MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = (x + 5) 2-2 1) Solve the problem. 2) The population of a city has varied considerably in recent years. The data in the table relates the population P to time t, in years, where t = 0 corresponds to Use a quadratic function fitted to the data to predict the population in ) Year, t Population, P 1990(t = 0) 59, (t = 1) 48, (t = 2) 43, (t = 3) 49, (t = 4) 41, (t = 5) 39, (t = 6) 42,500 A) 119,000 B) 48,431 C) 81,202 D) 58,964 1

2 Use the factor theorem to decide whether or not the second polynomial is a factor of the first. 3) 9x4 + 37x3-4x2 + x + 4; x + 4 3) A) No B) Yes Sketch the graph of the parabola. 4) y = x2-4 4) Use the remainder theorem and synthetic division to find f(k). 5) k = -3; f(x) = -x3-2x ) A) -15 B) -42 C) -12 D) 12 2

3 Use synthetic division to decide whether the given number k is a zero of the given polynomial function. 6) 1 2 ; f(x) = 2x 4-21x3 + 3x + 1 6) A) Yes B) No Find the zeros of the polynomial function and state the multiplicity of each. 7) 5x(x - 7)3 (x2-4) 7) A) Multiplicity 1 : 0 Multiplicity 2 : - 4 Multiplicity 3 : 7 C) Multiplicity 1 : 0 Multiplicity 2 : ±2 Multiplicity 3 : -7 B) Multiplicity 1 : ±2 Multiplicity 3 : 7 D) Multiplicity 1 : 0 Multiplicity 1 : ±2 Multiplicity 3 : 7 Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. 8) Zeros of -4, i, -i and P(-3) = 60 8) A) P(x) = -6x3-24x2-6x - 24 B) P(x) = -6x3-24x2 + 6x + 24 C) P(x) = 6x3 + 24x2 + 6x + 24 D) P(x) = 6x3 + 24x2-6x - 24 Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. 9) f(x) = x3-2x2-11x ) A) -4, 3 + 4i, 3-4i B) -4, 1 + 2i, 1-2i C) -4, 3 + 2i, 3-2i D) -4, i, i Use a graphing calculator to find the coordinates of the turning points of the graph of the polynomial function in the indicated domain interval. Give answers to the nearest hundredth. 10) f(x) = -x4 + 10x2-9; [2, 3] 10) A) (0, 9) B) (2.24, 16.00) C) (3, ) D) (-3, 16.00) 3

4 Sketch the graph of the polynomial function. 11) f(x) = (x + 4) ) Find any vertical asymptotes. 12) f(x) = 5x + 9 2x ) A) x = 1 2 B) x = - 2 C) x = 2 D) x = 5 2 Determine which of the rational functions given below has the following feature(s). 13) The horizontal asymptote is y = 3 13) A) f(x) = 3x - 1 x + 9 B) f(x) = 3 x - 3 C) f(x) = x + 3 x + 9 D) f(x) = 3x 2-1 x + 9 4

5 14) The x-axis is its horizontal asymptote of which function? 14) A) f(x) = 4x - 1 B) f(x) = x + 4 C) f(x) = 9 D) f(x) = x - 4 x + 9 x + 9 x - 4 x Find the horizontal asymptote of the given function. 15) g(x) = x 2 + 8x - 5 x - 5 A) y = 0 B) y = 8 C) None D) y = ) 16) h(x) = 4-9x - 3x + 7 A) y = - 3 B) y = 3 C) None D) y = 0 16) Give the equation of the oblique asymptote, if any. 17) f(x) = x 2-9x + 3 x + 9 A) y = x - 18 B) x = y + 9 C) None D) y = x ) 5

6 Graph the function. x ) f(x) = x2 + x ) If f is one-to-one, find an equation for its inverse. 19) f(x) = -5x ) A) f-1(x) = x + 1 B) f -1(x) = x C) f-1(x) = x D) f-1(x) = x - 1 If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one." 20) {(6, -7), (11, -6), (9, -5), (7, -4)} 20) A) Not one-to-one B) {(-7, 6), (-6, 11), (-5, 9), (-4, 7)} C) {(6, -7), (11, -6), (9, -5), (7, -4)} D) {(7, 6), (6, 11), (-5, 9), (-4, 7)} 6

7 Write the expression as a sum, difference, or product of logarithms. Assume that all variables represent positive real numbers. 21) loga(8x5y) 21) A) loga8 + 5logax + logay B) loga(8 + x5 + y) C) (loga8 )(logax) (logay) D) loga8 + (logax) 5 + logay Give all solutions of the nonlinear system of equations. 22) x2 - y2 = 39 x - y = 3 A) {(8, -5)} B) {(-8, 5)} C) {(8, 5)} D) {(-8, -5)} 22) Find the values of the variables for which the statement is true, if possible. 23) 7-1 = x y z A) x = -7; y = -8; z = -9 B) x = 8; y = 9; z = 7 C) x = 7; y = 8; z = 9 D) x = -7; y = 9; z = 8 23) Perform the operation or operations when possible. 24) A) B) C) D) ) The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. 25) objective function = 7x + 3y; maximum 25) A) 100 B) 36 C) 44 D) 16 7

8 For the function as defined that is one-to-one, graph f and f-1 on the same axes. 26) f(x) = 4x 26) 8

9 Use the graph of f to sketch a graph of the inverse of f using a dashed curve. 27) 27) Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. 28) 1 2 log 2 x log 2 x log 2 x 28) A) log 2 (x7) B) 7 6 log 2 (x 8) C) log 2 ( x17/6) D) log 2 (x9/2) 9

10 Graph the function. 29) f(x) = 1 x 2 29) Write in logarithmic form. 30) 42 = 16 30) A) log 2 16 = 4 B) log 16 4 = 2 C) log 4 2 = 16 D) log 4 16 = 2 10

11 Graph the exponential function using transformations where appropriate. 31) f(x) = 3x ) Solve the equation. 32) log3 81 = x 32) A) {243} B) {27} C) {4} D) {84} Find the value of the determinant ) A) -252 B) 243 C) 234 D) ) 11

12 Graph the function. Give the domain and range. 34) f(x) = log 3 x 34) A) domain: (, 0); range: (, ) B) domain: (0, ); range: (, ) C) domain: (, 0); range: (, ) D) domain: (0, ); range: (, ) 12

13 Match the function with its graph. 35) f(x) = log2 (x - 1) 35) Write the expression as a sum, difference, or product of logarithms. Assume that all variables represent positive real numbers. 36) log ) A) log4 4 + log4 4 - log4 5 B) (log4 4) 1 2 log log4 5 C) log log 4 4 log4 5 D) log log log4 5 Solve the equation and express the solution in exact form. 37) ln(21x - 7) = ln 8 37) A) B) C) D) Solve the equation. 38) ln ex - ln e4 = ln e5 38) A) {20} B) {-1} C) {1} D) {9} 13

14 Solve the problem. 39) What is the rate on an investment that doubles $5051 in 9 years? Assume interest is compounded quarterly. A) 3.9% B) 5.8% C) 9.7% D) 7.8% 39) 40) A sample of 750 grams of radioactive substance decays according to the function A(t) = 750e-.048t, where t is the time in years. How much of the substance will be left in the sample after 10 years? Round your answer to the nearest whole gram. A) 0 grams B) 1 gram C) 464 grams D) 39 grams 41) How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the function A(t) = 500e-.202t, where t is the time in years? Round your answer to the nearest hundredth year. A) years B) years C) 3.43 years D) years 40) 41) Use the given row transformation to change the matrix as indicated ) ; -6 times row 3 added to row 1 42) A) B) C) D) Write the system of equations associated with the augmented matrix. Do not solve. 43) A) -2x - 5y = 8 B) -5x - 2y = 8 C) -2x - 5y = 0 2x + 8y = 9 2x + 8y = 9 2x + 8y = 0 D) -2x - 5y = 8 8x + 2y = 9 43) Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. 44) 3x + y = 8 44) 9x + 4y = 20 A) {(-4, 4)} B) {(-4, -4)} C) {(4, -4)} D) Find the partial fraction decomposition for the rational expression. 45) -2x2 + 3x + 8 (x + 1)2(3x + 4) A) C) 4 3x (x + 1) x x x (x + 1)2 B) D) 4 3x (x + 1)2 + 2 x x (x + 1) x ) 46) -6 x2(x2 + 7) 46) A) 6 x (x2 + 7) B) -6 x2 + 6 (x2 + 7) C) -6 7x (x2 + 7) D) 6 7x (x2 + 7) 14

15 Graph the solution set of the system of inequalities. 47) x - 2y 2 x + y 0 47) Find the indicated matrix. 48) Let A = 1 3 and B = 2 5 A) Find 4A + B. 48) -1 6 B) C) D)

16 Find the size of the matrix. 49) A) 6 B) ) C) 1, -9, 2, 2, -9, -2 D) 2 3 Solve the problem. 50) Since 1990, the number of new books published each year has been growing at a rate that can be approximated by a quadratic function. The table shows the number of books published in the United States for selected years. Use the regression feature on your calculator to determine the quadratic which best fits the data. 50) Years 1990 = Books Published (1000 s)