NAME:Math 103 Exam 2 retake12 December 2008100 pointsInstructions:1. This exam has 6 pages (including this one), which contain 8 problems and one bonus problem.Please check that you have all of the pages.2. Answer all of the following questions clearly and completely. Justify all of your answers.3. You may not use a book or any notes for this exam.4. Give your answer to each problem completely and clearly in the space provided. You may usethe back of the exam pages for scratch work; however, if you want this work to be considered,make note of it in the space provided for the problem.5. Erase or cross out work you do not wish to be graded.6. Credit, partial or full, will be given only if sufficient steps leading to the answers are shown.7. You have 50 minutes to complete this exam.Page 1Problem 1. (17 points) Let f (x) = 4x3− 12x2− 7x + 15.(a) (3 points) What is the maximum possible number of zeros f may have? Why?(b) (8 points) Find all possible rational zeros of f . (You do not have to test them to see if they’reactually zeros.)(c) (6 points) Without graphing, prove that f must have a zero somewhere between x = 3 andx = 4.Problem 2. (10 points) Let f (x) = 9x5− 3x + 2 and g(x) = 4 − x. Find (f ◦ g)(x). You do nothave to simplify.Page 2Problem 3. (10 points) Let f be the piecewise-defined function given byf(x) =−x2+ 4, if x < 0;√x, if 0 ≤ x < 4.Graph y = f (x).−11−11−22−22−33−33−44−44−55−55xyProblem 4. (10 points) Give a polynomial function h(x) that satisfies all of the following condi-tions:• The zeros of h are x = 1 and x = −3 (and no other real values of x).• The graph of h crosses the x-axis at x = 1 but just touches the x-axis at x = −3.• The degree of h is at least 5.You do not need to simplify your expression for h.Page 3Problem 5. (16 points) You have a total of 1800 feet of fencing. You want to build a rectangularcorral with two halves, as shown in the picture below.wh(a) (5 points) Write an equation that relates the variables w and h. [ Hint: Use the fact that thetotal length of the fencing should be 1800 feet. ](b) (5 points) Express the total area A of the corral (both halves together) as a function of w.[ Hint: Your answer from part (a) should help you. ](c) (2 points) You should have a quadratic function as your answer to part (b). Its graph is aparabola. Does this parabola open up or open down? Why?(d) (4 points) What dimensions w and h will give you the largest enclosed area?Page 4Problem 6. (20 points) Here we go again. LetR(x) =(x + 4)(−3x + 1)(x − 1)(x + 4).(a) (3 points) What is the domain of R?(b) (3 points) If 0 is in the domain of R, what is the y-intercept of R?(c) (3 points) Write R in lowest terms.(d) (3 points) Find the x-intercept(s) of R.(e) (4 points) Find any vertical asymptotes of R.(f) (4 points) Find the horizontal or oblique asymptote of R.Page 5Problem 7. (5 points) Is the following graph the graph of a function? If so, what function? If not,why not?−2 1 2 3 4 6−3−21245xyProblem 8. (12 points) Solve the following inequality for x.(x − 5)2(x + 3) ≤ 0Bonus problem (+3 points). Explain why every polynomial of odd degree must have at least onereal zero. [ Hint: Think about the end behavior of the graph. ]Page
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