Mathematics 134 – Intensive Calculus for Science 2 Things to Know – Final Exam April 30, 2001

General Information

• The final exam will be given in our regular class room 8:30am - 11:30am on Saturday, May 5.

• This will be a comprehensive exam – it will cover all of the material from Chapters 6-10 of the text that we have studied this semester. See the detailed list of topics below.

• The final will be similar in format to the three full period midterm exams, but roughly twice as long.

• If you are well prepared and work steadily, it should be possible to complete the exam in about 2 hours. But you will have the full-three hour period to work on the exam if you need that much time.

• If there is interest, we can up a daytime or evening review session during study week. I am not free Thursday evening, but most other times are possible.

Topics to Be Covered

1) The Fundamental Theorem of Calculus: If F (x) is an antiderivative of a continuous function f(x) on [a, b], then

∫_{a}^{b} f(x) dx = F (b) − F (a)

2) Antiderivatives graphically and numerically

3) The power, sum, and constant multiple rules for antiderivatives

4) Indefinite integrals by substitution, parts and using the table of integrals

5) Improper integrals (integrals over infinite intervals, and integrals of functions with discontinuities in the interval [a, b]) – be able to identify which kind of improper integral you have, and set up the appropriate limits to evaluate them/determine whether they converge or not

6) Integrals for computing volumes of solids with known cross-sections: V = ∫_{a}^{b} A(x) dx, where A(x) is the cross-section area function. Volumes of solids of revolution are a special case of this.

7) Integrals for computing total mass/center of mass of wires and plates of given shapes, given the mass density function

8) Density functions of distributions, cumulative distribution function, median, mean.

9) Taylor polynomials and Taylor approximations

10) Taylor series and interval of convergence

11) “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}.

12) 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

Suggestions on How to Study

A) Reread your class notes and work through the examples we did in class – everything

on the final will be very close to something we did along the way somewhere! B) Look over your quizzes and exams, especially any questions you had difficulty with.

Read the comments and corrections – they’re there to help you figure out what you did wrong and how to solve the problem correctly.

C) Also look at your graded problem sets and the write-ups from the group discussion

days. D) Look over the practice exams for the three full-period midterms. E) Don’t panic! There is a lot of material here, but not all that many key ideas. Every-

thing on the final will be very close to something we did along the way somewhere!