Assignments App of Int



AP Calculus Assignments: Differential Equations

|Day |Topic |Assignment |

|1 |Exponential growth |HW Differential Equations – 1 |

|2 |Separable Differential Equations |HW Differential Equations – 2 |

|3 |Practice **QUIZ** |HW Differential Equations – 3 |

|4 |Slope Fields |HW Differential Equations – 4 |

|5 |Domains of DEs |HW Differential Equations – 5 |

|6 |Practice **QUIZ** |HW Differential Equations – 6 |

|7 |Review |HW Differential Equations – Review |

|8 |***TEST*** | |

AP Calculus HW: Differential Equations - 1

1. A population of bacteria in your toilet bowl is growing at a rate proportional to the number present. Suppose there are initially two million bacteria in your toilet and 24 hours later there are 2.4 million

(I don't want to know how you determined this).

a. Find the population of bacteria in your toilet as a function of time in days.

b. Find the number of bacteria in your toilet after two weeks.

c. What was the average rate of change of the bacteria population over those two weeks?

d. What was the average population of bacteria during those two weeks?

e. Suppose your toilet becomes unusable when the bacteria population reaches 10 billion. When will you have to either clean your toilet or move out of your apartment?

2. In certain cases involving resistance to motion, it is reasonable to assume that the resistance force is proportional to the velocity of the object: Fr = -kv where k is the constant of proportionality. (Why is the force negative?) Since F = ma = m[pic], it follows that in the absence of other forces, the rate of change of velocity is proportional to the velocity: [pic]= -αv (where α has been substituted for k/m).

a. Find an expression for the velocity v as a function of time in this case assuming an initial velocity of vo.

b. What is [pic]?

c. Find an expression for the total distance traveled, (s, as a function of time.

d. What is [pic]?

e. A young ice skater starts a glide with an initial velocity of 8ft/s. Ten seconds later, her speed is down to 4.852 ft/s. Find her speed as a function of time and find the total distance she glides.

3. You are out driving your car when you run over a nail and puncture your tire. As air leaks out of your tire, the rate of change of the tire pressure is proportional to the tire pressure. The tire pressure initially was 35 psi and at the instant the tire was punctured its pressure began decreasing at a rate of 0.28 psi/min.

a. Write and solve the differential equation that relates your tire pressure to time.

b. What will be the pressure in the tire 10 minutes after it is punctured?

c. If the car is safe to drive as long as the tire pressure remains above 20 psi, how long after the tire is punctured can you continue to drive?

4. Suppose the resistance force on a falling object is proportional to the velocity of the object. Then combined with the force of gravity, we get an equation of motion [pic] (where v is measured in ft/s).

a. Suppose terminal velocity for a particular object is –160 ft/s. Find the value of k in the above equation. (Hint, what is [pic]when you reach terminal velocity?)

b. In the movie Terminal Velocity, Nastassja Kinski gets locked in the trunk of a Cadillac which Charlie Sheen then drives out the back of a C-130 aircraft. (He does this fully aware that the C-130 is airborne at the time.) Find an expression for the Caddy's vertical velocity as a function of time assuming its initial velocity was 0. (Hint: factor out k before separating the variables.)

c. How long does it take the Caddy in part b to reach 99% of its terminal velocity and how far has it fallen in that time?

d. In the movie, the Caddy took two minutes and six seconds to reach the ground where it erupted in a suitably cinematic fireball. How high was the C-130 flying?

AP Calculus HW: Differential Equations - 2

Find the particular solution for each of the following equations.

1. [pic], [pic] 2. [pic], [pic] 3. [pic], [pic]

4. [pic], [pic]

NOTE: The AP often asks questions about differential equations where it is not necessary and maybe not even possible to actually solve the equation.

5. Verify that [pic] is the general solution to the differential equation [pic]. (Think about this: Without a calculator, how could you determine if x = 9 is a solution to [pic] ?)

6. Suppose [pic] is the particular solution to the differential equation [pic] where [pic]. Approximate the value of [pic].

AP Calculus HW: Differential Equations - 3

1. Consider the differential equation [pic].

a. Let [pic] be the particular solution to the given differential equation for 1 < x < 5 such that the line

y = –2 is tangent to the graph of f. Find the x-coordinate of the point of tangency and determine whether f has a local maximum, local minimum, or neither at this point. Justify your answer.

b. Let [pic] be the particular solution to the given differential equation for –2 < x < 8, with the initial condition [pic]. Find [pic].

2. The function f is differentiable for all real numbers. The point (2, ¼) is on the graph of y = f(x), and the slope at each point (x, y) on the graph is given by [pic].

a. Find [pic] and evaluate it at the point (3, ¼).

b. Find [pic] by solving the differential equation [pic] with initial condition[pic].

3. A coffee pot has the shape of a cylinder with radius 5 inches, as shown in the figure. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of [pic] cubic inches per second. (The volume of a cylinder with radius r and height h is [pic].)

a. Show that [pic].

b. Given that h = 17 at time t = 0, solve the differential equation [pic]

for h as a function of t.

c. At what time t is the coffeepot empty?

4. Let [pic] be the particular solution to the differential equation [pic] where [pic].

a. Write an equation for the tangent line to f at the point where x = 2.

b. Determine if f is concave up or concave down where x = 2 and justify your answer.

c. Evaluate [pic] .

AP Calculus HW: Differential Equations - 4

For problems #1 and 2,

a. Sketch the slope field for the given differential equation on dot paper.

b. Find the general solution of the differential equation.

1. [pic] 2. [pic]

For problems #3 and 4,

a. Sketch the slope field for the given differential equation on dot paper.

b. Sketch the particular solution that satisfies the initial condition.

c. Find the particular solution analytically.

3. [pic], y(1) = 0 4. [pic], y(0) = 3

1. 2.

3. 4.

AP Calculus HW: Differential Equations - 5

1. a. Solve for y(x): [pic], y(1) = 4

b. Explain why x = 5 is not in the domain of the solution.

For each of the following, find the particular solution and give the domain of the solution.

2. [pic], y(–1) = 4 3. [pic], y(5) = 3 4. [pic], y(0) = 3

5. [pic], [pic] 6. [pic], y(0) = 1

7. [pic],y(0) = 0

AP Calculus HW: Differential Equations - 6

1. The slope field for the equation [pic] is shown at right.

a. Find (approximately) the point (0, 1) on the diagram.

b. Sketch the solutions passing through the points

i (0, 2) ii. (0, 0) iii. (0, 1)

2. Which of the following slope fields corresponds to the equation [pic]?

(1) (2) (3)

3. Which of the following slope fields represents a differential equation of the form [pic]?

(1) (2) (3)

(4) (5)

(This assignment continued on the next page.)

4. Match the slope fields below with their differential equations from the choices below.

(1) [pic] (2) [pic] (3) [pic] (4) [pic] (5) [pic]

(6) [pic] (7) [pic] (8) [pic] (9) [pic]

a. b. c.

d. e. f.

g. h. i.

5. a. On dot paper, sketch the slope field for

the differential equation [pic].

b. Sketch the particular solution having

initial condition y(1) = 3.

c. Find the particular solution to the equation

analytically.

d. Find the domain of the solution.

6. A common population growth model says that the rate of change is jointly proportional to the populations and the difference between some constant and the population: [pic].

a. On the dot paper on the next page, graph the equation for k = 1/6 and M = 7.

b. Sketch the particular solution corresponding to P(0) = 1.

c. What happens to the population as t ( (?

d. What would happen to the population if P(0) > 7?

e. Use the identity [pic] to solve the differential equation analytically with P(0) = 1.

HW - 1

1a. p = 35e-.008t b. 32.3 psi c. About 70 minutes

2a. P ( 2e0.18232t b. 25.678 million c. 1.691 million/day d. 9.277 million e. 46.715 days

3a. v = voe-αt b. 0 c. s(t) = d. e. 160 ft

4a. 0.2 b. v = -160(1 – e-.2t) c. 2892 ft d. 19360 ft

5b. Estimated time of death is 8:15 AM. Lisa may want a lawyer.

HW - 2

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic]

6a. [pic] b. r/k

HW - 3

1a. x = 3; f has a rel min since [pic] and [pic] b. [pic]

2a. –1/16 b. [pic]

3a. [pic] ( [pic] ( [pic]

b. [pic] c. [pic] seconds

4a. y = –7(x – 2) b. Down b/c [pic] c. –7/4

HW - 4

1. 2. 3. 4.

[pic] [pic] y = ln|x| [pic]

HW – 5

1. a.[pic] b. y(5) = 0 which would make the original equation undefined.

2a. y = 2 + 2|x| b. x < 0 3a. [pic] b. (2, ()

4a. [pic] b. (–e, () 5a. [pic] b. [pic]

6a. [pic] b. (-1, () 7a. y = sin x b. [pic]

HW - 6

1. 2. (1) 3. (2)

4a. (2) b. (4) c. (5) d. (3)

e. (1) f. (6) g. (8) h. (7) i. (9)

5. a,b. c. [pic] d. x > 0

6a.b.

c. 7

d. Decrease to 7

e. [pic]

AP Calculus Review: Applications of Integration

1. Find the solution to the differential equation [pic]satisfying the condition [pic].

2. a. Draw the slope field for the differential

equation [pic].

b. Sketch the solution curve that passes through the

point (1, 1).

c. Sketch the solution curve that passes through the

point (–2, 0).

d. Verify that [pic] is the general solution

to the differential equation.

e. Find the value of C for the particular solution

that has initial condition [pic].

3. Solve the differential equation [pic] if [pic] and state its domain.

4. Suppose [pic] is the particular solution to the differential equation [pic] where [pic].

a. Determine if f has a relative maximum or a relative minimum at x = 3 and justify your answer.

b. Evaluate [pic]

Answers

1. [pic]. y(6) = –4 ( C = 100.

x2 + 4y2 = 100 ( [pic] where we choose the negative root to satisfy the initial condition.

2a,b,c. See graph at right.

d. [pic].

But also[pic] so y' = x + y

e. [pic]

3. [pic] ( [pic] ( ln|y| = 2ln|x| – x + C

( y = Ax2e–x. y(1) = 2 ( A = 2e so y = 2x2e1–x

Since y' is not defined at x = 0, and the solution must include

x = 1, the domain is x > 0.

4a. [pic] and [pic] so f has a rel min at x = 3

b. 1/9

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(Use the slope field at right to help understand the domain.)

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