MATH 115 MIDTERM EXAM NAME: _______________________________ November, 2016 Instructor: S. Sands I have completed this assignment myself, working independently and not consulting anyone except the instructor. INSTRUCTIONS The exam is worth 100 points. There are 13 problems. This exam is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your exam no later than Sunday, November 20. Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the exam, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work. If you have any questions, please contact me by e-mail. 1. (6 pts) (A) (B) (C) (D) (a) Which of these graphs represents a function? Answer(s): ___________ (b) Which of these graphs represents an even function? Answer(s): ____________ (c) Which of these graphs represents a one-to-one function? Answer(s): ____________ 2. (3 pts) Convert to a logarithmic equation: 6x = 1296. A. log 6 x=1296 B. log x 6=1296 C. log x 1296=6 D. log 6 1296=x (no explanation required) 3. (3 pts) Which of the following is TRUE about the line through the points (2, -5) and (-2, -5)? (no explanation required) A. B. The slope is negative. The slope is 0. 2. ______ 3. _______ C. D. The slope is positive. The slope is undefined. 4. (10 pts) Emily is a salesperson who must decide between two monthly income options. Option A: Salary of $1040 per month, plus 7.2% of monthly sales (in dollars) or Option B: Salary of $1425 per month, plus 5.6% of monthly sales (in dollars) (a) State a linear function f (x) that represents the Option A total income, given monthly sales of x dollars. (b) State a linear function g(x) that represents the Option B total income, given monthly sales of x dollars. (c) Emily wants to determine the following: For what amount of monthly sales is Option A the better choice rather than Option B? To answer the question, should you solve the inequality f (x) > g(x) or the inequality f (x) < g(x) ? Write the appropriate inequality, show how to solve it algebraically, and state your answer to the question in a sentence. 5. (10 pts) Let f (x) = x2 - 6x + 2 and g(x) = 4x - 7. (a) Find the composite function ( f o g )( x ) and simplify the results. Show work. (b) Find ( f o g )(2 ) . Show work. 6. (8 pts) Let 5 x 2 20 f ( x) 2 x 36 . (no explanations required) (a) State the x-intercept(s). (b) State the y-intercept(s). (c) State the vertical asymptote(s). (d) State the horizontal asymptote. f ( x )= 7. (10 pts) Let (a) Find f 1 3 x5 2 x9 , the inverse function of f. Show work. (b) What is the domain of f ? (c) What is the domain of the inverse function? 8. (6 pts) The graph below represents a rational function y = f (x) . (no explanations required) (a) State the horizontal asymptote(s). (b) State the vertical asymptote(s). (c) The graph represents which of the following functions? Choice: _____ A. f ( x )= x 24 x 23 x4 B. f ( x )= x24 x 2+3 x4 f ( x )= x2 x 3 x4 f ( x )= x 2 x +3 x4 C. D. 2 2 9. (6 pts) The graph below represents a polynomial function y = f (x) . (no explanations required) (a) Is the degree of the polynomial odd or even? (b) State the interval(s) on which the function is decreasing. (c) State the value of f (2). 10. (6 pts) Solve the equation. Check all proposed solutions. Show work in solving, and state your final conclusion. x 6 27 2 0 x 9 x 9x 11. (6 pts) Solve the equation. Check all proposed solutions. Show work in solving, and state your final conclusion. 4 x +33=x +7 12. (9 pts) Let f (x) = 2e - x 1. Answers can be stated without additional work/explanation. (a) Which describes how the graph of f can be obtained from the graph of y = ex ? Choice: ________ A. Reflect the graph of B. Reflect the graph of C. Stretch the graph of D. Stretch the graph of y = ex across the x-axis, stretch vertically by a factor of 2, and shift downward by 1 unit. y = ex across the y-axis, stretch vertically by a factor of 2, and shift downward by 1 unit. y = ex vertically by a factor of 2, shift to the left by 1 unit, and shift downward by 1 unit. y = ex vertically by a factor of 2, shift to the right by 1 unit, and shift downward by 1 unit. (b) What is the y-intercept? (c) Which is the graph of f ? GRAPH A (below) GRAPH C (below) GRAPH B (below) GRAPH D (below) 13. (17 pts) On a particular fall day, the outdoor temperature was recorded periodically and a quadratic function was used to model the data. y = f (t) = 0.28t2 + 7.56t + 23.97 for 0 t 24 where t = time of day (in hours) and y = temperature (in degrees) REMARK: Time is measured in hours elapsed since midnight. For instance, t = 4 means 4 am. t = 22 means 10 pm. (a) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and find the time when it occurred. Report the time to the nearest quarter hour (i.e., __:00 or __:15 or __:30 or __:45). (For instance, a time of 18.25 hours is reported as 6:15 pm in the usual time notation.) Report the maximum temperature to the nearest tenth of a degree. Show algebraic work. (b) Find and interpret the average rate of change of f over the interval [16, 20]. Show work. (c) Use the quadratic polynomial y = 0.28t2 + 7.56t + 23.97 together with algebra to approximate the time(s) of day when the outdoor temperature y was 68 degrees. That is, solve the quadratic equation 68 = 0.28t2 + 7.56t + 23.97 . Show work in solving. (Show how the quadratic formula is used, with the appropriate values substituted.) Round the results to the nearest tenth. Write a concluding sentence to report the time(s) to the nearest quarter-hour, in the usual time notation. (Use more paper as needed.)