Review Sheet for Midterm 2
Math 32, Spring 2004
Benjamin Johnson
The second midterm will be held Wednesday, April 14, from 8:10AM to 9:00AM in 60 Evans. The exam will cover all of chapters 5 through 7, plus sections 8.1 and 8.2 from chapter 8.
To do well on this exam you should be able to do at least each of the following:
Chapter 5: Exponential and Logarithmic Functions
5.1 (Exponential Functions)
Sketch a graph of a function such
as . Specify the domain and range of a given exponential
function.
5.2 (The Exponential Function )
Sketch a graph of an exponential function with base e.
5.3 (Logarithmic Functions)
Sketch a graph of a logarithmic
function. Specify the domain and range
of a given logarithmic function.
Convert an expression involving a logarithmic function into an
equivalent expression involving an exponential function (i.e. ).
5.4 (Properties of Logarithms)
State and use all the major
properties of logarithms given this chapter to simplify expressions involving
logarithms, (e.g. ,
etc…). Express a logarithm to any
desired base using the change of base formula, (i.e.
).
Solve simple equations with a variable in the exponent using logarithms.
5.5 (Equations and Inequalities with Logs and Exponents)
Solve equations involving logarithms, by converting logarithmic equations to exponential equations. Solve equations with variables in exponents by simplifying and taking the log of both sides of the equation. Solve inequalities using logarithms and exponents. (Remember to intersect the solution set you obtain with the domain of the original equation to obtain your final answer).
5.6 (Compound Interest)
Use the various formulas presented in this section to compute the amount obtained from a principal invested at a nominal interest rate r, compounded annually, n times per year, or continuously. Given a nominal interest rate and a method of compounding, compute the effective rate. Find the doubling time given a nominal rate r and a method of compounding interest.
5.7 (Exponential Growth and Decay)
Given an application problem involving exponential growth or decay, determine the growth rate, or the decay rate, respectively. Find the doubling time, or half-life, given an exponential growth or decay function. Solve an application problem from this section similar to those on your homework or the ones done in class.
Chapter 6: Trigonometric Functions of Angles
6.1 (Trigonometric Functions of Acute Angles)
Evaluate the sine and cosine of an angle in a right triangle, given the length of at least 2 of the sides.
6.2 (Algebra and the Trigonometric Functions)
Simplify algebraic expressions containing trigonometric functions.
6.3 (Right Triangle Applications)
Find the lengths of various sides of a right triangle using the trigonometric functions.
6.4 (Trigonometric Functions of Angles)
Compute the sine and cosine
functions for all angles that are multiples of or
. (You do not need to use reference angles to
do this; you should use the definitions of sine and cosine based on points on
the unit circle). Identify the
positions of various angles on the unit circle.
6.5 (Trigonometric Identities
State and apply the trigonometric
identity . State and apply the definitions of the 4
other trigonometric functions in terms of sine and cosine. Prove a trigonometric identity, either by
algebraically simplifying one side of the equation to obtain the other, or by
simplifying both sides of the equation to obtain identical simplified forms.
Chapter 7: Trigonometric Functions of Real Numbers
7.1 (Radian Measure)
Specify the location of angles measured in radians on the unit circle. Convert degrees to radians and vice versa. State and apply the relationship between radian measure of an angle and arc length on a circle with a given radius.
7.2 (Radian Measure and Geometry)
State and apply the formulas for
arc length, ( ), area of a sector, (
), area of a triangle, (
), and area of a segment (
).
State and apply the relationship between linear speed
and angular speed
on a circle (
).
7.3 (Trigonometric Functions of Real Numbers)
Evaluate the sine and cosine of
real numbers that are multiples of or
. Specify the domain, range, and period of the
sine and cosine functions. Simplify
expressions involving trigonometric functions using basic trigonometric
identities and algebra.
7.4 (Graphs of the Sine and Cosine functions)
Sketch a graph of the sine and cosine functions quickly and accurately.
7.5
(Graphs of and
)
Given constants A, B, and C,
sketch a graph of the functions and
. Explain how the constants A, B, and C,
relate to the amplitude, period and phase shift of the above types of
functions. Given a description of a
periodic function including its value at zero, amplitude, period, and phase
shift, write a formula of the above type which models the given description.
7.6 (Simple Harmonic Motion)
Given an equation describing simple harmonic motion, state the amplitude, period, and frequency. Interpret these quantities in terms of the motion of an object connected to a spring.
7.7 (Graphs of the Tangent and the Reciprocal Functions)
Sketch a graph
of any of the six trigonometric functions, or slight variations (e.g. 
Chapter 8: Analytical Trigonometry
8.1 (The Addition Formulas)
Use the addition formulas for sin and cos to prove identities.
8.2 (The Double-Angle Formulas)
Use the double-angle formulas for sin and cos to prove identities.
If you have mastered the above, you can expect to do very well on at least 80% of the exam. There may also be at least one creatively designed problem on the exam, attempting to assess your ingenuity and genuine understanding of the material.
Additional resources:
http://math.berkeley.edu/~benjamin/mt2s.pdf
These are links on my website to a second midterm (and solutions) given by Seth Dutter, a current GSI for this class who taught Math 32 this past summer. You can expect your exam to be somewhat similar both in content and in difficulty.
- Your homework problems are one of your best resources. You should attempt to re-work any homework problem that you did not answer correctly the first time. Use the solution sets handed out in your section to help you accomplish this.
- The review exercises at the end of the chapters are excellent resources, especially the chapter tests. The answers to all the chapter test questions (even the even-numbered ones) are given in the back of your book. You should try to work out as many of these test questions as you can.
- The Student Learning Center offers free tutoring for this class. Drop-in tutoring at the Cesar Chavez Student Center Atrium is available Monday through Thursday from 9AM until 4PM and on Friday from 9AM until noon. Also there is drop-in tutoring available in Moffitt Library (Room 450A) Monday, Tuesday, and Wednesday, from 6PM until 8PM. See http://slc.berkeley.edu for more information about the Student Learning Center.
- Your Discussion Section leaders and your instructor are available during our respective office hours to answer your questions regarding course material. Feel free to take advantage of these opportunities to obtain further assistance.
GOOD LUCK!!