrt { 1 + \ln x }$ 
H) $y = \sqrt { x } ( 1 + x )$

Question 5

Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 100 fruit flies are present initially and 200 are present after 5 days, how many will be present after 10 days?

Accepted Answer

The answer of Assume that the rate of growth of...

Question 6

Which of the following is a solution of the differential equation $\frac { d y } { d x } - 4 = 0$

Accepted Answer

A)  $y = e ^ { - 4 x }$ 
B)  $y = 4 x$ 
C)  $y = e ^ { 2 x ^ { 2 } }$ 
D)  $y = \sin 2 x$ 
E)  $y = e ^ { 2 x }$ 
F) $y = 2 x ^ { 2 }$ 
G) $y = \frac { 1 } { 4 x + 1 }$ 
H) $y = e ^ { 4 x }$ 
A)  $y = e ^ { - 4 x }$ 
B)  $y = 4 x$ 
C)  $y = e ^ { 2 x ^ { 2 } }$ 
D)  $y = \sin 2 x$ 
E)  $y = e ^ { 2 x }$ 
F) $y = 2 x ^ { 2 }$ 
G) $y = \frac { 1 } { 4 x + 1 }$ 
H) $y = e ^ { 4 x }$

Question 7

Suppose that a certain population grows according to an exponential model.(a) Write the differential equation for this situation with a relative growth rate of k = 0.01. Produce a solution for the initial condition t = 0 (in hours) and population P = 1 (in thousands).(b) Find the population when t = 10 hours, t = 100 hours, and t = 1000 hours.(c) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(d) As the time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& 1.105 thousand, @#IMG-DLM& 2.718 thousan

Question 8

Suppose $\frac { d y } { d x } = x + y$ , $y ( - 1 ) = - 2$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 0 )$ .

Accepted Answer

A) 3.5 
B) 5.5 
C)  $- 3.5$ 
D)  $- 5.5$ 
E) 2.5
F)6

G) $- 2.5$ 
H) $- 6$ 
A) 3.5 
B) 5.5 
C)  $- 3.5$ 
D)  $- 5.5$ 
E) 2.5
F)6

G) $- 2.5$ 
H) $- 6$

Question 9

The following data approximate the results obtained by subjecting Hela-S cells to 250 kvp x-rays: $$\begin{array} { | l | c | c | c | c | c | c | c | c | c | c | c | } 
\hline \text { Dose (rads) } & 0 & 100 & 150 & 200 & 250 & 300 & 400 & 450 & 500 & 550 & 600 \
\hline \text { Fraction Surviving } & 1 & 0.5 & 0.4 & 0.3 & 0.17 & 0.13 & 0.03 & 0.02 & 0.01 & 0.006 & 0.005 \
\hline
\end{array}$$ Assume that these data fit an exponential model.(a) Find the appropriate exponential model.(b) Add another line to the table using your population model for the given doses of radiation.(c) Compare the model entries to the given data and explain any discrepancy.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) Answer

Question 10

Consider the differential equation $\frac { d y } { d t } = y ^ { 3 } - 2 y ^ { 2 } - 15 y$ .(a) What are the equilibrium solutions?
(b) For what values of $y$ is $y ( t )$ increasing?
(c) For what values of $y$ is $y ( t )$ decreasing?
(d) Use the information from above to sketch the direction field for the given differential equation.

Accepted Answer

(a) @#IMG-DLM& , @#IMG-DLM& ,

Question 11

Suppose a population growth is modeled by the logistic differential equation with the carrying capacity 2000 and the relative growth rate k = 0.06 per year. If the initial population is P(0) = 500, and P(10).

Accepted Answer

A) 309 
B) 756 
C) 310 
D) 757 
E) 308
F)755
G)307
H)800 
A) 309 
B) 756 
C) 310 
D) 757 
E) 308
F)755
G)307
H)800

Question 12

Which of the following is a solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } - 6 \frac { d y } { d x } + 9 y = 0$

Accepted Answer

A)  $y = e ^ { - 3 x }$ 
B)  $y = e ^ { 2 x }$ 
C)  $y = x e ^ { - 3 x }$ 
D)  $y = x e ^ { 3 x }$ 
E)  $y = e ^ { - 2 x }$ 
F) $y = e ^ { - 3 x } + x$ 
G) $y = x e ^ { 3 x } + x$ 
H)None of these 
A)  $y = e ^ { - 3 x }$ 
B)  $y = e ^ { 2 x }$ 
C)  $y = x e ^ { - 3 x }$ 
D)  $y = x e ^ { 3 x }$ 
E)  $y = e ^ { - 2 x }$ 
F) $y = e ^ { - 3 x } + x$ 
G) $y = x e ^ { 3 x } + x$ 
H)None of these

Question 13

Find the orthogonal trajectories of the family of curves $x ^ { 2 } + y ^ { 2 } = k$ . Then draw several members of each family on the same coordinate plane.

Accepted Answer

The answer of Find the orthogonal trajectories of the family...

Question 14

A tank contains 10 gallons of water and 4 pounds of a chemical, $Z$ , per gallon. To decrease the concentration of $Z$ , pure water is added to the container at a rate of 2 gallons per minute, and the well stirred mixture is drawn from the tank at the same rate.(a) Find the amount of chemical $Z$ in the tank at time $t$ .(b) How much chemical $Z$ is present after 10 minutes?
(c) How long does it take for the concentration of chemical $Z$ to be reduced to 0.1 pound per gallon?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 15

A bacteria culture starts with 200 bacteria and in 1 hour contains 400 bacteria. How many hours does it take to reach 2000 bacteria?

Accepted Answer

A)  $\ln 400$ 
B)  $\ln 10$ 
C)  $10$ 
D)  $\ln 1600$ 
E)  $\ln 2000$ 
F) $\ln 200$ 
G) $5$ 
H) $\frac { \ln 10 } { \ln 2 }$ 
A)  $\ln 400$ 
B)  $\ln 10$ 
C)  $10$ 
D)  $\ln 1600$ 
E)  $\ln 2000$ 
F) $\ln 200$ 
G) $5$ 
H) $\frac { \ln 10 } { \ln 2 }$

Question 16

A ball is thrown directly upward from the ground with an initial speed $24$ ft/s.(a) Determine the differential equation that the position function $y ( t )$ satisfies.(b) What are the initial conditions?
(c) How high does the ball rise?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 17

The graph of a direction field for the differential equation $\frac { d y } { d x } = 1 - y$ is given below:   (a) Sketch a solution curve that satisfies the given condition, but without solving the differential equation:
(i) $y ( 0 ) = 1$ (ii) $y ( 2 ) = 1$ (iii) $y ( 0 ) = 0$ 
(b) Solve the differential equation for each of the conditions in part (a). Compare your answers to the curves you produced in part (a).
(c) What is the relationship between the curves (i) and (ii) in part (a)? Explain why this occurs.

Accepted Answer

(a) @#IMG-DLM& (i) y(0) = 1 (i

Question 18

Make a rough sketch of a directional field for the autonomous differential equation $y ^ { \prime } = f ( y )$ where $f$ is the function graphed below.

Accepted Answer

The answer of Make a rough sketch of a directional...

Question 19

In 1970, the Brown County groundhog population was 100. By 1980, there were 900 groundhogs in Brown County. If the rate of population growth of these animals is proportional to the population size, how many groundhogs might one expect to see in 1995?

Accepted Answer

The answer of In 1970, the Brown County groundhog population...

Question 20

(a) Solve the differential equation $\frac { d P } { d t } = \frac { b k P e ^ { - k t } } { 1 - b e ^ { - k t } }$ , with b = 2 and k = 0.1, and $P _ { o }$ = 1.(b) Sketch a graph of the solution you produced for part (a) and discuss the major characteristics of this monomolecular growth curve.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& P (t) is a

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ with P(0) = 10. Find the population after 500 years.

A thermometer is taken outside from a room where the temperature is 72 $^\circ$ F. Outdoors, the temperature is 48 $^\circ$ F. After one minute, the thermometer reads 55 $^\circ$ F. After how many minutes does the thermometer read 50 $^\circ$ F?

A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after $t$ hours is:

Find the solution of the initial-value problem $y ^ { \prime } = \frac { \ln x } { x y }$ , $y ( 1 ) = 2$ .

Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 100 fruit flies are present initially and 200 are present after 5 days, how many will be present after 10 days?

Which of the following is a solution of the differential equation $\frac { d y } { d x } - 4 = 0$

Suppose $\frac { d y } { d x } = x + y$ , $y ( - 1 ) = - 2$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 0 )$ .

Suppose a population growth is modeled by the logistic differential equation with the carrying capacity 2000 and the relative growth rate k = 0.06 per year. If the initial population is P(0) = 500, and P(10).

Which of the following is a solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } - 6 \frac { d y } { d x } + 9 y = 0$

Find the orthogonal trajectories of the family of curves $x ^ { 2 } + y ^ { 2 } = k$ . Then draw several members of each family on the same coordinate plane.

A bacteria culture starts with 200 bacteria and in 1 hour contains 400 bacteria. How many hours does it take to reach 2000 bacteria?

A ball is thrown directly upward from the ground with an initial speed $24$ ft/s.(a) Determine the differential equation that the position function $y ( t )$ satisfies.(b) What are the initial conditions? (c) How high does the ball rise?

In 1970, the Brown County groundhog population was 100. By 1980, there were 900 groundhogs in Brown County. If the rate of population growth of these animals is proportional to the population size, how many groundhogs might one expect to see in 1995?

(a) Solve the differential equation $\frac { d P } { d t } = \frac { b k P e ^ { - k t } } { 1 - b e ^ { - k t } }$ , with b = 2 and k = 0.1, and $P _ { o }$ = 1.(b) Sketch a graph of the solution you produced for part (a) and discuss the major characteristics of this monomolecular growth curve.

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Exam 7: Differential Equations

Suppose a population growth is modeled by the logistic equation dPdt=0.01P−0.0001P2\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }dtdP​=0.01P−0.0001P2 with P(0) = 10. Find the population after 500 years.

A thermometer is taken outside from a room where the temperature is 72 ∘^\circ∘ F. Outdoors, the temperature is 48 ∘^\circ∘ F. After one minute, the thermometer reads 55 ∘^\circ∘ F. After how many minutes does the thermometer read 50 ∘^\circ∘ F?

A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after ttt hours is:

Find the solution of the initial-value problem y′=ln⁡xxyy ^ { \prime } = \frac { \ln x } { x y }y′=xylnx​ , y(1)=2y ( 1 ) = 2y(1)=2 .

Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 100 fruit flies are present initially and 200 are present after 5 days, how many will be present after 10 days?

Which of the following is a solution of the differential equation dydx−4=0\frac { d y } { d x } - 4 = 0dxdy​−4=0

Suppose dydx=x+y\frac { d y } { d x } = x + ydxdy​=x+y , y(−1)=−2y ( - 1 ) = - 2y(−1)=−2 . Use Euler's method with step size h=0.5h = 0.5h=0.5 to approximate y(0)y ( 0 )y(0) .

Suppose a population growth is modeled by the logistic differential equation with the carrying capacity 2000 and the relative growth rate k = 0.06 per year. If the initial population is P(0) = 500, and P(10).

Which of the following is a solution of the differential equation d2ydx2−6dydx+9y=0\frac { d ^ { 2 } y } { d x ^ { 2 } } - 6 \frac { d y } { d x } + 9 y = 0dx2d2y​−6dxdy​+9y=0

Find the orthogonal trajectories of the family of curves x2+y2=kx ^ { 2 } + y ^ { 2 } = kx2+y2=k . Then draw several members of each family on the same coordinate plane.

A bacteria culture starts with 200 bacteria and in 1 hour contains 400 bacteria. How many hours does it take to reach 2000 bacteria?

A ball is thrown directly upward from the ground with an initial speed 242424 ft/s.(a) Determine the differential equation that the position function y(t)y ( t )y(t) satisfies.(b) What are the initial conditions? (c) How high does the ball rise?

Make a rough sketch of a directional field for the autonomous differential equation y′=f(y)y ^ { \prime } = f ( y )y′=f(y) where fff is the function graphed below.

In 1970, the Brown County groundhog population was 100. By 1980, there were 900 groundhogs in Brown County. If the rate of population growth of these animals is proportional to the population size, how many groundhogs might one expect to see in 1995?

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0001 P ^ { 2 }$ with P(0) = 10. Find the population after 500 years.

A thermometer is taken outside from a room where the temperature is 72 $^\circ$ F. Outdoors, the temperature is 48 $^\circ$ F. After one minute, the thermometer reads 55 $^\circ$ F. After how many minutes does the thermometer read 50 $^\circ$ F?

A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after $t$ hours is:

Find the solution of the initial-value problem $y ^ { \prime } = \frac { \ln x } { x y }$ , $y ( 1 ) = 2$ .

Which of the following is a solution of the differential equation $\frac { d y } { d x } - 4 = 0$

Suppose $\frac { d y } { d x } = x + y$ , $y ( - 1 ) = - 2$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 0 )$ .

Which of the following is a solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } - 6 \frac { d y } { d x } + 9 y = 0$

Find the orthogonal trajectories of the family of curves $x ^ { 2 } + y ^ { 2 } = k$ . Then draw several members of each family on the same coordinate plane.

A ball is thrown directly upward from the ground with an initial speed $24$ ft/s.(a) Determine the differential equation that the position function $y ( t )$ satisfies.(b) What are the initial conditions? (c) How high does the ball rise?