Mathematics 134 – Intensive Calculus for Science 2 Exam 3 – Things To Know April 23, 2001

General Information

As announced in the course syllabus, the third full-period midterm exam of the semester will be given in class on Friday, April 27. The format will be similar to that of the midterm exams last semester and Exams 1 and 2 this time. The exam will cover the material since the last exam (Quizzes 7,8,9), including the following material from Chapters 9 and 10 of the text:

1) Taylor polynomials and Taylor approximations

2) Taylor series and interval of convergence

3) “Shortcuts” (substitution, algebraic manipulations, term-by-term integration and differentiation) for computing Taylor series from the known series for

e^{x}, sin(x), cos(x), \frac{1}{(1 − x)}, (1 + x)^{p}.

4) Differential equations a) What it means for a function to be a solution of a differential equation

b) Graphical meaning of solutions via slope fields

c) Approximate values of solutions via Euler’s Method

d) Formulas for solutions via separation of variables

e) Applications to growth/decay problems

A copy of the table of integrals from our text will be provided.

There will be a review for the exam in class on Thursday, April 26. I will also try to set up an evening session with Jodi Marani on Thursday for last minute questions.

Sample Exam

Caution: The following problems indicate only the kinds of topics that will be used in, the approximate amount of computation required for, and the level of difficulty of the questions that will appear on this exam. The actual exam questions will look different from these and the exam itself will be slightly shorter. It is important to understand the methods involved in the solutions to these problems as well as the answers. Do other practice problems besides these ones if you want to be well-prepared.

I. A) Compute the 4th degree Taylor polynomial for f(x) = ln(1 + x) at a = 0, using the

definition of Taylor polynomials.

B) Use your polynomial to approximate the value of ln(1.1). What is the error in your approximation?

II. A) Using our “shortcut” rules, determine the first three nonzero terms in the Taylor series

at a = 0 of the functions cos(x) and (1 − x^{2})^{\frac{1}{2}} . B) Using the Taylor series, determine which of the two functions in part A is larger for

small values of x (say x < .1). Explain how you can tell.

III. All parts of this question refer to the differential equation

\frac{dy}{dx} = 4 − y

A) Sketch the slope field of this equation, showing the slopes at points on the lines

y = 0, 1, 2, 3, 4, 5, 6 B) On your slope field, sketch the graph of the solution of the equation with y(0) = 1. C) Using separation of variables, solve the differential equation with the initial condition

y(0) = 1.

IV. The population of fish in a lake is attacked by a microscopic water-borne parasite at t = 0, and as a result the population declines at a rate proportional to the square root of the population from that time on. A) Express this statement about the rate of growth of the population P as a differential

equation. B) Use n = 4 steps of Euler’s Method to approximate the value of the solution of your

differential equation with P (0) = 200 at t = 1. C) Use separation of variables to find the formula for the general solution of your differ-

ential equation. D) At t = 0 there were 900 fish in the lake; 441 were left after 6 weeks. When did the

fish population disappear entirely?

Also see Problem Sets 6,7,8, the last three quizzes, and the Review Problems from the ends of Chapters 9,10 in the text for other practice problems.