Midterm Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor completely. If the polynomial cannot be factored, say it is prime. 10x 2-95x + 225 2. Solve the problem. Two friends decide to meet in Chicago to attend a Cub's baseball game. Rob travels 189 miles in the same time that Carl travels 171 miles Rob's trip uses more interstate highways and he can average 6 mph more than Carl. What is Rob's average speed? 3. Find the real solutions, if any, of the equation. Use the quadratic formula. x 2 + x + 1 = 0 4. Write the expression in the standard form a + bi. 5. Solve. If necessary, round to the nearest tenth. The zoo has hired a landscape architect to design the triangular lobby of the children's petting zoo. In his scale drawing, the longest side of the lobby is 17 cm. The shortest side of the lobby is 3 cm. The longest side of the actual lobby will be 46 m. How long will the shortest side of the actual lobby be? a. 260.7 m c. 0.1 m b. 8.1 m d. 2.6 m 6. Use synthetic division to find the quotient and the remainder. x 5-3x 4-13x 3 + 18x 2-17x + 12 is divided by x - 5
7. The graph of a function is given. Decide whether it is even, odd, or neither. 8. Determine algebraically whether the function is even, odd, or neither. f(x) = 9. The graph of a function f is given. Use the graph to answer the question. Find the numbers, if any, at which f has a local maximum. What are the local maxima? 10. Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local minima. If necessary, round answers to two decimal places. f(x) = x 3-3x 2 + 1; (-5, 5)
11. Find the average rate of change for the function between the given values. f(x) = ; from 2 to 8 12. Find and simplify the difference quotient for the given function. f(x) = 13. Find the indicated intercept(s) of the graph of the function. y-intercept of f(x) = 14. Find the domain of the rational function. F(x) =. 15. Find the indicated intercept(s) of the graph of the function. y-intercept of f(x) = 16. Solve the inequality algebraically. Express the solution in interval notation. >
17. Express as a single logarithm. ln - ln + ln (x 2-14x + 49), x > 0 18. Suppose that ln 2 = a and ln 5 = b. Use properties of logarithms to write each logarithm in terms of a and b. ln 19. Find the intercepts of the function f(x). f(x) = 4x 4-40x 3 + 101x 2-10x + 25 20.Find the vertical asymptotes of the rational function. F(x) = 21. Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3 5x = 3.8 22. Determine where the graph of f is below the graph of g by solving the inequality. f(x) = x 4 + 1 g(x) = x + 1
23. Use the given zero to find the remaining zeros of the function. f(x) = x 3-2x 2 + 5x + 26; zero: 2 + 3i 24. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. log 6 13 log 13 216 25. Solve the problem. The formula P = 14.7e -0.21x gives the average atmospheric pressure, P, in pounds per square inch, at an altitude x, in miles above sea level. Find the average atmospheric pressure for an altitude of 2.3 miles. Round your answer to the nearest tenth. 26. Solve the problem. A bacterial culture has an initial population of 10,000. If its population declines to 4000 in 4 hours, what will it be at the end of 6 hours? Assume that the population decreases according to the exponential model. 27.Solve the problem. In a town whose population is 3000, a disease creates an epidemic. The number of people, N, infected t days after the disease has begun is given by the function N(t) =. Find the number of infected people after 10 days. 28. State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. f(x) = 13x 3-5x 2-6 29. State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. 6(x - 1) 11 (x + 1) 5 30. Form a polynomial whose zeros and degree are given. Zeros: 1, multiplicity 2; 5, multiplicity 1; degree 3
31. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = 4(x - 7)(x - 5) 4 32.Solve the problem. Which of the following polynomial functions might have the graph shown in the illustration below?
33. Determine the maximum number of turning points of f. f(x) = -x 2 (x + 7) 3 (x 2-1) 34. Use the horizontal line test to determine whether the function is one-to-one. 35.For the given functions f and g, find the requested composite function value. f(x) =, g(x) = x 2 + 9; Find (g f)(-2). 36. Find the domain of the composite function f g. f(x) = ; g(x) = x + 3 37. For the given functions f and g, find the requested composite function value. f(x) =, g(x) = 5x; Find (f g)(3). 38. Find the domain of the rational function. G(x) =
39.Graph the function as a solid line or curve and its inverse as a dashed line or curve on the same axes. f(x) = a. c. b. d. 40.Use the graph to determine the domain and range of the function.
41. Evaluate the expression using the values given in the table. (f g)(4) 42.Evaluate the expression using the values given in the table. f(g(-5))
43. Decide whether or not the functions are inverses of each other. f(x) =, g(x) = 44. The function f is one-to-one. Find its inverse. f(x) = 45. Approximate the value using a calculator. Express answer rounded to three decimal places. 3.7 46. Solve the equation. 3 1 + 2x = 27 47.Change the exponential expression to an equivalent expression involving a logarithm. 7 2 = 49 48. Find the domain of the function. f(x) = 3 - ln(7x) 49. Find and simplify the difference quotient of f,, for the function. f(x) = 6x + 4
50. The graph of a logarithmic function is shown. Select the function which matches the graph. 51. The graph of a piecewise-defined function is given. Write a definition for the function.
52.Determine whether the graph is that of a function. If it is, use the graph to find its domain and range, the intercepts, if any, and any symmetry with respect to the x-axis, the y-axis, or the origin. 53. Solve the problem. Find (-3) when f(x) = 3x - 4 and g(x) = 3x 2 + 14x + 3. 54. Solve the problem. Find (f - g)(4) when f(x) = -3x 2 + 2 and g(x) = x + 6.
55. Solve the inequality. Express your answer in set notation. x 2 > 16 56. Find the domain of the function. f(x) = 57. Solve the problem. Given f(x) = and ( )(x) =, find the function g. 58. Find the domain of the function. f(x) = ln 59. Simplify the expression. (5-3 ) -1 60. An expression that occurs in calculus is given. Factor completely. 3(x + 5) 2 (2x - 1) 2 + 4(x + 5) 3 (2x - 1) 61.An expression that occurs in calculus is given. Write the expression as a single quotient in which only positive exponents and/or radicals appear.
62. An expression that occurs in calculus is given. Write the expression as a single quotient in which only positive exponents and/or radicals appear. 6x 3/2 (x 3 + x 2 ) - 8x 5/2-8x 3/2 63. An expression that occurs in calculus is given. Write the expression as a single quotient in which only positive exponents and/or radicals appear. 64. Convert the angle in radians to degrees. 65. In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact value of the indicated trigonometric function of t. (, ) Find sin t. 66.Find the exact value. Do not use a calculator. cot 0 67. Find the exact value. Do not use a calculator. tan 68. Find the exact value. Do not use a calculator. sin (22 )
69. Find the exact value of the expression if = 45. Do not use a calculator. f( ) = sin Find 9f( ). 70. Find the exact value of the expression. Do not use a calculator. tan + tan 71. Solve the problem. Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are 37 and 21. How far apart are the hikers? Round your answer to the nearest whole meter. 72.Solve the triangle. 73. Solve the triangle.
74. Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. C = 35, a = 18.7, c = 16.1 75. Solve the triangle. a = 6, b = 14, c = 15 76. Find the exact area under the curve on the interval 0<x<4 when f(x)=x+5. 77. Find the exact area under the curve on the interval 1<x<6 when f(x)=3 78) Estimate the area under ( ) on the interval 2<x<5 using 3 left hand rectangles. 79) Estimate the area under ( ) on the interval 1<x<5 using 8 right hand rectangles 80) Estimate the area under ( ) on the interval 1<x<4 using 3 trapezoids.