Midterm Review Sheet PDF

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This is a midterm review sheet and the professor was Elisabeth Jaffe....

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MAT 206 – Midterm Review Sheet Equations of a Line 1. Find slope and x and y-intercepts of the line 4x + 8y = 64 2. Find slope and x and y-intercepts of the line 2x + 3y = 26 3. Find slope and x and y-intercepts of the line y = 3x + 7 4. Find slope and x and y-intercepts of the line y – 2 = 4(x – 5) 5. Find equation of a line given a y-intercept of (0, 5) and a slope of 10. 6. Find equation of a line given a y-intercept of (0, -2) and a slope of -1/3. 7. Find equation of a line given the points (2, 3) and (5, 8). 8. Find equation of a line given the points (-4, 1) and (-6, -10). Parallel and Perpendicular Lines 9. Given a line with a slope of 2, find the equation of a line that is parallel and passes through the point (0, 3). 10. Given a line with a slope of 1/5, find the equation of a line that is parallel and passes through the point (2, -8). 11. Given a line with an equation of 2x – 6y = 24, find the equation of a line that is parallel and passes through the point (0, -6). 12. Given a line with a slope of -2/3, find the equation of a line that is perpendicular and passes through the point (-3, 2). 13. Given a line with a slope of 4, find the equation of a line that is perpendicular and passes through the point (1, -6). 14. Given a line with an equation of 3x + 5y = 30, find the equation of a line that is perpendicular and passes through the point (0, 7). Solving Functions 15. If h(r) = 2r - 9, find h(8) 16. If g(t) = -3t3 + 4t2 – 8t + 1, find g(-2) 17. If r(x) = 7x – 8, find r(3x) 18. If h(r) = 3r2 + 5r - 7, find h(x + 2) Solving for Roots 19. Find the roots of 2x2 – 13x – 7 20. Find the roots of 3x2 + 2x – 16 21. Find the roots of 2x2 – 16x + 8 22. Find the roots of x3 + 9x2 – x – 9 23. Find the roots of 4x3 + 8x2 – 9x – 18 24. Find the roots of x3 – 5x2 + 4x - 20 Finding Domain 25. Find the domain of

√2𝑥+1 𝑥−7

26. Find the domain of

3𝑥−4 √𝑥+3

27. Find the domain of

4𝑥+2 𝑥−5

28. Find the domain of √6𝑥 + 3

Increasing / Decreasing 29. Show the intervals on the graph where it is increasing, where it is decreasing, and where it is constant

30. Show the intervals on the graph where it is increasing, where it is decreasing, and where it is constant

Difference quotient 31. Find difference quotient of f(x) = x2 + 2x 32. Find difference quotient of f(x) = 2x2 - 3 33. Find the difference quotient of f(x) = 3x2 – 7x Average rate of change 34. Find average rate of change from x = 1 to x = 5 of f(x) = 2x3 – 3x2 + 9 35. Find average rate of change from x = -2 to x = 3 of f(x) = 4x2 – x + 10 36. Find average rate of change from x = -4 to x = -1 of f(x) = x3 + 4x2 – 6x + 5 Composition of Functions 37. Find f(g(x)) if f(x) = 2x – 7 and g(x) = x2 – 3 38. Find g(f(x)) if f(x) = x2 + 3x – 1 and g(x) = x – 3 39. Find h(p(3)) if h(x) = 2x + 4 and p(x) = 7x 40. Use composition of functions to determine if f(x) and g(x) are inverses when f(x) = 3(x + 2) 𝑥−2 and g(x) = 3 41. Find f(f-1(x)) when f(x) = 3x + 9 Transformations of Functions 42. Vertically shrink f(x) by a factor of 5 when f(x) = 7x - 8 43. Shift f(x) up 4 and left 7 when f(x) = 2x2 - 6 44. Reflect f(x) in the y-axis when f(x) = 4x3 – 7x2 + 8 45. Horizontally shrink f(x) by a factor of 2 when f(x) = 3x - 8 46. Reflect f(x) in the origin when f(x) = -4x + 10 47. Vertically stretch f(x) by a factor of 4 and then shift it 4 right and 2 down when f(x) = x2 - 3 Quadratic Functions 48. Put in vertex form and find the vertex of 3a2 – 24a + 9 49. Put in vertex form and find the vertex of 4p2 – 24p + 12 50. Put in vertex form and find the vertex of x2 + 8x - 4 End Behavior 51. Determine the degree of the polynomial and if the leading coefficient is positive or negative of the function below:

52. Determine the degree of the polynomial and if the leading coefficient is positive or negative of the function below:

53. 54. 55. 56.

Determine end behavior of f(x) = -7x4 + 9x3 – 4x + 8 Determine the end behavior of f(x) = 3x3 + 2x – 4 Determine the end behavior of f(x) = 6x6 + 2x5 – 3x3 + 4x – 10 Determine the end behavior of f(x) = -4x5 + 2x2 + 3x

Remainder / Factor Theorem 57. Using the remainder theorem, find the remainder when you divide polynomials 58. Using the remainder theorem, find the remainder when you divide polynomials 59. Using the factor theorem, determine if the divisor is a factor of the dividend 60. Using the factor theorem, determine if the divisor is a factor of the dividend

2𝑥 2 +3𝑥−7

𝑥+2 −5𝑥3 −2𝑥+1

𝑥−1 3𝑥 2 −3𝑥−6 𝑥−2 4𝑥 3 −12𝑥+2 𝑥+1

Find the polynomial 61. Given that a polynomial has a degree = 3 and roots of x = 3, x = 2, and x = -1, find the polynomial 62. Given that a polynomial has a degree = 4 and roots of x = 5, x = -2, and x = 5+2i, find the polynomial 63. Given that a polynomial has a degree = 2 and roots of x = -4+2i, find the polynomial 64. Given that a polynomial has a degree = 5 and roots of x = -2, x = 3+3i, and x = -5-i, find the polynomial Finding Asymptotes 65. Find the horizontal, slant, and vertical asymptote(s) if they exist 4𝑥 2 𝑥+2

66. Find the horizontal, slant, and vertical asymptote(s) if they exist 𝑥 2 −3𝑥−4 2𝑥 2 +𝑥−1

67. Find the horizontal, slant, and vertical asymptote(s) if they exist 3𝑥

𝑥 2 +2𝑥−3

68. Find the horizontal, slant, and vertical asymptote(s) if they exist 2𝑥 3 −𝑥 2 −2𝑥+1 𝑥 2 +3𝑥+2

69. Find the horizontal, slant, and vertical asymptote(s) if they exist 𝑥 2 +2𝑥−8 𝑥+2

Partial Fractions 70. Write the partial fraction decomposition (find the constants) 71. Write the partial fraction decomposition (find the constants)

4𝑥 2 −1

2𝑥(𝑥+1)2 3𝑥+6 𝑥 3 +2𝑥

72. Write the partial fraction decomposition (don’t find the constants) 73. Write the partial fraction decomposition (don’t find the constants)

2𝑥 3 −4𝑥 2 −15𝑥+5 𝑥 2 −2𝑥−8 𝑥 3 −𝑥+3

𝑥 2 +𝑥−2

Exponential functions (Compounding Interest) 74. If you invest $10000 in a bank account with an APR of 5% that compounds continuously, how much do you have after 6 years? 75. If you invest $11500 in a bank account with an APR of 1.2% that compounds quarterly, how much do you have after 3 years? 76. If you invest $200000 in a bank account with an APR of 3% that compounds daily, how much do you have after 10 years? Expanding and Condensing Logarithmic Expressions

77. Using the properties of logarithms, expand ln

𝑥2

𝑦2 𝑧3 𝑦

78. Using the properties of logarithms, expand log 2 (𝑥 4 √ 𝑧3 ) 79. Using the properties of logarithms, expand log 4 11𝑏 2 𝑐 80. Using the properties of logarithms, condense 3log3x + 1/4log3y – 4log3z 81. Using the properties of logarithms, condense 2(ln x + 4ln y) – 3ln z

Solving Logarithmic and Exponential Equations Variables in exponents Solve for x: 82. 82x+1 = 32x - 10 83. 4(3x) = 20 84. 3x + 8 = 76 - x 85. 4ex = 91 Solving equations where the bases of the logs are equal Solve for x: 86. log (3x – 4) = log (x – 10) 87. log3(5x + 7) = log3(9x – 8)

Solving equations requiring multiplication of logarithms Solve for x: 88. ln(x + 1) – ln(x – 2) = ln x 89. log4x – log4(x – 1) = ½ 90. log2x + log2(x + 2) = log2(x + 6)...