MATH 115 Precalculus Fall, 2016, V1.4 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 3x 17 . 1. ______ [-10/3, 8] [-8, 10/3] (-, -10/3] (-, -10/3] [8, ) 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. B. C. D. f (x) = 5x4 - x3 - x2 - 5 f (x) = -2x3 + 8x2 - x + 3 f (x) = -7x6 + 3x2 + x f (x) = 6x3 + 2x2 - x - 1 vvvv 3. Express as an equivalent expression: 3 log x + log 1 log (z 4) A. B. 3. ________ log 3 + 5 C. log D. log Page 1 of 7 MATH 115 Precalculus 4. Determine the interval(s) on which the function is increasing. A. (-1, 3) B. (- 1.3, 1.3) C. , 1 and 3, D. (-2.5, 1) and 4.5, 5. Which of the functions corresponds to the graph? Fall, 2016, V1.4 4. ______ 5. ______ A. f ( x ) = e x 1 B. f ( x ) = 1 e x C. f ( x ) = 2 e x D. f ( x ) = e x Page 2 of 7 MATH 115 Precalculus Fall, 2016, V1.4 6. Which of the functions corresponds to the graph? 6. ______ A. f (x) = 2(1 - cos x) B. f (x) = 2 + cos x C. f (x) = 3 - cos x D. f (x) = 2 - sin x SHORT ANSWER: 7. Points (-1, 2) and (-7, 6) 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. A salesperson earns a base salary of $1,540 per month and a commission of 7.6% on the amount of sales. If the salesperson has a paycheck of $4,819.40 for one month, what was the amount of sales for the month? Answer: _____________ Page 3 of 7 MATH 115 Precalculus Fall, 2016, V1.4 10. Water initially at 190 F. is left in a room of temperature 70 F to cool. - 0.096 t After t minutes, the temperature T of the water is given by T(t) = 70 + 120e Find the temperature of the water 10 minutes after it is left to cool. (Round to the nearest degree.) Answer: ________ 1 11. Given the function = 4 , find a formula for the inverse function. 5 Answer: __________________ 12. (a) State the reference angle associated with 135. Answer: ________ (b) Convert 135 to radians. Leave the answer in terms of . Answer: ________ 13. Given y = 6 sin(4x - ), state the (a) period Answer: ________ (b) phase shift Answer: ________ 14. Solve the trigonometric equation (cos x)(2sin x + 1) = 0 in the interval [0, 360). Answer: _________________ 15. Find the exact value of arcsin sin (b) Find the exact value of arccos tan (a) Answer: ________ Answer: ________ 2 16. For the parabola given by (y + 5) = 8(x - 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, 2016, V1.4 3x 2 + 3 . x2 4 (a) State the domain. Answer: _________________ (b) State the horizontal asymptote. Answer: _________________ (c) State the vertical asymptote(s). Answer: _________________ (d) Which of the following represents the graph of f ( x) = 3x 2 + 3 ? x2 4 GRAPH A. GRAPH B. GRAPH C. GRAPH D. Answer: ______________ Page 5 of 7 MATH 115 Precalculus Fall, 2016, V1.4 SHORT ANSWER, with work required to be shown, as indicated. 18. Find the equation for a line which passes through the points (4, -3) and (6, - 7). Write the equation in slope-intercept form. Show work. 19. Find the exact solutions and simplify as much as possible: 2x2 = 6x + 3. Show work. 2 20. Let f (x) = 9x - 5 and g(x) = x - 2. (a) Find the composite function ( f o g )( x) and simplify. Show work. (b) Find ( f o g )(1) . Show work. 21. A projectile is launched from a platform 24 feet high with an initial velocity of 80 feet per second. The height h of the projectile at t seconds after launch is given by h = -16t2 + 80t + 24 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. = 0. Show work. 23. Suppose that cos = 4/5 and that is a Quadrant IV angle. (a) Find the exact value of sin . Show work. (b) Find the exact value of sin(2 ). Show work. 2 24. Prove the identity 1 - (sin x - cos x) = sin(2x) Page 6 of 7 MATH 115 Precalculus Fall, 2016, V1.4 25. From a point 42 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 55.7. 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 = 44, B = 62, and c = 30.0. Find the length of side a, rounded to the nearest tenth. Show work. = 6, -12 and = 8, 4. 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 Math 115 Final Examination Fall, 2016 1 Math 115 PreCalculus Name______________________________ Final Examination: Fall, 2016 Instructor __________________________ Answer Sheet Instructions: 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. Record your answers and work in this document. There are 28 problems. Problems #1-6 are multiple choice. Record your choice for each problem. Problems #7-17 are short answer. Record your answer for each problem. Problems #18-28 are short answer with work required. When requested, show all work and write all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either case, show work neatly and correctly, following standard mathematical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may attach extra pages. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Your exam will receive a zero grade unless you complete the following honor statement. Please sign (or type) your name below the following honor statement: I have completed this final examination myself, working independently and not consulting anyone except the instructor. I have neither given nor received help on this final examination. Name _____________________ Date___________________ Math 115 Final Examination Fall, 2016 MULTIPLE CHOICE. Record your answer choices. 1. 2. 3. 4. 5. 6. SHORT ANSWER. Record your answers below. 7. (a) (b) (c) 8. 9. 10. 11. 12. (a) (b) 13. (a) (b) 2 Math 115 Final Examination Fall, 2016 14. 15. (a) (b) 16. (a) (b) (c) 17. (a) (b) (c) (d) SHORT ANSWER with Work Shown. Record your answers and work. Proble m Number Solution Answer: Work: 18 3 Math 115 Final Examination Answer: Work: 19 Answers: (a) (b) Work for both parts: 20 Fall, 2016 4 Math 115 Final Examination Answers: (a) (b) Work for both parts: 21 Answer: Work: 22 Fall, 2016 5 Math 115 Final Examination Answers: (a) (b) Work for both parts: 23 Proof: 24 Fall, 2016 6 Math 115 Final Examination Answer: Work: 25 Answer: Work: 26 Fall, 2016 7 Math 115 Final Examination Answers: (a) (b) Work for both parts: 27 Answers: (a) (b) Work for part (b): 28 Fall, 2016 8