Practice Test 3 (longer than the actual test will be) 1. Solve the following inequalities. Give solutions in interval notation. (Expect 1 or 2.

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1 MAT 115 Spring 2015 Practice Test 3 (longer than the actual test will be) Part I: No Calculators. Show work. 1. Solve the following inequalities. Give solutions in interval notation. (Expect 1 or 2.) a. x 2 7x 18 0 b. x 3 16x 2 0 c. x 3 16x < 0 2. Draw the graph of the function y = log 2 x. Plot the x-intercept and three additional points. Include any asymptotes.

2 3. State the following trig identities: a. Pythagorean Identities b. Even and odd identities for sine, cosine, and tangent: sin(-x) = cos(-x) = tan (-x) = c. Cofunction identites for sine and cosine: sin ( π 2 θ) = cos (π 2 θ) = d. Double-angle identities for sine and cosine: sin(2x) = cos(2x) = cos(2x) = cos(2x) = Extra credit: be able to solve for sin 2 x and cos 2 x in terms of cos(2x). 4. Solve the following trig equations. Give all solutions in the interval [0, 2π). a. 2 cos 2 x + cos x 1 = 0 b. sin 2 x sin x = 0

3 5. State the range of the following inverse trig functions using interval notation. a. sin -1 (x) b. cos -1 (x) c. tan -1 (x) 6. Evaluate each of the following inverse trig functions. (Expect 6-8 on the test.) a. sin -1 (1) b. cos -1 (1) c. tan -1 (1) d. sin -1 (0) e. cos -1 (0) f. tan -1 (0) g. sin -1 ( 1) h. cos -1 ( 1) i. tan -1 ( 1) j. sin -1 ( 1 2 ) k. cos-1 ( 1 2 ) l. tan-1 ( 1 3 ) m. sin -1 ( 1 2 ) n. cos-1 ( 1 2 ) o. tan-1 ( 1 3 ) p. sin -1 ( 3 2 ) q. cos-1 ( 3 2 ) r. tan-1 ( 3) s. sin -1 ( 3 2 ) t. cos-1 ( 3 2 ) u. tan-1 ( 3) v. sin -1 ( 2 2 ) w. sin-1 ( 2 2 ) x. cos-1 ( 2 2 )

4 7. Consider the polynomial function y = x 3 + 8x x. (Expect one of these on the test.) a. Draw arrows to indicate its end behavior. b. Find the y-intercept. c. Find the x-intercept(s). State their multiplicities, and whether the graph crosses or touches and bounces off the x-axis at each. d. Use the information above to sketch the graph of the polynomial. 8. Consider the polynomial function y = x 4 10x x 2. a. Draw arrows to indicate its end behavior. b. Find the y-intercept. c. Find the x-intercept(s). State their multiplicities, and whether the graph crosses or touches and bounces off the x-axis at each. d. Use the information above to sketch the graph of the polynomial.

5 9. Evaluate each of the following: (There will be about ten of these on the test.) a. 2-4 b. 9 3/2 c. 27 4/3 d. 9-1/2 e f. 2 1/2 g. 5 0 h. e 0 i. e 1 j. log 2 16 k. log 9 3 l. log 2 (. 5) m. log 2 1 n. log 2 0 o. log 2 ( 4) p. log 2 ( 1/2) q. log 36 6 r. log 6 (1/36) s. log(.001) t. log 1 u. log 10,000 v. ln 0 w. ln 1 x. ln e 10. Let f(x) = x and g(x) = 4x + 3. a. Find the function f(g(x)). b. Find the domain of f(g(x)), and state it in interval notation.

6 11. If f (g(h(x))) = ln(sec x), what are f(x) and g(x)? f(x) = g(x) = 12. Consider the rational function y = 2x+12 x 4. (Expect one rational function on the test.) a. Find the domain, and state it using interval notation. b. Find the y-intercept. c. Find the x-intercept(s). Does the graph crosses or touches and bounces off the x-axis? d. Find any vertical asymptote(s). e. Determine the end behavior (horizontal asymptote). f. Use a number line and check signs on either side of vertical asymptotes and x-intercepts to determine where y is positive and where y is negative. g. Use all of the above information to sketch the graph of the function.

7 13. Consider the rational function y = x2 36 x 2 9. (There will be one of these problems.) a. Find the domain, and state it using interval notation. b. Find the y-intercept. c. Find the x-intercept(s). Does the graph crosses or touches and bounces off the x-axis? d. Find any vertical asymptote(s). e. Determine the end behavior (horizontal asymptote). f. Use a number line and check signs on either side of vertical asymptotes and x-intercepts to determine where y is positive and where y is negative. g. Use all of the above information to sketch the graph of the function.

8 Part II: Calculators Allowed. Show work. 14. Find the x 6 term in the expansion of (2x 3) Use the graphing calculator, synthetic division, and the quadratic formula to find all real and complex zeros of the polynomial 2x 3 10x x 22.

9 16. Determine the future value of a $20,000 investment at an annual interest rate of 3.5% compounded quarterly over 15 years. Give your answer rounded to the nearest cent. 17. Determine the doubling time of an investment at an annual interest rate of 2.75% compounded continuously. Give your answer rounded to the nearest.01 year. 18. A city s population is growing exponentially. If the population was 12,000 in 2000 and 14,500 in 2010, find the function P(t) = P 0 e kt that represents its population as a function of time, with t = time in years, with t = 0 representing the year 2000, rounding the value of k to 3 significant digits. Then use the function to estimate the population in 2015.

10 19. The radioactive isotope Yttrium-90 is used for the treatment of non-hodgkins lymphomia. It has a half-life of 64.0 hours. Find the value of its decay constant k in the radioactive decay function N(t) = N 0 e k t, rounded to three significant digits.

the input values of a function. These are the angle values for trig functions

the input values of a function. These are the angle values for trig functions SESSION 8: TRIGONOMETRIC FUNCTIONS KEY CONCEPTS: Graphs of Trigonometric Functions y = sin θ y = cos θ y = tan θ Properties of Graphs Shape Intercepts Domain and Range Minimum and maximum values Period