Math 107 questions# 4 Spring 2017 Instructor: Student Name: _______________________________ I have completed this assignment myself, working independently and not consulting anyone except the instructor. INSTRUCTIONS The quiz is worth 100 points. There are 13 problems, and one bonus question. This quiz 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 quiz no later than Sunday, 19 February. 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 quiz, 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. General quiz tips and instructions for submitting work are posted in the Quizzes module. Read each question carefully. If you have any questions, please feel free to contact me. 1. (4 pts) Solve the inequality x2 3x and write the solution set in interval notation. Show work. 2. (4 pts) Solve x +5 x 6 x +8 0 and write the solution set in interval notation. Show work. 2 3. (4 pts) For f (x) = x3 + 3x2 - 6, use the Intermediate Value Theorem to determine which interval must contain a zero of f. Show work. A. B. C. D. Between 0 and 1 Between 1 and 2 Between 2 and 3 Between 3 and 4 4. (4 pts) Translate this sentence about volume into a mathematical equation. The volume V of wood is directly proportional to the height h and the square of the girth g. 5. (8 pts) Look at the graph of the quadratic function and complete the table. No explanations required. Graph Fill in the blanks Equation State the vertex: The graph represents which of the following equations? ____________ Choice:____ State the range: _____________ A. y = -x2 + 2x + 6 State the interval on which the function is decreasing: B. y = -2x2 - 4x + 6 C. y = 2x2 + 4x - 6 _____________ D. y = x2 - 2x - 6 6. (6 pts) Each graph below represents a polynomial function. Complete the following table. No explanations required. Graph Graph A Graph B Is the degree of the polynomial odd or even? (choose one) Is the leading coefficient of the polynomial positive or negative? (choose one) How many real number zeros are there? 7. (12 pts) Let P ( x )=x 34.5 x 20.5 x +6. When factored, ( 32 ) ( x+ 4 ) P ( x )=( x1 ) x + . (a) State the domain. (b) Which sketch illustrates the end behavior of the polynomial function? A. B. vvvv C. vvvv D. vvvv (c) State the y-intercept: (d) State the real zeros: (e) State which graph below is the graph of P(x). vvvv Answer: ______ 8. (8 pts) Let GRAPH A (below) GRAPH B (below) GRAPH C (below) GRAPH D (below) 2x2 2 f ( x) 2 x 4 . (a) State the y-intercept. (b) State the x-intercept(s). (c) State the vertical asymptote(s). (d) State the horizontal asymptote. 9. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion. x 1 6 2 0 x 2 x 2x 10. (8 pts) Which of the following functions is represented by the graph shown below? Explain your answer choice (provide some work or state a theorem used to draw your conclusion). Be sure to take the asymptotes into account in your explanation. 10. _____ 2 A. B. C. f ( x )= x x +3 x f ( x )= x x 9 f ( x )= 2x x 3 x 2 2 2 D. f ( x )= 2 x 9 2 11. (8 pts) For z = 2 5i and w = 1 6i, find z/w. That is, determine 25 i 16 i and simplify as much as possible, writing the result in the form a + bi, where a and b are real numbers. Show work. 12. (8 pts) Consider the equation 25x2 + 1 = 6x. Find the complex solutions (real and non-real) of the equation, and simplify as much as possible. Show work. 13. (18 pts) The cost, in dollars, for a company to produce x widgets is given by C(x) = 6x + 3600 for x 0, and the price-demand function, in dollars per widget, is p(x) = 50 0.02x for 0 x 2500. In Quiz 2, problem #10, we saw that the profit function for this scenario is: P(x) = 0.02x2 + 44x 3600 (a) The profit function is a quadratic function and so its graph is a parabola. Does the parabola open up or down (explain why)? __________ (b) Find the vertex of the profit function P(x) using algebra. Show algebraic work. (c) State the maximum profit and the number of widgets which yield that maximum profit: The maximum profit is _______________ when ____________ widgets are produced and sold. (d) Determine the price to charge per widget in order to maximize profit. (e) Find and interpret the break-even points. Round answer to nearest whole number. Show work. 14. (6 pts) The faucet in Nancy's house can fill the sink in 6 minutes, while the drain can empty the same sink in 9 minutes. If her son, Tommy, turns the faucet on and leaves the drain open, how much time does Nancy have to discover the open faucet before the sink is filled and starts to overflow? In other words, how long will it take to fill the sink (in which case, Nancy will have a mess on her hands)