Midterm 2 Review Sheet

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4.1 Exponential functions: Definition of exponential functions, their domain and ... tions, graphing logarithmic functions, solving logarithmic equations, solving ...
Midterm 2 Review Sheet Some thoughts on preparing for the exam: To study for the exam, doing lots of problems helps, but it matters how you do them. Be wary of being too dependent on the book for looking up equations or rules. Try to do exercises without looking things up in your book or notes. If you get stuck or don’t know how to start, think about a problem for 5 minutes, and if you’re still stuck, then look in the book. Then try a similar problem, and again, try to solve it without looking anything up. Keep doing this until you can do that type of problem without looking anything up. It is easy to do problems and then not remember anything about how to do them–the only way you really know if you can do a type of problem is by doing it correctly on your own without looking anything up. Try some problems from the review exercises at the end of each chapter as well-the benefit of this is that you won’t know what section they’re from so you’ll have to figure out what rules/equations/concepts apply on your own. If you want to read something besides the book, see the Resources section of the course website–Paul’s Online Math Notes or Purple Math may be helpful and have lots of examples. The practice test posted on the website doesn’t cover everything (for example, there are no problems on solving triangles on it, but this will certainly be covered on our midterm), but it may be helpful. A good idea is to try taking the practice test under exam conditions: time yourself and don’t use any resources–see how you do. Feel free to email me with questions. Note on calculators: You’ll need a calculator to do the problems on solving triangles. On the exam, you will not need a calculator (and you will not be allowed to use one). Instead, if the problem requires knowledge of something like sin(22), I will provide that information. Without further ado, here’s a list of the topics we’ve covered and some recommended exercises. You don’t have to do all of them–pick the ones that you need work on, and do them. If you want even more exercises to work on for a specific section, send me an email and I can find some more.

4.1 Exponential functions: Definition of exponential functions, their domain and range, graphing exponential functions, graphing families of exponential functions (vertical and horizontal translations, stretching/shrinking, etc.), solving exponential equations Recommended exercises: 31, 35, 37, 49, 54, 71, 76, 80 4.2 Logarithms: Definition of logarithms, their domain and range, evaluating logarithmic functions, graphing logarithmic functions, solving logarithmic equations, solving exponential equations using logarithms Recommended exercises: 19, 26, 35, 38, 45, 62, 72, 81, 85, 93, 95, 98, 103, 105 4.3 Rules of logarithms: inverse rules of logarithms, logarithm of a product, logaritm of a quotient, logarithm of a power, using these rules to simplify expressions and solve equations Recommended exercises: 16, 25, 29, 44, 50, 55, 61 4.4 More exponential and logarithmic equations: Solving various logarithmic equations, including those with more than one logarithm, solving exponential equations using logarithms, see strategy for solving exponential and logarithmic equations at the bottom of page 387 Recommended exercises: 5, 10, 16, 19, 23, 29, 42, 45, 47 (See the review exercises on p. 399 for more problems) 1

5.1 Angles and their measurements Definition of an angle, degree measure of an angle, radian measure of an angle, converting from degrees to radians and from radians to degrees, quadrants Recommended exercises: 26, 55, 58, 68, 70, 78, 92 5.2 The Sine and Cosine functions Definition of sine and cosine, evaluating sine and cosine at multiples of 90 degrees, multiples of 45 degrees, multiples of 30 degrees, using reference angles to evaluate sine and cosine in other quadrants, the Pythagorean identity (p. 428), using it to find cosine of an angle if you know sine of that angle and vice versa. Recommended exercises: 6, 12, 13, 18, 23, 22, 25 (better yet, make a table of sine and cosine evaluated at all angles we know–try to do this without looking at the book, then check your answer), 91, 92, 93 5.3 Graphing sine and cosine The graphs of sine and cosine, the definition of a periodic function, the periods of sine and cosine, definition of amplitude and phase shift, graphing families of trig functions (horizontal and vertical translations, stretching/shrinking/reflecting, changing the period (e.g. cos(3x), graphing combinations of these transformations) Recommended exercises: 14, 18, 19, 26, 33, 37, 41, 43 5.4 The other trig functions: tan, cot, sec, csc The definitions of these functions in terms of sine and cosine, evaluating these functions, graphing tan(x), cot(x), sec(x), csc(x) by finding vertical asymptotes, zeros, and plotting points (to do this, you need to know all solutions to the equations cos(x) = 0, sin(x) = 0). The periods and fundamental cycles of these functions. Recommended exercises: graph tan, cot, sec, and csc without looking anything up, state the vertical asymptotes and x-intercepts of each of them, any of 5-30, 55, 61, 64, 77 5.5 The inverse trig functions Definition of arcsin, arccos, arctan (first, review the definition of an inverse function in 2.5), their domains and ranges, evaluating the inverse trig functions, graphing the inverse trig functions, evaluating compositions of functions (see example 8), the inverse of a general sine or cosine functions (see example 9) Recommended exercises: any of 21-26, 37-52 (choose a good number of these, at least one or two for each of the inverse trig functions), 70, 73, 75, 81, 99, 100 5.6 Right triangle trigonometry Trigonometric rations (”sohcahtoa”), solving a right triangle Recommended exercises: 13, 17, 27, 30, 33, 34 (you may need a calculator for these) (see p. 487 for more exercises from chapter 5) 6.1 Basic trig identities The difference between a trig identity and a (conditional) trig equation, identities from the definitions, reciprocal identities (if you know the definitions of tangent, secant, etc., you can deduce these from the definitions), Pythagorean identities, odd and even identities, verifying that an equation is an identity Recommended exercises: 10, 15, 29, 33, 47, 93, 100, 102 6.2 Verifying identities various tricks for verifying trig identities (see all of the examples in this section, e.g. factoring a trigonometric expression, combining fractions, etc.) Recommended exercises: 28, 30, 38, 48, 55, 58, 61, 63, 64, 74, 81 (really, any of 55-90 are good, get enough practice with these so that you have some intuition for what sorts of tricks work where) 6.3 Sum and difference identities Sine and cosine of a sum or difference, cofunction identities, using the sum/difference identities to evaluate the trig functions at angles that are a sum of angles you know (e.g. 75=45+30, -15=30-45) Recommended exercises: 37, 40, 43, 46, 47, 48 2

6.4 Double and half angle identities: The double angle identities, using them to find sine or cosine or tangent of twice of some angle, the half angle identities, using them to find sine or cosine or tangent of half of some angle (e.g. find sin(22.5)), using these equations to verify identities Recommended exercises: 9, 10, 12, 15, 45, 58, 59, 60 6.5 We skipped this section 6.6 Conditional trigonometric equations Finding all solutions to an equation of the form cos(x) = a, sin(x) = a, or tan(x) = a–find all solutions in the fundamental cycle, then add k times the period of the function to get all of them), the number of solutions to sin(x) = a, cos(x) = a, tan(x) = a in the fundamental cycle (may depend on what a is) Recommended exercises: any of 1-18, 34, 38, 42, 64, 68, 73 (for more exercises from chapter 6, see the review exercises on p.551) 7.1 Law of Sines Types of triangles (AAS, ASA, SSA, SAS, SSS), the law of sines, using the law of sines to solve AAS, ASA, and SSA triangles, figuring out how many triangles are possible in the SSA case and solving them Recommended exercises: any of 7-20, try to find a few of each kind of triangle and solve them (especially SSA-try to do one example of each kind of case that can arise) 7.2 Law of Cosines The law of cosines, using it to solve SSS and SAS triangles, general procedure for solving triangles (see p. 569) Recommended exercises: any of 7-16, try to do at least two each of SSS and SAS.

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