MATH 12001 Precalculus: Algebra & Trigonometry Spring 2003 Sections 2 & 3 Darci L. Kracht Name: Score: /100. 115 pts available EXAM 3 Tuesday, March 18, 2003 Part I: NO CALCULATORS. (You must turn this part in before taking your calculator out.) Section A: Short Answer. (15 questions at 2 points each.) (1) The following is the graph of a polynomial function, y = P (x). (You may assume that all intercepts and turns in the graph are depicted.) Is the degree of P (x) odd or even? ans(1) (2) Refer to the the graph of a polynomial function, y = P (x), in question (1). Is the leading coefficient of P (x) positive or negative? ans(2) (3) Refer to the the graph of a polynomial function, y = P (x), in question (1). Is it possible for the degree of P (x) to be less than 5? ans(3) (4) Suppose P (x) is a polynomial having 45 as a zero. Find a factor of P (x). ans(4) (5) Suppose P (x) is a polynomial having 3 as a zero. Find an x- intercept of the graph y = P (x). (List it as an ordered pair.) ans(5) (6) Suppose P (x) is a polynomial having 5 as a zero. What is the remainder when P (x) is divided by x 5? ans(6)
(7) List all potential rational zeros of the polynomial P (x) = 15x 7 6x 5 + 5x 3 4x 2 + 3x 2 as given by the Rational Zeros Theorem. (Do not test them.) ans(7) (8) Find a polynomial function P of degree 3 having zeros 0, 12, and 5 and leading coefficient 3. Leave it in factored form. (Do not expand.) ans(8) (9) Find the exact value: log 3 1 9 ans(9) (10) Find the exact value: log 300 log 3 ans(10) (11) Simplify: e 4 ln x ans(11) (12) Simplify: log b b x 1 ans(12) (13) Write an equivalent exponential equation: ln A = B ans(13) (14) Rewrite as a single logarithm: 3 ln x ln ( x 2 + 7 ) ans(14) (15) Give the domain of the function f(x) = log 2 (3 5x) in interval notation. ans(15)
Section B: Graphs. 1. (10 points) Let f(x) = 1 x 4 + 3. (a) The function f is a transformation of a basic rational function. Give the rule for this basic function. (b) Give a sequence of transformations from the basic function to f. (c) Carefully sketch the graph of f. Sketch asymptotes with dotted lines and label with their equations. Plot and label (with their ordered pairs) at least six points on the graph. 2. (15 points) Let f(x) = 2 x. (a) Carefully sketch the graph of f. Sketch asymptotes with dotted lines and label with their equations. Plot and label (with their ordered pairs) at least five points on the graph. (b) Give the formula for f 1. (c) Carefully sketch the graph of f 1 on the same coordinate system. Sketch asymptotes with dotted lines and label with their equations. Plot and label (with their ordered pairs) at least five points on the graph.
MATH 12001 Precalculus: Algebra & Trigonometry Spring 2003 Sections 2 & 3 Darci L. Kracht Name: EXAM 3 Tuesday, March 18, 2003 Part II: Long Answer. Show all work to receive credit. (Calculators permitted.) 1. (15 points) Factor the polynomial completely over the set of real numbers. Use the methods of Chapter 3, showing your reasoning clearly. P (x) = 6x 4 x 3 8x 2 + x + 2 2. (20 points) Solve the equation algebraically, giving both an exact answer and a decimal approximation rounded to four decimal places. State which solutions (if any) are extraneous and why. (a) e 6x + 7e 3x 18 = 0 (b) log x + log(x 21) = 2 factorization: P (x) =
3. (5 points) Use the Change of Base Formula and a calculator to evaluate, rounding to four decimal places. Show your steps. log 2 137 5. (5 points) Suppose $150 was invested by your greatgreat-great-grandmother in an account earning 6.5% annual interest on January 1, 1900. Suppose interest is compounded continuously. (a) Write the formula for A(t), the amount in the account after t years. 4. (10 points) Suppose terrorists steal a sample of weapons-grade Plutonium-239. The mass, in kilograms, of Plutonium-239 remaining in the sample after t years is given by the formula (b) Determine the amount in the account when you inherit it on January 1, 2004. m(t) = 4.1 e 0.0000285 t (a) How much Plutonium-239 was in the sample when it was stolen? (b) What is the half-life of Plutonium-239 (to the nearest thousand years)? 6. (5 points) How many times more intense was the 1978 Mexico City earthquake (magnitude 7.9) than the 1995 Kobe, Japan earthquake (magnitude 7.2)? Recall: M = log I S. (c) It is estimated that the minimum mass needed to construct a bomb from weapons-grade Plutonium-239 is 4 kg. How many years will it take for the stolen sample of Plutonium-239 to decay to a mass of 4 kg?.