Mathematics 132 – Calculus for Physical and Life Sciences 2 Exam 1 – Review Sheet February 13, 2008

General Information

As announced in the course syllabus, the first midterm exam of the semester will be given (“main seating”) at 6:00 p.m. on Wednesday, February 20. The format will be similar to that of the midterm exams last semester.

• Bring a photo ID to the exam.

• The exam will be designed to take an hour but you will have an extra 30 minutes to work and check your solutions.

• You will be given a TI-30 scientific calculator for the exam which does NOT have graphing capabilities so be prepared to answer questions without your personal calculator. (Note: Some of you may have one of these calculators purchased for use in Chemistry courses here. That is also OK.)

• Use of cell phones, I-pods, and all other electronic devices is not allowed during the exam. Please leave such devices in your room or put them away in your backpack (make sure cell phones are turned off).

What will be covered

The exam will cover the material since the start of the semester, including the following material from sections 5.1 through 5.8 of Stewart:

1) Riemann sums and the definition of the definite integral.

2) The Fundamental Theorem of Calculus: Part 1: If f(t) is continuous on [a, b] and F (x) = ∫_{a}^{x} f(t) dt, then F′(x) = f(x). Part 2: (the Evaluation Theorem) 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)

3) Antiderivatives graphically and numerically

4) Basic antiderivative rules: All rules coming from basic derivative formulas: Know ∫ x^{n} dx, ∫ a^{x} dx, ∫ sin(x) dx, ∫ cos(x) dx, ∫ \frac{1}{x^{2}+1} dx, ∫ \frac{1}{\sqrt{1−x^{2}}} dx, and so forth, plus the sum, and constant multiple rules

5) Integrals by substitution

6) Integrals by parts: Selecting appropriate u and dv, computing du and v, using the parts formula ∫ u dv = uv − ∫ v du, then finishing the integral on the right. Recall that this might involve using parts again, or another method such as substitution.

7) Integration by trigonometric substitution.

8) Integration of rational functions by partial fractions (see summary distributed in class).

9) Integrals by the table: Recognizing the appropriate entry, using reduction formulas repeatedly if necessary. Be aware that a preliminary substitution might be necessary to take an integral to one of the forms in the table.

Note: Some problems may ask you to carry out a particular integration method on a problem. Others may leave the choice up to you. Be prepared for both types of questions! There will be a review for the exam in class on Tuesday, February 19.

Review Problems

The Review Problems 1 - 46 at the end of Chapter 5 are good for preparation for this exam. It’s not necessary to work all of them. But you should try a good selection and practice choosing a method at least for most of the integrals in problems 9 - 46.

Sample Exam Questions

I. Let

f(x) = \{ \begin{matrix}1 & if 0 ≤ x ≤ 3 \\ x − 2 & if 3 ≤ x ≤ 5 \\ 13 − 2x & if 5 ≤ x ≤ 8 \end{matrix}

A) Sketch the graph y = f(x). In the rest of the parts, F (x) = ∫_{0}^{x} f(t) dt, where f is the

function from part A. B) Assuming F (0) = 0, Compute F (1), F (2), F (3), F (4), F (5), F (6), F (7), F (8) given the

information in the graph of f. C) Are there any critical points of F? If so, find them and say whether they are local

maxima, local minima, or neither. If not, say why not. D) Sketch the graph y = F (x) if F (0) = 0, and also if F (0) = 2.

II. Find the derivatives of the following functions

A) f(x) = ∫_{0}^{x} sin(t)/t dt.

B) g(x) = ∫_{5}^{x^{3}} tan^{4}(t) dt. C) h(x) = ∫_{−3x}^{5x} e^{t^{2}} sin(t) dt.

III.

A) Compute ∫ 5x^{4} − 3\sqrt{x} + e^{x} + \frac{2}{x} dx

B) Apply a u-substitution to compute ∫ x(4x^{2} − 3)^{3/5} dx

C) Apply a u-substitution to compute ∫_{1}^{2} e^{sin(πx)} cos(πx) dx

D) Do you need partial fractions to compute

∫ \frac{t^{2} + 1}{t^{3} + 3t + 3} dt?

Explain, and give a simpler method. E) Apply integration by parts to compute ∫ x^{2}e^{−2x} dx

F) Apply partial fraction decomposition to compute

∫ \frac{1}{x(x − 1)(x + 2)} dx

G) Which trigonometric substitution would you apply to compute ∫ \frac{1}{u\sqrt{a^{2}−u^{2}}} du? What trigonometric integral do you get after making the substitution? Complete the derivation of the integral.

IV. Compute each of the integrals below using some combination of basic rules, substitution, integration by parts, the table of integrals, partial fractions, and trigonometric substition. You must show all work for full credit.

A)

∫ \frac{x^{3}}{x^{2} + 4x} dx

B)

∫ (x^{2} + 2x)^{3/2} dx

C)

∫ \frac{e^{\sqrt{sin(x)}} cos(x)}{\sqrt{sin(x)}} dx

D)

∫ \frac{dz}{z^{2}\sqrt{z^{2} + 6z + 10}}