MATH 115 Precalculus Spring, 2015, V1.2

Transcription

1 MULTIPLE CHOICE 1. Solve, and express the answer in interval notation: 6 5x A. (, 2] [2/5, ) B. [2/5, 2] C. (, 2/5] D. (, 2/5] [2, ) 2. Which of the following polynomials has a graph which exhibits the end behavior of upward to the left and upward to the right? 2. A. f (x) = 3x 6 x 4 x 2 5 B. f (x) = 6x 3 + 9x 2 x + 3 C. f (x) = 5x 4 + 3x 2 + x D. f (x) = 2x 3 + 4x 2 x 1 vvvv 3. Express in equivalent form: 3 log (x + 2) + log 1 log z 3. A. log ( x + 8 ) z 6 log(x) + 1 B. log(z) (x + 2) 3 C. log ( ) z D. log (3x + 7 z) 3 Page 1 of 7

2 4. Determine the interval(s) on which the function is increasing. 4. A. (, 1) B. ( 2, 2) C. ( 4.5, 1) and (2.5, ) D. (, 3) and (1, ) 5. Which of the functions corresponds to the graph? 5. A. f ( x) = e x + 2 B. f ( x) = e x + 3 C. f ( x) = e x + 3 D. f ( x) = e x + 4 Page 2 of 7

3 6. Which of the functions corresponds to the graph? 6. A. f (x) = 2 cos x + 1 B. f (x) = cos x + 2 C. f (x) = sin(2x) + 1 D. f (x) = 2 sin x + 1 SHORT ANSWER: 7. Points ( 2, 2) and ( 8, 6) are endpoints of the diameter of a circle. (a) What is the exact length of the diameter? (Simplify as much as possible) (b) What is the center of the circle? (c) What is the equation of the circle? 8. Find the value of the logarithm: log 2 ( ) A salesperson earns a base salary of $1,620 per month and a commission of 6.9% on the amount of sales made. If the salesperson has a paycheck of $3, for one month, what was the amount of sales for the month? Page 3 of 7

4 10. A can of soda at 80 F. is placed in a refrigerator that maintains a constant temperature of 37 F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by T(t) = e t Find the temperature of the soda 10 minutes after it is placed in the refrigerator. (Round to the nearest degree.) 11. Given the function f(x) = 1 x + 2, find a formula for the inverse function. 6 3 _ 12. (a) State the reference angle associated with 267. (b) Convert 267 to radians. Leave the answer in terms of π. 13. Given y = 5 sin(6x π), state the (a) period (b) phase shift 14. Solve the trigonometric equation (cos x)(2sin x 1) = 0 in the interval [0, 360 ). 15. (a) Find the exact value of sin -1 (sin Srr ) 6 (b) Find the exact value of cos -1 (tan 7rr ) For the parabola given by (x 3) 2 = 8(y + 2), find the following: (a) direction parabola opens (to the left, right, up, or down) (b) vertex (c) focus Page 4 of 7

5 17. Let f (x) = 2 x. 3 + x (a) State the domain. (b) State the vertical asymptote(s). (c) State the horizontal asymptote. (d) Which of the following represents the graph of f (x) = 2 x 3 + x? GRAPH A. GRAPH B. GRAPH C. GRAPH D. Page 5 of 7

6 SHORT ANSWER, with work required to be shown, as indicated. 18. Find the equation for a line which passes through the points (8, 7) and ( 2, 9). Write the equation in slope-intercept form. 19. Find the exact solutions and simplify as much as possible: 25x = 20x. 20. Let f (x) = 5x 2 9 and g(x) = x 2. (a) Find the composite function ( f o g)(x) and simplify. (b) Find ( f o g)( 2). 21. A projectile is launched from a platform 30 feet high with an initial velocity of 112 feet per second, The height h of the projectile at t seconds after launch is given by h = 16t t + 30 feet. (a) How many seconds after launch does the projectile attain maximum height? (b) What is the maximum height? x Solve: + = 0. x + 2 x Suppose that sin θ = 12/13 and that θ is a Quadrant II angle. (a) Find the exact value of cos θ. (b) Find the exact value of sin 2θ. 24. Prove the identity (sin x + cos x) 2 = 1 + sin(2x) Page 6 of 7

7 25. From a point 47 feet from the base of a redwood tree, the angle of elevation to the top of the tree is Find the height of the tree to the nearest tenth of a foot. (sketch is not to scale) 26. For the triangle ABC, we are given that A = 42, B = 58, and c = Find the length of side b, rounded to the nearest hundredth. 27. Let u = 4, 8 and v = 4, 2. (b) (c) Calculate the dot product u v. Determine the angle between u and v. Round the result to the nearest degree. 28. An ellipse has the equation (x - S) 2 + 2S (y + 1) 2 = 1 81 (a) Is the major axis horizontal or vertical? (b) Find the exact values of the foci of the ellipse. Page 7 of 7