Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic functions) Factoring and finding zeros of a polynomial (using synthetic division and other factoring techniques, you ll be given at least one zero to get started) Listing out all potential zeroes (using the rational zeroes theorem) graphing polynomials (including end behavior, test points and zeroes) using synthetic division to evaluate a polynomial at a specific point. (using the remainder theorem) basic arithmetic of complex numbers Finding vertical and horizontal asymptotes and sketching a graph of a rational function computations with logarithms computations using the laws of logarithms Part 2 (calculator) Chapters 1 and 2 Same as midterm Chapter 3 factoring and finding zeros of a polynomial (numbers may not work out nicely, for example you may need to use the quadratic formula, or some zeroes will be imaginary. However, you may use a graphing calculator to help find the zeros) dividing polynomials (both regular and synthetic) Finding a polynomial that meets certain conditions (for example, find a 4 th degree polynomial that has zeroes 2i and 4, with 4 having a multiplicity 2) Chapter 4 finding the inverse of a function checking if a function is one-to-one graphing the inverse of a function exponential functions (domain, range, end-behavior, etc) applications of exponential functions (such as compound interest) calulating logarithms with a calculator applying properties of logarithms re-writing logarithm expressions (using definitions and/or laws of logarithms) solving equations (exponential and logarithmic) Sample questions are on the following pages
Sample Questions (These are only sample questions from chapters 3 and 4). See the midterm review for sample questions from chapters 1 and 2) Part 1: 1) Sketch a graph the function f(x) = x 3 + 2x 2 5x 6 (hint: 2 is a zero). Factor the function, then sketch a graph. label the axes, and indicate the location of all the zeros. 2) Factor the polynomial into linear factors (including any imaginary numbers). f(x) = x 4 x 3 + 7x 2 9x 18 3) Let f(x) = 3x 5 4x 3 + 5x 2 + x 10. Use synthetic division to evaluate f(1) 4) Compute the following, and give the answer in the form a + bi: a) (2 3i) (4 + 7i) b) (8 + 5i) (2 3i) c) i 47 5) Find the vertical and horizontal asymptotes, then sketch a graph of the following: f ( x) x 2 9 2 2x 7x4 6) Sketch a graph of the following functions. Include any horizontal and vertical asymptotes a) f(x) = e x+1 b) f(x) = -2 ln(-x) 7) Compute the following:
a) log93 b) 1 ln e c) log(7400) (hint: log 7.4 = 0.86923) 8) Matching: Place the appropriate letter next to each function. One graph will not be used. Use your knowledge graphs of polynomials. f(x) = -2x 3 11x 2 8x f(x) = x 5 + 3x 4-5x 3-15x 2 + 4x + 12 f(x) = x 4 + x 3 +2x 2 + 5x + 2 f(x) = -2x 4 11x 2 8x + 3 f(x) = -0.2x 5-1.3x 4 + 2x 3-3x 2 + 6x + 30 A B C D E F
Part 2: 1) Find all the real zeros of the polynomial P(x) = 4x 3 6x 2 + 1. Use the quadratic formula if necessary. 2) Factor the polynomial into linear factors, including any imaginary numbers. Give all the zeroes, and the multiplicity of each zero. f(x) = x 4-625 3) Find a polynomial with integer coefficients that satisfies the given conditions: P has degree 3, and zeroes 2 and 1+ i 4) Divide 4x 3 +7x + 12 by 2x + 1. Give the answer in the form Q(x) + R(x)/D(x) 5) Use the definition of logarithms to re-write the following: a) Write in exponential form: ln(x + 2) = 4 b) Write in logarithmic form: 4 3 = 64 c) Solve for x: logx25 = 2 6) Solve the following equations: a) 450 = 200(1.25) x+1 b) log 6 ( x 3) log 6 ( x 2) 2 7) Calculate the following: a) log 47 b) ln 12 c) log6 50 (use the change-of-base formula)
8) Break the following expression into simpler logarithms: ln z x 4 y 9) is the function f(x) = x 2 one-to-one? Why or why not? 10) Find the inverse of f(x) = 4x 3 + 2 11) Give the domain, range and asymptotes of the function f(x) = -3e x+1 12) the population of a country has been growing exponentially. In 1990, the population was 125 million. In 2005 the population was 140 million. a) find a function in the form f(x) = Ce kt that models the population growth. Let t = the years since 1990. b) Use the model from part (a) to predict the year the population will be 160 million.