MATH 115 Precalculus Fall, 2017, V3.4 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: | 8 - 7x | 6. A. B. C. D. 1. ______ (-, 2] [2/7, ) (-, 2/7] [2, ) (-, 2/7] [2/7, 2] 2. Which of the following polynomials has a graph which exhibits the end behavior of downward to the left and upward to the right? 2. ______ A. B. C. D. f (x) = - 4x3 + x + 8 f (x) = - 3x4 - 7x2 - x f (x) = 9x6 + 2x3 - 4x - 1 f (x) = 6x5 - 5x4 - 1 3. Write as an equivalent expression: A. 6 log( x) log(4 y ) B. x2 + 9 log 4 y C. ( x + 3) 2 log y4 D. log ( 2 x + 7 4 y ) vvvv 2 log (x + 3) - 4 log y + log 1 3. _______ Page 1 of 7 MATH 115 Precalculus 4. Determine the interval(s) on which the function is increasing. A. (-, - 4.5) and (-1, 2.5) B. (-1, 2.5) C. (- 2, 2) D. (-3, 1) 5. Which of the functions corresponds to the graph? Fall, 2017, V3.4 4. ______ 5. ______ A. f ( x ) = 2 + e x B. f ( x ) = 2 e x C. f ( x ) = 2e x D. f ( x ) = e2x Page 2 of 7 MATH 115 Precalculus Fall, 2017, V3.4 6. Which of the functions corresponds to the graph? 6. ______ A. f (x) = sin(2x) + 1 B. f (x) = 2 sin x + 1 C. f (x) = 2 cos x + 1 D. f (x) = 2 - cos x SHORT ANSWER: 7. Points (-7, 1) and (-5, 9) 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: ___________________________ \u0007 8. Find the value of the logarithm: log \u0005 \u0006 . \b Answer: ____________ 9. A salesperson earns a base salary of $1,540 per month and a commission of 7.6% on the amount of sales made. If the salesperson has a paycheck of $4,371 for one month, what was the amount of sales for the month? Answer: _____________ Page 3 of 7 MATH 115 Precalculus Fall, 2017, V3.4 10. A can of soda at 78 F. is placed in a refrigerator that maintains a constant temperature of 40 F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by T(t) = 40 + 38 e - 0.058 t Find the temperature of the soda 10 minutes after it is placed in the refrigerator. (Round to the nearest tenth of a degree.) Answer: _________ 4 9 11. Given the function \u000e = 3 2 , find a formula for the inverse function. Answer: __________________ 12. (a) State the reference angle associated with 240. Answer: ________ (b) Convert 240 to radians. Leave the answer in terms of . Answer: ________ 13. Given y = 3 sin(6x - ), state the (a) period Answer: ________ (b) phase shift Answer: ________ 14. Solve the trigonometric equation \u00152 cos \u000e 1\u001a\f2 cos \u000e + 1\u000e = 0 in the interval [0, 360). Find the exact values of the solutions in degrees. Answer: _________________ 15. (a) Find the exact value of arccos \u0006sin (b) Find the exact value of arcsin \u0006sin !" \u0005 \u0005" ! Answer: ________ Answer: ________ 2 16. For the parabola given by (x - 5) = 8(y + 2), 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, V3.4 2 x 2 18 . x 2 25 (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) = 2 x 2 18 ? x 2 25 GRAPH A. GRAPH B. GRAPH C. GRAPH D Answer: ______________ Page 5 of 7 MATH 115 Precalculus Fall, 2017, V3.4 SHORT ANSWER, with work required to be shown, as indicated. 18. Find the equation for a line which passes through the points (1, -7) and (5, -9). Write the equation in slope-intercept form. Show work. 2 19. Find the exact solutions and simplify as much as possible: 3x + 2 = 8x. Show work. 2 20. Let f (x) = 3x + 7 and g(x) = x 3. (a) Find the composite function ( f o g )( x ) and simplify. Show work. (b) Find ( f o g )( 2) . Show work. 21. A projectile is launched from a platform 20 feet high with an initial velocity of 144 feet per second, The height h of the projectile at t seconds after launch is given by h = -16t2 + 144t + 20 feet. (a) How many seconds after launch does the projectile attain maximum height? (b) What is the maximum height? 22. Solve: #$\b #$ + \u0007\b # % $ \u0007\b = 0. Show work. Show work. Show work. 23. Suppose that cos = 1/5 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 - (cos x - sin x) = sin(2x) Page 6 of 7 MATH 115 Precalculus Fall, 2017, V3.4 25. From a point 65 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 49.3. 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 = 48, B = 62, and c = 35.00. Find the length of side b, rounded to the nearest hundredth. Show work. '( = 9, 6 and ) '( = 2, 3. 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 # $ +\u000e% + , - !\u000e% \u0007\b =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