MAT123 Fall 2003 Review for Midterm II

3.6 Understand the vocabulary of rational functions: vertical and horizontal asymptotes. Be able to locate vertical asymptotes by determining the roots of the denominator. Example 3 page 311. Be able to locate horizontal asymptotes by dividing numerator and denominator by the highest power of x in the denominator same example. Box, page 313. Exercises 8,9,10,11. No slant asymptotes on this test! Be able to graph a rational function from its asymptotes and intercepts, using a table to keep track of the sign as x increases past roots of numerator and denominator. Example 3 again, Example 4 p. 313; Exercises 25,27,31,37,39.

4.1 Understand the two behaviors of exponential functions y=ax. When a > 1 and when a < 1 (Box, page 337). Note that the "base" a is ALWAYS A POSITIVE NUMBER. Be familiar with the family of exponential functions in Figure 6 p. 340. Understand how to use transformations to deduce the graphs of complicated functions from simpler ones Example 6 p. 340, Exercises 23,25,27,29. Understand compound interest as an example of an exponential function: if $1 is invested at 5%=.05 interest per year, then after 1 year you have $1.05, after 2 years $(1.05)2, after 3 years $(1.05)3, etc. Understand that when you compound n times a year, you divide the rate by n and multiply the power by n (Box, page 342). Exercises 49,51. Understand that in the limit, as n goes to infinity, you get "continuous compounding" and that this corresponds to multiplying your initial investment by ert, where r is the rate per year and t is the time in years Exercise 51,53.

4.2 Understand the definition of logarithms as inverses of exponentials. logax is the power of a which gives x. That is, y = logax is completely equivalent to x = ay. (Boxes on p.349). Examples 1, 2 p.305, Exercises 1-28.. Know the elementary "properties of logarithms" (Box, p.535), in particular when the base is e and the logarithm is the natural logarithm ln(x). (Box, p.355). Understand how to apply the usual transformations (af(x), f(x+c), f(x)+d) to the graph of y=ln(x). Example 9, Exercises 45-48.

4.3 Understand the Laws of logarithms (Box, p.359) and be able to use them to expand, simplify or evaluate expressions Examples 1-4, Exercises 13,15,17 27. Know how the Change of base formula y = logb(x) = loga(x)/loga(b) can be derived from the exponential form x = by by taking loga of both sides and rearranging, rather than memorizing the Formula (Box at top of p.362) Example 5, Exercises 51,53,55.

4.4 This section gives practical steps for solving exponential equations (Guidelines, p.365) and logarithmic equations (Guidelines, p.368). Always remember to check your answers! Exercises 25, 33, 40.

6.1 Understand the terminology of angles: vertex, initial side, terminal side. Remember that angles are measured counterclockwise (see negative angles on p.476). Be able to convert degrees to radians and vice-versa (Box, p.474). Understand that the radian measure of an angle is the length of the arc the angle intercepts, when the vertex is placed at the center of a circle of radius 1. And that in a circle of radius r the lenths s of the arc intercepted by a central angle of θ radians is s = rθ (Box, p.477). Example 4, Exercises 5,7,11,13.

6.2 Understand the definitions of sinθ, cosθ and tanθ when θ is one of the acute angles in a right triangle (Box, p.483 - top line). We will not stress sec, csc and cot. Be able to do computations like Example 1. Be familiar with the "special triangles" with angles (45o, 45o, 90o) and (30o, 60o, 90o) - know that the latter is half of an equilateral trangle! - and be able to calculate sin, cos, tan of 30o, 45o and 60o from the geometry (see Figs. 5 and 6 p.484 and Table 1 p.485). Be able to "solve" a right triangle given one acute angle and one side or given all three sides. Examples 4, 5, Exercises (sin,cos and tan only) 1,11,31,43,45,51.

6.3 The side angles in a right triangle are acute angles. Know how to generalize the definitions of sin, cos and tan to any angle. Understand that if the angle is in standard position (p.475) then the sign of the three functions depends on which quadrant the terminal edge falls in (Box, p. 494). Be able to use the concept of "reference angle" (Box, p.496) and the sign rule to calculate sin, cos and tan of any angle. Examples 3a, 4a. Understand how to use the identities tanθ = sinθ/cosθ and the "Pythagorean Identity" sin2θ + cos2θ = 1 (Box p. 498; just concentrate on these two). In particular be able to express sinθ in terms of cosθ if you know what quadrant θ is in (Example 5). Know how to calculate the area of a triangle from two sides and the included angle (Box, p.500). Exercises 1,5,35,43,50.

Use Review material starting on p.521 for further practice. Concept Check 1-10a, Exercises 1-12, 15-25, 29-53 (sin, cos and tan only).

Nov 7, 2003