Study Guide for the Exams

Summary:  The course covers a lot of ground, but some topics are definitely more important than others. This page gives you some guidance in preparing for your exams.

Caution:  This page lists the most important topics, which should account for most of the points on an exam. With the best will in the world, it’s not practical to list every single topic from the course. While there is no intention to trick you, if something is not listed here don’t assume that there will be no questions at all about it.

See also:  How to Take a Math Test for advice on preparation and test taking

Precalculus Topics

You will not be tested directly on these topics, but you need them to solve most of your calculus problems. If you never learned them or you’re rusty, you need to make up this material.

• distance between two points in the plane
• slope of a line (section P.2)
• equation of a line in point-slope form (section P.2)
• function notation, including things like f(x+7) (section P.3)
• simplifying and factoring expressions involving radicals and exponents
• solving linear and quadratic equations; solving higher-order equations by factoring
• definitions of sine, cosine, tangent for any angle
• values of sine, cosine, tangent for any multiple of π/6 or π/4 (any multiple of 30° or 45°)
• reciprocal trig identities; tangent/cotangent trig identities; Pythagorean trig identities

Some handouts can help you.

Chapter 1. Limits

• concept of a limit, especially graphically (section 1.2)
• You are not responsible for ε-δ proofs of limits.
• limits by direct substitution, when it’s appropriate (pages 57–59)
• limits by reducing fractions and rationalizing (pages 61–62)
• removable and nonremovable discontinuities (page 69), their meaning and how to find them
• one-sided limits (page 70)
• limit “equals ∞” or “equals −∞”, as opposed to “does not exist” (section 1.5)
• vertical asymptotes, meaning and how to find (section 1.5)

Suggested review problems (pages 88–89, odd and even numbers): 5–6, 7–10 (skip the ε-δ proof), 11–26, 27c, 28c, 29–46, 51–70

Chapter 2. Differentiation

• concept of derivative as slope of tangent line, which is limiting case of slope of secant lines (section 2.1)
• You are not responsible for finding derivatives by the definition using the limit of the difference quotient.
• notation for derivatives (page 97)
• higher-order derivatives, concept and notation (page 123)
• relationship of position, velocity, acceleration (page 123)
• finding derivatives of algebraic and trig functions (sections 2.2–2.4, summarized on page 133) — most important are the chain rule, power rule, and product and quotient rules
• rewriting functions to match a differentiation rule
• finding equation of tangent line
• derivative of implicit function (section 2.5)
• related-rate problems (section 2.6; guidelines page 145)

Some handouts can help you.

Suggested review problems (pages 153–155, odd and even numbers): 5–12, 15–39, 41–80, 89–92, 97–104, 105–106 (ignore normal line), 107–111

Chapter 3. Applications of Differentiation

• critical numbers (section 3.1)
• where function is increasing and decreasing (section 3.3), concave up and concave down (section 3.4)
• finding minima and maxima by the Second Derivative Test (section 3.4) or the First Derivative Test (section 3.3)
• minima and maxima on a closed interval: remember to check endpoints (section 3.1)
• points of inflection, how to recognize graphically and how to find (section 3.4)
• limits “at infinity” for rational functions and horizontal asymptotes of same (section 3.5)
• finding slant asymptotes (page 204)
• analyzing graphs of functions (section 3.6) — importance minor but >0
• optimization problems (section 3.7; guidelines page 212)
• propagated error, relative error, percent error (section 3.9)
• You are not responsible for section 3.8.

Some handouts can help you.

Suggested review problems (pages 235–237, odd and even numbers): 1, 3–4, 15–30, 33–44, 49, 53–55, 65–66, 68–70, 75–76, 81–84

Chapter 4. Integration

• antiderivative or indefinite integral of powers and certain trig functions (section 4.1)
• rewriting functions to match integrable forms
• finding particular solutions of differential equations (section 4.1)
• summation formulas (section 4.2)
• You are not responsible for section 4.2 finding upper and lower sums by limits.
• relation of definite integral to area(s) (section 4.3)
• result of interchanging limits of integration (page 270) or breaking an integral into two (same page)
• using the Fundamental Theorem of Calculus to evaluate definite integrals (section 4.4)
• finding average value of a function (section 4.4)
• changing variable to integrate by substitution (section 4.5)
• shortcut integration of even and odd functions (page 296)
• You are not responsible for section 4.6.

Some handouts can help you.

Suggested review problems (pages 307–309, odd and even numbers): 1–10, 17–18, 27–50, 55–76, 79–80

Chapter 5. Logarithms and Exponentials

• rewriting ln expressions using properties (section 5.1)
• ln in derivatives (section 5.1) and integrals (section 5.2)
• guidelines for integration (page 327)
• integrals of trig functions (page 329)
• inverse function: knowing whether one exists, verifying a claimed inverse, and finding the inverse function (section 5.3)
• You are not responsible for derivative of an inverse function (pages 336–337).
• derivatives and integrals using e^x (section 5.4)
• You are not responsible for hand sketching graphs of logs and exponentials, though you should be able to recognize them and know their properties.
• You are not responsible for sections 5.5 through 5.10.

Some handouts can help you.

Suggested review problems (pages 405–407, odd and even numbers): 3–30, 35–36, 39–60, 65–76

Chapter 6. Applications

• area between curves, including dealing with points of intersection (section 6.1)
• volumes of solids of revolution by the disk or washer method (section 6.2) and the shell method (section 6.3)
• You are not responsible for sections 6.4 through 6.7.

Suggested review problems (pages 476–477, odd and even numbers): 1–19, 21–32