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FCS, Mr. GarciaAP Calculus ABDate: 2/16/18

AP Calculus AB Chapter 6 Review.

This could be used as a study guide for the quiz on 6.1, 6.2, 6.4.

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What must you know for chapter 6.1, 6.2, 6.4 (Sections 6.3, 6.5, and Euler’s Method are BC topics). See practice problems below.

Section 6.1 Differential equations and Slope Fields

Given a differential equation, you must know how to find the general solution and the particular solution if an initial condition (a point) is given.

Example 1. Find the general and particular solution to the differential equation, whose graph passes through the point (1. 0).

Another way to pose the question in example one is to ask to verify if the answer to the differential equation, satisfies the d. e. (differential equation).

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Section 6.2 Solving integrals using U-Substitution

Example 2.Evaluate the integral,

Let u = cos(x) then du = -sinxdx

Substitute u

Now substitute du for –sinxdx

Now, evaluate the integral=

Now, substitute back

Section 6.4 Solving differential equations by separating the variables and

The topics of growth and decay.

the answer to this differential equation is

Where is the initial amount, k is the rate of growth (-k is decay) and t is the time.

Half-life =

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Chapter 6 mixed problems for practice.

  1. Write the differential equation that has as a general solution.
  1. Evaluate the integral,
  1. The decay equation for a radioactive element is .
  1. Write down what each variable represents.
  1. How long will it take this element to decay 70% of the original amount?
  1. Solve the initial value problem and y = 0 when x = 0, explicitly.
  1. Solve the differential equation, .
  1. Evaluate the integral,
  1. Verify that is the solution to .
  1. Evaluate the integral,.
  1. The Sm-151 isotope of samarium can be modeled by the differential equation, , where t is measured in years. What is its half-life?
  1. The decay equation for radon-222 gas is known to be , with t in days. About how long will it take the amount of radon in a sealed sample of air to decay to 90% of its original value?
  1. Use the separation of variables to solve the initial value problem and y =1 when x = e.
  1. Solve these 2 equations: a) b)
  1. Solve the initial value problem, , and y = 0, when x = 0.
  1. Find the general solution for .
  1. Evaluate the indefinite integral,
  1. Use 4 rows around the origin to create a slope field for . See page 323 in your textbook if you need help.
  1. Use your calculator to create a slope field for the differential equation in the previous question by entering the solution to your calculator and adding a list. See bottom of page 322 textbook for help.

19. Also do quizzes on pages 349, but skip #3 (BC concept) and 372, but skip #3 and 4.For more problem exercises continue on to the next page.

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More exercises

20. Evaluate the integrals analytically and then use your calculator to confirm and not the other way around!

a) b)

21. Solve the initial value problems analytically

a) , y(1) = 1b)

22. Construct a slope field in the space below for the differential equation given. Use points (0, -1), (0, 0), (0, 1), (0, 2); (-1, -1), (-1, 0), (-1, 1), (-1, 2); (1, -1), (1, 0), (1, 1), (1, 2).

23. Find the amount of time required for $10,000 to double if the 6.3% annual interest is compounded (a) annually, (b) continuously.