MATH 115 Precalculus Fall, 2017, V1.1 MATH 115 FINAL EXAMINATION This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Record your answers and work on the separate answer sheet provided. There are 28 problems. Problems #1-6 are Multiple Choice. Problems #7-17 are Short Answer. (Work not required to be shown) Problems #18-28 are Short Answer with work required to be shown. MULTIPLE CHOICE 1. Solve, and express the answer in interval notation: A. B. C. D. 7 4x 5 . 1. ______ (-, 1/2] [3, ) (-, 1/2] [1/2, 3] [1/2, ) 2. Which of the following polynomials has a graph which exhibits the end behavior of upward to the left and downward to the right? 2. _______ vvvv A. B. C. D. 3 f (x) = 4x + x + 8 f (x) = -5x6 - 9x2 - 5 f (x) = 7x4 - 3x2 - x f (x) = -8x5 + 7x3 - x - 5 3. Write as an equivalent expression: log x + log 1 - 3 log y A. log ( x + 1 3 y ) B. x +1 log 3y C. x +1 log y D. x log 3 y 3. ______ 3 Page 1 of 7 MATH 115 Precalculus 4. Determine the interval(s) on which the function is decreasing. A. (-, - 4) and (1, ) B. (- 4, 1) C. (- 5.5, -2) D. (-, -2) 5. Which of the functions corresponds to the graph? Fall, 2017, V1.1 4. ______ 5. ______ A. f ( x ) = e x B. f ( x ) = e x 1 C. f ( x ) = e x + 1 D. f ( x ) = e x + 1 Page 2 of 7 MATH 115 Precalculus Fall, 2017, V1.1 6. Which of the functions corresponds to the graph? 6. ______ A. f (x) = cos(2x) + 2 B. f (x) = sin x + 3 C. f (x) = 2 cos x + 1 D. f (x) = 3 - sin x SHORT ANSWER: 7. Points (1, -3) and (7, 1) are endpoints of the diameter of a circle. (a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________ (b) What is the center of the circle? Answer: ____________ (c) What is the equation of the circle? Answer: ___________________________ 8. Find the value of the logarithm: log . Answer: ____________ 9. Paul, a resident of Metropolis, pays Metropolis an annual tax of $50 plus 1.7% of his annual income. If Paul paid $1,019 in tax, what was Paul's income? Answer: _____________ Page 3 of 7 MATH 115 Precalculus Fall, 2017, V1.1 10. A can of soda at 82 F. is placed in a refrigerator that maintains a constant temperature of 35 F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by T(t) = 35 + 47 e - 0.058 t Find the temperature of the soda 20 minutes after it is placed in the refrigerator. (Round to the nearest tenth of a degree.) Answer: ________ 5 1 11. Given the function = , find a formula for the inverse function. 6 4 Answer: __________________ 12. (a) State the reference angle associated with 225. Answer: ________ (b) Convert 240 to radians. Leave the answer in terms of . Answer: ________ 13. Given y = 5 sin(6x - ), state the (a) period Answer: ________ (b) phase shift Answer: ________ 14. Solve the trigonometric equation (cos x)(2 sin x + 1) = 0 in the interval [0, 360). Answer: _________________ 15. (a) Find the exact value of arccos tan (b) Find the exact value of arcsin sin Answer: ________ Answer: ________ 2 16. For the parabola given by (x - 1) = -8(y - 3), find the following: (a) direction parabola opens (to the left, right, up, or down) Answer: ___________ (b) vertex Answer: ___________ (c) focus Answer: ___________ Page 4 of 7 MATH 115 Precalculus 17. Let f ( x) = Fall, 2017, V1.1 2 x . 3+ x (a) State the domain. Answer: _________________ (b) State the vertical asymptote(s). Answer: _________________ (c) State the horizontal asymptote. Answer: _________________ (d) Which of the following represents the graph of f ( x) = 2 x ? 3+ x GRAPH A. GRAPH B. GRAPH C. GRAPH D. Answer: ______________ Page 5 of 7 MATH 115 Precalculus Fall, 2017, V1.1 SHORT ANSWER, with work required to be shown, as indicated. 18. Find the equation for a line which passes through the points (4, -1) and (8, -9). Write the equation in slope-intercept form. Show work. 19. Find the exact solutions and simplify as much as possible: 3x2 + 6 = 10x. Show work. 2 20. Let f (x) = 4x + 8 and g(x) = x 1. (a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g )(3) . Show work. 21. A projectile is launched from a platform 10 feet high with an initial velocity of 48 feet per second, The height h of the projectile at t seconds after launch is given by h = -16t2 + 48t + 10 feet. (a) How many seconds after launch does the projectile attain maximum height? (b) What is the maximum height? 22. Solve: Show work. Show work. x + 10 32 + 2 = 0 . Show work. x+2 x 4 23. Suppose that sin = 5/13 and that is a Quadrant II angle. (a) Find the exact value of cos . Show work. (b) Find the exact value of sin(2 ). Show work. 2 24. Prove the identity (cos x + sin x) - 1 = sin(2x) Page 6 of 7 MATH 115 Precalculus Fall, 2017, V1.1 25. From a point 48 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 52.3. Find the height of the tree to the nearest tenth of a foot. Show work. (sketch is not to scale) 26. For the triangle ABC, we are given that A = 40, B = 63, and c = 38.0. Find the length of side b, rounded to the nearest hundredth. Show work. = 9, 3 and = 2, 6. 27. Let (b) . Show work. Calculate the dot product (c) and . Round the result to the nearest degree. Show work. Determine the angle between 28. An ellipse has the equation + =1 (a) Is the major axis horizontal or vertical? (b) Find the exact values of the foci of the ellipse. Show work. Page 7 of 7