Chapter 5 Outline

PAGE 1

AP Calculus BC

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Practice Quiz: Slope Fields & Eulers Method

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Consider the differential equation passing through the point (0, 1).

1.Sketch the slope field for this differential equation for the indicated points.

2. Use Eulers method with a step size to

Estimate the solution at:

(a) f(0.1)

(b) f(0.2)

(c) f(0.3)

(d) f(0.4)

(e) f(0.5)

3.Plot the estimated values on the slope field.

4.Use separation of variables to find the exact solution to the differential equation. Sketch the differential equation on the graph above.Free Response Questions: Slope Fields and Eulers Method1. Consider the differential equation

A) On the axes provided, sketch the slope field of the given differential equation.

B) Let be the particular solution to the differential equation subject to the initial condition . Use Eulers Method, starting at x=0, with a step size of 0.1, to approximate . Show the work that leads to the answer.

C) Find an equation or inequality that reflects when the solution to the differential equation would be concave up.

2. Consider the differential equation .

A) On the axes provided, sketch the slope field of the given differential equation.

B) Let be the particular solution to the differential equation subject to the initial condition . Use Eulers method, starting at x=0, with a step size of 0.1, to approximate . Show the work that leads to the answer.

C) Find the particular solution to the differential equation with the initial condition

BC CALCULUS CH 6 Revisited DIFFYQ

1. Given the differential equation

a. Make the slope field on the grid to the left

b. Draw a possible curve through (0,1) c. Use Eulers method with an initial condition of (0,3) and a to estimate the y value at x = 1 ( 1, ) x y

03

0.5

1

d. Is this answer an over or underestimate? Explain

e. Use separation of variables to solve the differential equation with initial condition (0,3) to find the exact value at x = 1 ( 1, )Review Hodgepodge

1. Given find the equation of the family or curves which pass through the point (ln 1, 0).

2.

3. Find the particular solution of the equation that satisfies the condition (1) = 6.

4. The values of x where the graph as illustrated of is concave downward is

5. An object moves along the x-axis so that its velocity at anytime t 0 is given by . Find the total displacement of the particle from t = 0 to t = 2.

6. Differentiate:

7. Evaluate:

EMBED Equation.DSMT4 8. Find the partial fractions for (You do NOT have to integrate).

9. Given a function is defined by . Find all the relative maximum and/or relative minimum points.

10. Given, then

Scavenger Hunt through Section 6.4Look through pg. 330-340 to find the following. 1. Find the equation relating the half-life of an element to k, the rate constant.

2. Find the equation known as Newtons Law of Cooling. 3. Find the general differential equation for a quantity that increases or decreases at a rate proportional to the amount present.

4. Find the general equation for the amount of money earned by an account, if interest is added k times per year. (ex: monthly, weekly, or daily)

5. In Newtons Law of Cooling, what does stand for?

What does stand for?

What does stand for?

6. What equation represents the resisting force on an object in motion?

7. If you separate and integrate the differential equation , using the initial condition, when , what is the result?

8. Although it may be surprising, half-life is a constant that depends only on ________________________ and not the _________________________________.

9. Answer John Napiers question in problem #30 on page 339. What does happen (part a)?

Part b)? How long will it take your money to triple?

Part c)? How much can you earn in one year (relative to how much you put in).

10. Solve #28 on page 339. SHOW ALL WORK on the back, including integration. How many grams of glucono lactone will be left after the first hour?

Newtons Law of Cooling1. A cup of coffee at temperature is placed on a table in a room at . The differential equation for its temperature at time is. After 10 minutes the temperature (in ) of the coffee is

(A) 96

(B) 100

(C) 105

(D) 110

(E) 115

2. Approximately how long does it take the temperature of the coffee in the previous question to drop to ?

(A) 10 min

(B) 15 min

(C) 18 min

(D) 20 min

(E) 25 min

3. According to Newtons law of cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. Suppose a corpse at a temperature of arrives at a mortuary where the temperature is kept at. Then the differential equation satisfied by the temperature of the corpse hr later is

(A)

(B)

(C)

(D)

(E)

4. If the corpse in the previous question cools to in 1 hour, then its temperature is given by the equation

(A)

(B)

(C)

(D)

(E)

Free Response5. According to Newtons law of cooling, the temperature of an object decreases at a rate proportional to the difference between its temperature and that of the surrounding air. If a roast at room temperature is put into a freezer, and if, after 2 hours, the temperature of the roast is :

(a) What is its temperature after 5 hours?(b) How long will it take for the temperature of the roast to fall to?Population Growth- Logistic Growth Models

1. Suppose that the growth of a population y = y(t) is given by the logistic equation

a) What is the population at time t=0?

b) What is the carrying capacity L? (Hint: It is not 60. Carrying capacity )

c) What is the constant k?

d) When does the population reach half of the carrying capacity?

e) Write an initial value differential equation problem whose answer would above given logistic equation.2. Suppose that the growth of a population y = y(t) is given by the logistic equation

a) What is the population at time t=0?

b) What is the carrying capacity L? (See hint for 1b)

c) What is the constant k?

d) When does the population reach 75% of the carrying capacity?

e) Write an initial value differential equation problem whose answer would above given logistic equation.3. Suppose that a population y(t) grows in accordance with the logistic model

a) What is the carrying capacity? (Hint: When a logistic function approaches capacity, the rate slows down and approaches zero. For what y-values does )

b) What is the value of k?

c) For what value of y is the population growing most rapidly? (Hint: You find the maximum of any function by finding its derivative and critical points. The derivative of is . In other words, find the 2nd derivative. Where it equals zero is where has its maximum and minimums.)

4. Suppose that a population y(t) grows in accordance with the logistic model

a) What is the carrying capacity? (See hint for 3a)

b) What is the value of k?

c) For what value of y is the population growing most rapidly? (See hint for 3c)

Logistic Growth Problems: BC Calculus Review

1. Which of the following statements characterize(s) the logistic growth or a population whose limiting value is L?

I. The rate of growth increases at first

II. The growth rate attains a maximum when the population equals

III. The growth rate approaches 0 as the population approaches L.

A) I only B) II only C) I and II only D) II and III only E) I, II, and III

2. Which of the following differential equations is not logistic?

A) B) C)

D) E) (where k and A are constants)

3. Suppose P(t) denotes the size of an animal population at time t and its growth is described by the differential equation. The population is growing fastest

A) initially B) when P=500 C) when P=1000 D) when

E) when

4. Because of limited food and space, a squirrel population cannot exceed 1000. It grows at a rate proportional both to the existing population and to the attainable additional population. If there were 100 squirrels 2 years ago, and 1 year ago the population was 400, about how many squirrels are there now?

5. Suppose a flu-like virus is spreading through a population of 50,000 at a rate proportional both to the number of people already affected and to the number still unaffected. If 100 people were infected yesterday and 130 are affected today:

a) write an expression for the number of people N(t) infected after t days

b) determine how many will be affected a week from today

Growth and Decay:1. If radium decomposes at a rate proportional to the amount present, then the amount left after yr, if is present initially and is the negative constant of proportionality, is given by

(A)

(B)

(C)

(D)

(E)

2. The population of a city increases continuously at a rate proportional, at any time, to the population at that time. The population doubles in 50 years. After 75 years the ration of the population to the initial population is

(A)

(B)

(C)

(D)

(E) none of these

3. If a substance decomposes at a rate proportional to the amount of the substance present, and if the amount decreases from 40 g to 10 g in 2 hr, then the constant of proportionality is

(A)

(B)

(C)

(D)

(E)

FREE RESPONSE

4. The population of a country is growing at a rate proportional to its population. If the growth rate per year is 4% of the current population, how long will it take for the population to double?

5. The bacteria in a certain culture increases continuously at a rate proportional to the number present.

(a) If the number triples in 6 hours, how many will there be in 12 hours?

(b) In how many hours will the original number quadruples?

6. Radium-226 decays at a rate proportional to the quantity present. Its half-life is 1612 years. How long will it take for one quarter of a given quantity of radium-226 to decay?

7. At a yearly rate of 5% compounded continuously, how long does it take for an investment to triple?

8. One important method of dating fossil remains is to determine what portion of the carbon content of a fossil is radioactive isotope carbon-14. During life, any organism exchanges carbon with its environment. Upon death this circulation ceases, and the in the organism then decays at a rate proportional to the amount present. The proportionality factor is 0.012% per year.

When did an animal die if an archaeologist determines that only 25% of the original amount of is still present in its fossil remains?

BC Calculus Chapter 6 Review

1. Which of the following is a solution of the differential equation ?

A. y = e-4xB. y = 4xC.

D. y = e2xE. y = e4x2. Which of the following is a solution of the differential equation ?

A. y = e-4xB. y = 4xC.

D. y = e2xE. y = e4x3. Solve the differential equation subject to the initial condition y(0) = 2. From your solution, find the value of y(1).

A.

B.

C.

D.

E.

4. A slope field is given at the right. Which of the following represents its differential equation?

A.

B.

C.

D.

E.

5. A bacteria culture starts with 200 bacteria and triples in size every half hour. After 2 hours, how many bacteria are there?

A. 17800

B. 16200

C. 23500

D. 19300

E. 15700

6. When a child was born, her grandparents placed $1000 in a savings account at 10% interest compounded continuously, to be withdrawn at age 20 to help pay for college. How much money is in the account at the time of withdrawal?

A. $2718.28

B. $14778.11

C. $10873.12

D. $7389.05

7. If you use Euler's Method and two steps with x = 0.1 for the differential equation with initial value y(0) = 1. When x = 0.2, then y is approximately equal to

A. 0.11

B. 1.100

C. 1.210

D. 1.464

8. Which of the following statements characterize the logistic growth of a population whose limiting value is L?

I.The rate of growth increases to begin with.

II.The growth rate attains a maximum when the population equals .

III.The growth rate approaches 0 as the population approaches L.

A. I onlyB. II onlyC. I and IID. II and III E. I, II and III

Free Response

9. Suppose that a population grows according to a logistic model.

(a) Write the differential equation for this situation with k = 0.01 and carrying capacity of 60 thousand.

(b) Solve the differential equation in part (a) with the initial condition t = 0 (hours) and population P = 1 thousand

(c) Find the population for t = 10 hours, t = 100 hours, and t = 1000 hours. Round to the nearest hour.

(d) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?

(e) As the time t increases without bound, what happens to the population?

(f) Sketch the graph of the solution of the differential equation.

10. (2002 BC 5) Consider the differential equation

a. let y = f(x) 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 y = g(x) be the particular solution to the given differential equation for -2 < x < 8, with the initial condition g(6) = -4. Find y = g(x).

Answers: 1. A 2. C 3. B 4. D 5. B 6. D 7. C 8. E

9. a. b. c. P(10) ~ 1.103; P(100) ~ 2.643; P(1000) ~ 59.840

d. The population will reach 2 thousand in about 71 hours. The population will reach 30 thousand in about 408 hours. The population will reach 55 thousand in about 648 hours. e. 60 thousand f.

10. a. The x-coordinate of the point of tangency is x = 3 (where ). Because f is continuous for 1 < x < 5, there is an interval containing x = 3 on which y < 0. On this interval, is negative to the left of x = 3 and positive to the right of x = 3. Therefore f has a local minimum at x = 3. b.

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