Math Midterm Ch 1-4 Review Answers

Pre-Calculus Midterm Exam Review Chapter 1: Functions and Their Graphs 1. Evaluate the function at each specified value of the independent variable and simplify. f ( x)  x 2  1 a. f (3) b. f ( x  1) c. f (b 3 ) 1a.)_____________ 1b.)_____________ 1c.)_____________ 2. Evaluate the function at each specified value of the independent variable and simplify. 2 x  1 f ( x)   2 x  2 a. x  1 x  1 f (2) b. f (1) c. f (0) d. f (2) 2a.)_____________ 2b.)_____________ 2c.)_____________ 2d.)_____________ 3. Find the domain of the given function. Use interval notation! a. f ( x)  x 1 x2 b. g ( x)  x2 x2 1 c. f ( x)  x 2  16 3a.)_____________ 3b.)_____________ 3c.)_____________ 4. Use a graphing utility to graph the function and determine the open intervals on which the function is increasing, decreasing, or constant (use interval notation!). Identify any relative minimum or relative maximum values of the function. a. f ( x)  x 3  3x b. f ( x)  x 2  9 4a.) increasing:____________ decreasing:____________ constant:______________ rel. min.:______________ rel. max.:______________ 4b.) increasing:____________ decreasing:____________ constant:______________ rel. min.:______________ rel. max.:______________ 5. Determine algebraically if the function is even, odd, or neither. Verify using a graphing utility. a. f ( x)  x 2  6 5a.)________________ b. f ( x)  x 2  6 x  4 5b.)________________ c. f ( x)  x 3  3x 5c.)________________ 6. g ( x)  x and h( x)  x . Describe the sequence of transformations from f ( x), g ( x), or h(x) to j (x) . 1 j ( x)  ( x  2) 2  6 b. a. c. j ( x)   x  5  7 j ( x)  2 x  3 2 Given f ( x )  x , 2 a.)________________________________________________________________________________________ b.)________________________________________________________________________________________ c.)________________________________________________________________________________________ 7. Given f ( x)  3  2x , g ( x)  x and h( x )  3 x 2  2 . Find the indicated values. ( f  g )(4) a. b. ( f  h)(5) c. ( fh)(1) 7a.)________________ 7b.)________________ 7c.)________________ g  1 h d. e. ( g  f )(2) f. ( f  h)(4) 7d.)________________ 7e.)________________ 7f.)________________ 8. Find the inverse of the function algebraically. f ( x)  a. 7x  3 8 f ( x)  4 x 3  3 b. c. f ( x)  x  10 8a.)________________ 8b.)________________ 8c.)________________ 9. Show that f (x) and g (x) are inverses. f ( x)  x  1 , g ( x)  x 2  1, x  0 10. Given the graph a. y 1 f ( x) 2 y  f (x) at the right, sketch the following: b. y   f (x) c. y  f ( x  2) d. y  f (x ) Chapter 2: Polynomials and Rational Functions 11. Express each complex number in standard form a.  12   48 b. 3  2i 2 a  bi . c. 1 4  7i 11a.)_______________ 11b.)_______________ 11c.)_______________ d. 3  2i e. 3  2i i 23 f. 43  4i   5i1  i  11d.)_______________ 11e.)_______________ 11f.)_______________ 12. Solve the following equations for x. a. 12x 2  12  25x b. x 2  4x  7 c. x 2  6x  18  0 12a.)_______________ 12b.)_______________ 12c.)_______________ 13. If P ( x)  4 x  5 x  1, use synthetic substitution to find: 3 a. 2 P(2) b. 1 P  2 c. P(i) 13a.)_______________ 13b.)_______________ 13c.)_______________ 14. If -2 is a zero of the polynomial P ( x)  2 x  x  k , find the value of k. 15. A polynomial 3 14.)________________ P(x) is divided by x  2 . The quotient is 2x 2  x  2 and the remainder is -1. Find P(x) . 15.)______________________ 16. Two roots of the equation x 4  x 3  5x 2  x  6  0 are x  2 and x  3 . Find the remaining roots. 16.)________________ 17. Find all real zeros of f ( x)  2 x  7 x  4 x  27 x  18 . Use the rational root theorem and synthetic division. 4 3 2 17.)________________ 18. Sketch the graph of the equation (without a graphing calculator). Use the zeros of the function and the end behavior of the function. f ( x)   x 2 ( x  1)( x  2) 19. A rectangular enclosure, sub-divided into three congruent pens as shown, is to be made using a barn as one side and 120 meters of fencing for the rest of the enclosure. Find the value of x that gives the maximum area for the enclosure. 19.)________________ BARN 2x 3  2x 2  4x  4  0 20. Solve for x. 20.)________________ 21. Find all real and imaginary roots of 22. Find a cubic equation with integral coefficients having the solutions 23.  x  2 if x  -1   2  Graph f ( x)  x if - 1  x  1 1  if x  1   2x 4  3x 3  7 x 2  7 x  5  0 21.)________________ 1  i, and 3 . 22.)________________ 24. Sketch the graph of the rational function by hand. Identify any x-intercepts, the y-intercept, horizontal asymptotes, vertical asymptotes, and slant asymptotes. a. f ( x)  x3 x2 x-intercept(s) = _______ y-intercept = _________ H.A. = ______________ V.A. = ______________ S.A. = ______________ b. f ( x)  2x 2 x2  4 x-intercept(s) = _______ y-intercept = _________ H.A. = ______________ V.A. = ______________ S.A. = ______________ c. f ( x)  x2  x 1 x 3 x-intercept(s) = _______ y-intercept = _________ H.A. = ______________ V.A. = ______________ S.A. = ______________ 25. Determine whether x  xy  y  6 has symmetry in: Circle the correct answer (s) 2 a. the x-axis 26. 2 b. the y-axis c. the line y=x d. the origin Sketch the graph of the quadratic function. Identify the vertex & intercept(s). f ( x )  ( x  4) 2  4 vertex = _________________ x-intercept(s) = ___________ y-intercept = _____________ Chapter 3 and Exponents Packet Exponential Functions and Logarithmic Functions 27. Evaluate the following exponential expressions: a. 3 5  310 32 b. 4 2  40  1 c. 95 32 d. 2 4  4 3 2 3 27a.)_______________ 27b.)_______________ 27c.)_______________ 27d.)_______________ 28. Solve each exponential equation: a. 38 x  272 x b. 4 32  2 3x 28a.)_______________ 28b.)_______________ 29. A gallon of gasoline cost $3.09 two years ago. Now it costs $1.78. To the nearest percent, what has been the annual rate of decrease in cost? 29.)________________ x 30. 1 Graph y  3 and y    on the same set of axes. 3 x How are the graphs related? 31. Suppose $1,200 is invested at an interest rate of 9.6%. how much is the investment worth after 2 years if interest is compounded: a. Monthly? b. Continuously? 31a.)_______________ 31b.)_______________ 32. Given a. log 40  1.6021, using rules of logarithms, find the value of: {without a calculator!} log 4 b. log.25 32a.)_______________ 32b.)_______________ 33. Find the exact value of: without a calculator! 33a.)_______________ a. b. log 27 3 log2 16 c. 5log4 64 33b.)_______________ 33c.)_______________ 34. Express the following in terms of logb M and logb N a. logb 3 M3 N2 b. logb M 3 N 2 34a.)__________________________________ 34b.)__________________________________ 35. Solve log2 x  2  log2 x  5  1 . 35.)________________ 36. Solve 3 x  31 to the nearest hundredth. 36.)________________ 37. If 250 mg of ibuprofen has a half-life of 3 hours, then how much ibuprofen is in a person’s bloodstream after (a) 7 hours? (b) after t hours? 37a.)_______________ 37b.)_______________ 38. Simplify: a. 35  33 3 2 38a.)_______________ b. 3 81 8 7 38b.)_______________ c. 6x 4 3  2x 2x 1 3 5 3 38c.)_______________ 39. 40. Solve for x: a. log2 x  4 39a.)_______________ b. log3 x 2  5  log3 4 x 2  2 x  39b.)_______________ c. log5 25 5  x 39c.)_______________ d. log 6 4  log 6 3  log 6 2  x 39d.)_______________ Solve for x: a. e 5 x  2 x3 40a.)_______________ b. 5 x1  23x6 40b.)_______________ c. e 2 x  3e x  2  0 40c.)_______________ d. 9 x 3  81 27 x 40d.)_______________ Chapter 4 [4.1-4.6]: Trigonometry 41. 42. 43. A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 240 . 41.)________________ The circular blade on a saw rotates at 2400 revolutions per minute. a. Find the angular speed in radians per second. 42a.)_______________ b. The blade has a radius of 4 inches. Find the linear speed of a blade tip in inches per second. 42b.)_______________ A satellite in circular orbit 1125 km above a planet makes one complete revolution every 120 minutes. Assuming that he planet is a sphere of radius 6400 km, find the linear speed of the satellite in kilometers per minute. Round your answer to the nearest whole number. 43.)________________ 44. 45. 46. A truck is moving at a rate of 90 km per hour and the diameter of its wheels is 1.25 meters. Find the angular speed of the wheels in radians per minute. 44.)________________ Evaluate (if possible) the six trigonometric functions if   2 . 3 45.  2  sin    3   2  cos    3   2  csc    3   2  sec    3   2  tan     3   2  cot    3  Evaluate the trigonometric function. a.  3  sin    4  b.  7  csc   6  c.  5  tan    3  46a.)_______________ 46b.)_______________ 46c.)_______________ d. sec 4  e.  5  cot   2  f.  13  cos   4  46d.)_______________ 46e.)_______________ 46f.)_______________ 47. Find the exact values of the six trigonometric functions of the angle  shown in the figure. 18  12 48. 47. sin   csc   cos   sec   tan    cot   Simplify each expression. a. sin x cos x 1  cos 2 x b. sin x  cos x2  sin x  cos x2 48a.)_______________ 48b.)_______________ c. tan 2 x 1 sec x  1 d. sec x  sin x tan x 48c.)_______________ 48d.)_______________ 49. Find two solutions of the equation. Give your answers in degrees 0    360 and radians 0    2  without using a calculator. a. cot   3 3 b. sec  2 49a.)_____________________ 49b.)_____________________ 50. 51. 52. 53. John stands 150 meters from a water tower and sights the top at an angle of elevation of 36 . How tall is the tower? 50.)________________ The Aerial run in Snowbird, Utah has an angle of elevation of 20.2 . Its vertical drop is 2900 feet. Estimate the length of the run. 51.)________________ In a sightseeing boat near the base of the Horseshoe Falls at Niagara Falls, a passenger estimates the angle of elevation to the top of the falls to be 30 . If the Horseshoe Falls are 173 feet high, what is the distance from the boat to the base of the falls? 52.)________________ Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. 3 cos   , sin   0 7 54. 53. sin   csc   cos   sec   tan    cot   The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. 8,  15 54. sin   csc   cos   sec   tan    cot   Answer the questions regarding the amplitude, period, phase shift and vertical shift of each trigonometric function y  a cosb  c   d or y  a sinb  c   d . Then graph two full periods of the trigonometric function. Be sure to label your axes accurately! 55. 56. 57. y  cos 4  2   y  3 sin   1 2   y  3 cos    4 2  a= b= c= d= amplitude = period = phase shift = vertical shift = a= b= c= d= amplitude = period = phase shift = vertical shift = a= b= c= d= amplitude = period = phase shift = vertical shift = 58.  x   y  4 cos    3 a =  2 2 b= c= d= period = phase shift = vertical shift = a= b= c= d= amplitude = period = phase shift = vertical shift = a= b= c= d= amplitude = period = phase shift = vertical shift = amplitude = Graph two periods of the trigonometric function. 59. 60. y  3 tan 2 1 y  cot  1 2 Answer Key 1a. 2b. 10 -1 Midterm Review Packet x 2  2x  2 1b. 2c. 1c. 2d. 2 b6  1 2a. 3a. 3b.  ,  3c.  ,  4  4,  4a. decreasing: 4a. constant: none 4a. 4a. rel. max.: 4b. increasing: 4b. 4b. 5a. 6a. constant: none 4b. 4b. rel. max.: none even 5b. neither 5c. odd f (x) has a horizontal shift 2 units left, reflects over the x-axis and has a vertical shift 6 units down 6b. 6c. 7a. 3,  rel. min.:  3, 0; 3, 0 f (x) has a vertical stretch of 2 and a vertical shift up 3 units f (x) has a horiz. shift 5 units right, reflects over the x-axis, a vertical shrink of ½ and a vertical shift 7 units down -7 7e. 8b.  1, 1  1, 2 4a. -3  ,  2   2,  increasing:  ,  1  1,  rel. min.: 1,  2 decreasing:  ,  3 6 7 f 1 ( x)  3 x 3 4 7b. 70 7c. 7f. -97 8a. 8c. f 1 ( x)  x 2  10, x  0 5 f 1 ( x)  7d. 8x  3 7 1 5 9. show that f g ( x)   g  f ( x)   x 10. See graphs 11a.  2i 3 11b. 5  12i 11c. 11d.  1 4 3  i 7 7 11e. i 11f. 17  11i 12a. 12b.  2  11 12c. 3  3i 13a. 13 13b. 13c. 6  4i 14. 18 15. P( x)  2 x 3  3x 2  4 x  3 16. i 17.  1, 2,  3,  3 2 18. See graph 20.  1,  2 21. 22. 23. See graph 24a. 1 1,  ,  1  2i 2 x-intercept: 3, 24b. 24c.  19. 4 7  i 65 65 3 4 , 4 3 1 4 15 meters f ( x)  x3  5 x 2  8 x  6 0 ; y-intercept:  0, 3  ; H.A.: y = 1; V.A.: x = 2; S.A.: none  2 x-intercept: 0, 0 ; y-intercept: 0, 0 ; H.A.: y = 2; V.A.: x = 2 and x = -2; S.A.: none x-intercept: none; y-intercept:  1  ; H.A.: none; V.A.: x = 3; S.A.: x + 2 0,    25. c and d 26. 27a. 27 27b. 28a. -1 28b. 31a. $1452.89 31b. 33a. 4 34b. 3 log b M  2 log b N 37b.  3 vertex: 4,  4 ; x-intercept: 2, 0; 6, 0 ; y-intercept: 0, 12 27c. 729 27d. 5 8 29. 24.1% 30. See graph $1454.00 32a. 0.6021 32b. -0.6021 33b. 1 3 33c. 125 34a. 2 logb M  logb N 3 35.  3  57 2 36. 3.13 37a.  49.61  1 3 y  250  2 38a. 81 38b. 64 38c. 3  x3 x 39a. 1 16 39b. 39d. 1 40a. -0.483 40b. 40d. -2 41. 28 inches 3 42a. 43. 394 km/min 44. 2400 rad/min 45. 46a.  16 17 3 2 t 5 , 1 3 39c.  5.412 80 rad/sec 40c. 42b. 3  2  sin    2  3  1  2  cos    2  3   csc    sec    2  tan     3  3  2  cot   3  3  3 2 3 2 3    3  2   2 3  5 2 0, 0.693(ln 2)  1005in/sec 2 2 46b. 46f. -2  2 2 48c. sec x 49b. 45, 315, 53. 55.  7 4 , 47. 3 13 sin   csc   13 13 3 cos   2 13 sec   13 13 2 tan    3 2 50. 46d. cot   48a. 1 cot x 2 3 cos x 108.98 meters 49a. 120, 300, 51. 8398.54 feet 46e. 0 48b. 2 2 5 , 3 3 52. 299.64 feet 4 2 10 7 10 csc    7 20 cos    3 7 sec    2 10 3 cot   a=1  3 48d. sin    tan    46c. b=4 54. 7 3 sin    15 17 csc    17 15 cos   sec   8 17 tan     3 10 20 c=0 d = -2 55. – 60. See graphs 17 8 15 8 cot    8 15 Amp = 1 Per =  Phase Shift = 0 Vert. Shift = -2 Vert. Shift = 1 2 1 2 56. a = -3 b= 57. a=3 b =1 58. a=4 b=  d=1 Amp = 3 Per = 4 Phase Shift = 0 c=  d=4 Amp = 3 Per = 2 Phase Shift =   Vert. Shift = 4 c = -1 d = -3 Amp = 4 Per = 4 Phase Shift = -1 Vert. Shift = -3 c=0 d=0 Amp = none Per =  Phase Shift = none Vert. Shift = none Phase Shift = none Vert. Shift = 1 c=0 2 2 2 59. a = -3 b=2 2 60. a= 1 2 b=1 c=0 d=1 Amp = none Per = 