Math 1200 Exam 4A Fall Name There are 20 questions worth 5 points each. Show your work in a neat and organized fashion. Award full credit fo

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6 Math 1200 Exam 4A Fall Name There are 20 questions worth 5 points each. Show your work in a neat and organized fashion. Award full credit for clarity of expression and orderly presentation of solution. Q16B 1. Use long division to divide (6x 3 5x 2 3) (3x + 2). Q17B 2. Given f(x) = x 4 + 3x 3 7x x 10, use the Remainder Theorem to evaluate f( 5). Q18B 3. Based on your answer to Problem 2, is (x + 5) a factor of f(x)? Explain your reasoning. Yes. This is because x + 5 completely divides f(x) without any remainder. Thus, by the remainder theorem (x + 5) is a factor of f(x) 5 points

7 Q20B 4. For the function f(x) = 5x2 1, identify all of its asymptotes (vertical, horizontal, or slant). x 2 x 6 For vertical asymptotes: x 2 x 6 = 0 (x + 2)(x 3) = 0 thus x = 3 and x = 2 are the lines of Vertical asymptotes 2.5 points Since the degree of the leading term of both the numerator and denominator are the same, the horizontal asymptote is given by y = 5x2 x2 = points Q19B 5. Suppose that y varies jointly as w and x and inversely as the square of z. Write a variation model using k as the constant of variation. The model is given by y = wxk z 2, where k is the constant of variation. 5 points Q13B Q14B 6. Use the definition of a one-to-one function to determine whether f(x) = x is one-to-one. Students are supposed to show that if f(a) = f(b), then a = b for a one to one function. a = b points for correctly evaluating the function at x=a and x=b and setting up the equation. a 3 = b 3 3 a 3 3 = b 3 2 points for simplifying the expression to this step. Thus a = b showing that f(x) is a one-to-one function. 1 point for concluding 7. Using function composition, determine whether the functions f(x) = 2x 7 and g(x) = x+7 2 are inverses of each other. Students are required to show that the composite functions (fog)(x) = x and (gof)(x) = x for f(x) and g(x) to be inverse functions of each other. (fog)(x) = 2 ( x+7 ) 7 = x = x 2 points 2 (gof)(x) = 2x 7+7 = 2x = x. 2 points 2 2 thus f(x) and g(x) are inverse functions of each other. 1 point Use the graph at the right to answer questions 8 and

Q11B 8. (a) As x, f(x) 1. 2.5 points (b) As x, f(x) 1. 2.5 points Q12B 9. (a) As x 3, f(x). 2.5 points (b) As x 3 +, f(x). 2.5 points Q9B 10.

06, t = 10. Since the interest rate is compounded monthly, n = 12 A = P (1 + r n )nt 2 points A = 5,000 (1 +.06 12 )12 10 2 points 1 point 11.

8 Q11B 8. (a) As x, f(x) points (b) As x, f(x) points Q12B 9. (a) As x 3, f(x). 2.5 points (b) As x 3 +, f(x). 2.5 points Q9B 10. Suppose that $5000 is invested with a 6% interest rate, compounded monthly. Write an expression for the total amount in the account after 10 years. Do not simplify your expression. P = 5,000, r =.06, t = 10. Since the interest rate is compounded monthly, n = 12 A = P (1 + r n )nt 2 points A = 5,000 ( ) points 1 point 11. Find the domain of g(x) = log(3 x) and write it in interval notation. Q15B Since logarithmic with argument 0 is undefined, we have 3 x > 0 x < 3. 3 points for setting the argument of the logarithm to be strictly positive Thus, the domain of g(x) is (, 3) 2 points for writing the domain in interval notation. Q1B 12. Factor g(x) = x 3 x 2 14x + 24 completely, given that 2 is a zero.

13. Write a polynomial of degree 3 with zeros 2, 2, and 7. Leave this polynomial in factored form. Q2B Let f(x) be a third-degree polynomial with zeros 2, 2 and 7.

a = 6 or a = 1 gives the boundary values. 2 points for finding the boundary values. From the sign chart, the interval (, 1) (6, ) is the solution set 1 point Q6B 15. 5 x 0.

9 13. Write a polynomial of degree 3 with zeros 2, 2, and 7. Leave this polynomial in factored form. Q2B Let f(x) be a third-degree polynomial with zeros 2, 2 and 7. Then x 2, x + 2 and x 7 are the factors of f(x). f(x) = (x 2)(x + 2)(x 7) 5 points For Problems 14 and 15, solve the inequalities and write your answer in interval notation. 14. a 2 > 5a + 6 (a 6)(a + 1) = 0 Q5B a 2 5a 6 > 0. a = 6 or a = 1 gives the boundary values. 2 points for finding the boundary values. From the sign chart, the interval (, 1) (6, ) is the solution set 1 point Q6B x 0. x+1 We have 5 x = 0 5 x = 0 x = 5. 1 point x+1 Also, we find the restricted value of the rational function which is x = 1. 1 point Again, from the sign chart, ( 1,5] is the solution set. 1 point Q3B 16. The graph of a function y = f(x) is given below. Is the function one-to-one? Justify your answer.

10 Clearly, the graph above fails the horizontal line test. Hence the function is not one-to-one. 5 points Q10B 17. The amount of a pain reliever that a physician prescribes for a child varies directly as the weight of the child. A physician prescribes 160 mg for a 40-lb child. Write a variation model and solve for the constant of variation k. Let w represents the weight of the child. Also let A represents the amount of pain reliever. A = wk. 2.5 points We have A=160mg, w=40, k=? k = 160. k = points 40 thus, the final model is A = 4w Q4B Find the inverse function of f(x) = x 5. 3 y = x 5 3 x = y 5 (x) 3 3 = ( y 5) 3 x 3 = y 5 y = x f 1 (x) = x point for replacing f(x) with y 1 point for interchanging x and y 2 points for simplifying the expression to arrive at this step 1 point for replacing y with inverse function notation Q7B 19. Evaluate (a) log 2 16 = 4 because 2 4 = 16 Alternatively, log 2 16 = x 2 x = 16 2 x = 2 4. thus x = points (b) ln(e 8 ) log e e 8 = x e x = e 8. Thus x = points Q8B 20. Graph the function f(x) = 2 x Award 2.5 points for correctly plotting the points on the cartesian plane.

Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph