1. (5 points) Use the change-of-base formula to evaluate the following logarithm. Round your answer to four decimal places.

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1 Millersville University Name Answer Key Department of Mathematics MATH 160, Precalculus, Final Examination December 14, 2011, 10:15A-12:15P Please answer the following questions. Answers without justifying work will receive no credit. Partial credit will be given as appropriate, do not leave any problem blank. Point values of exercises are given in parentheses. 1. (5 points) Use the change-of-base formula to evaluate the following logarithm. Round your answer to four decimal places. log = ln0.65 ln16 = (5 points) The half-life of radioactive actinium is years. If initially there are 100 grams of radioactive actinium, how many grams will be left after 15 years? After 15 years, 50 = 100e k(21.77) 1 2 = e k(21.77) ln 1 2 = 21.77k k = ln(1/2) e (15) grams.

2 3. (5 points) Use properties of logarithms to condense the following expression into a single logarithm. 3lnx ln(x+3)+2lny 3lnx ln(x+3)+2lny = lnx 3 ln(x+3)+lny 2 ( ) x 3 y 2 = ln x+3 4. (5 points) Solve the following equation. logx+log(x 15) = 2 2 = logx+log(x 15) 2 = log[x(x 15)] 10 2 = x(x 15) 0 = x 2 15x = (x 20)(x+5) x = 20

3 5. (5 points) Find the equation in slope-intercept form of the line passing through the points (2, 3) and ( 4,9). Slope, Line, m = 9 ( 3) 4 2 = 12 6 = 2. 2 = y ( 3) x 2 2(x 2) = y +3 2x+4 3 = y y = 2x (4 points) The quantity y varies inversely with x and when x = 3/2 quantity y = 32. Find a mathematical model in the form of an equation relating x and y. y = k x 32 = k 3/2 k = 48 y = 48 x

4 7. (1 point each) If f(x) = 1 x and g(x) = 2 x, find the following functions. (a) (f +g)(x) = 1 x +2 x (b) (f g)(x) = 1 x 2 x (c) (f g)(x) = ( ) 1 2 x = 2 x x (d) (f g)(x) = 1 g(x) = 1 2 x (e) (g f)(x) = 2 1 f(x) = 2 x = 2 x

5 8. (5 points) Expand the following product and use the fundamental trigonometric identities to simplify the result. (secx tanx) 2 (secx tanx) 2 = sec 2 x 2secxtanx+tan 2 x = 1+tan 2 x 2secxtanx+tan 2 x = 1+2tan 2 x 2secxtanx 9. (5 points) Find the exact value (no decimal approximations) of cos285. cos285 = cos( ) = cos240 cos45 sin240 sin45 ( = 1 ) ( ) ( )( ) = = 4

6 10. (5 points) Find the exact values (no decimal approximations) of all the solutions of the following equation in the interval [0, 2π). cos 2 x+sinx = 1 1 = cos 2 x+sinx 1 = 1 sin 2 x+sinx 0 = sin 2 x+sinx 0 = sinx(sinx 1) sinx = 0 or sinx = 1 x = 0, π, or π (5 points) Verify the following trigonometric identity. sec 2 xcotx cotx = tanx sec 2 xcotx cotx = tanx cotx(sec 2 x 1) = cotxtan 2 x = (cotxtanx)tanx = tanx =

7 12. (2 points each) Consider the conic section whose equation is (x 5) (a) Find the center of the conic section. Center: (5, 3) (y +3)2 36 = 1. (b) Find the vertices of the conic section. Vertices: (5, 3±6) or (5, 9) and (5,3). (c) Find the foci of the conic section. Since c 2 = a 2 b 2 = = 20 then Foci: (5, 3±2 5). (d) Name this type of conic section. This is the equation of an ellipse.

8 13. (2 points each) Suppose cscθ = 3/2 and cosθ < 0. Find the exact values (no decimal approximations) for the trigonometric functions. sinθ = cosθ = 3 tanθ = 2 5 = cotθ = 2 secθ = 3 5 = cscθ = 3 2

9 14. (5 points) Your football has landed on the roof of Wickersham Hall. When you are 25 feet from the base of the building, the angle of elevation to your football is 50. How high off the ground is your football? Using right triangle trigonometry, tan50 = h 25 h = 25tan feet. 15. (5 points) An airplaneis 90 miles south and120 miles west ofanairport. What bearing should the airplane take to fly toward the airport? Bearing θ = arctan ( )

10 16. (5 points) Use long division to find the quotient and remainder (if any) for the rational expression below. 3x 3 +4x 1 x x x 2 +1) 3x 3 +4x 1 3x 3 3x x Quotient: q(x) = 3x Remainder: r(x) = x 1 x (2 points each) Consider the function f(x) = 2x2 5x 12 x 2 16 (a) Find the x-intercepts (if any) for the function. Note that f(x) = (2x+3)(x 4) so that f(x) = 0 only when 2x+3 = 0. (x+4)(x 4) x-intercept: x = 3 2 (b) Find the y-intercept (if any) for the function. f(0) = = 3 4

11 (c) Find the horizontal asymptotes (if any) for the function. Since the degree of the numerator and the degree of the denominator are the same, the horizontal asymptote is y = 2 1 = 2. (d) Find the vertical asymptotes (if any) for the function. For x 4, f(x) = 2x+3 x+4 thus the vertical asymptote is x = 4.

12 18. (5 points) Find all the zeros of the following function. f(x) = 3x 3 +14x 2 7x 10 Therationalzerosofthefunctionmustbeintheset{±1,± 1 3,±2,±2 3,±5,±5 3,±10,±10 3 }. Note that f(1) = 0 so x 1 divides f(x). f(x) = (x 1)(3x 2 +17x+10) = (x 1)(3x+2)(x+5) 0 = (x 1)(3x+2)(x+5) The zeros are x = 1, x = 5, and x = 2/3.

13 Equations of the Conic Sections Parabola (x h) 2 = 4p(y k) (vertical axis) x = h (axis) y = k p (directrix) (h, k) (vertex) (h,k +p) (focus) (y k) 2 = 4p(x h) (horizontal axis) y = k (axis) x = h p (directrix) (h, k) (vertex) (h+p,k) (focus) Ellipse For both standard forms assume 0 < b < a. (x h) 2 a 2 + (y k)2 = 1 (major axis horizontal) b 2 c 2 = a 2 b 2 (h, k) (h±c,k) (center) (foci) (h ± a, k) (vertices) (x h) 2 b 2 + (y k)2 = 1 (major axis vertical) a 2 c 2 = a 2 b 2 (h, k) (h,k ±c) (h,k ±a) (center) (foci) (vertices) Hyperbola (x h) 2 a 2 (y k)2 = 1 (axis horizontal) b 2 c 2 = a 2 +b 2

14 (h, k) (h,k ±c) (h,k ±a) (center) (foci) (vertices) y = k ± b (x h) (asymptotes) a (y k) 2 a 2 (x h)2 b 2 (h, k) (h±c,k) = 1 (axis vertical) c 2 = a 2 +b 2 (center) (foci) (h ± a, k) (vertices) y = k ± a (x h) (asymptotes) b