AP Calculus Summer Homework

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1 Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. ln e 3 2. Write as the sum and/or difference of logarithms. Express powers as factors. (note: the 3/5 is an exponent acting on the argument of the log) ln 3/5, x > 7 3. Evaluate the expression using the values given in the table. f(g(-5)) 4. Solve the equation. e x - 3 = x Solve the problem. The bacteria in a 9-liter container double every 3 minutes. After 48 minutes the container is full. How long did it take to fill a quarter of the container? 6. Change the exponential expression to an equivalent expression involving a logarithm. e x = 6 1

2 7. Find the domain of the function. f(x) = ln(5 - x) 8. Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. f(x) = x Graph the function. f(x) = 10. Evaluate the expression using the given values. x = 5, y = Evaluate the expression using the given values. x = -2, y = 8 2

3 12. Evaluate the expression using the given values. - x = -8, y = Determine which value(s), if any, must be excluded from the domain of the variable in the expression. 14. Simplify the expression. (-5) Simplify the expression Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not 0. (8x 3 ) Simplify the expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not Factor completely. If the polynomial cannot be factored, say it is prime. x Factor completely. If the polynomial cannot be factored, say it is prime. x x Factor completely. If the polynomial cannot be factored, say it is prime. x 2 + 2x Factor completely. If the polynomial cannot be factored, say it is prime. x 2 + 4x

4 22. Factor completely. If the polynomial cannot be factored, say it is prime. 2x 2-2x Factor completely. If the polynomial cannot be factored, say it is prime. x An expression that occurs in calculus is given. Factor completely. 2(x + 6)(x - 5) 3 + (x + 6) 2 6(x - 5) Reduce the rational expression to lowest terms. 26. Reduce the rational expression to lowest terms. 27. Reduce the rational expression to lowest terms. 28. Express the graph shown using interval notation. Also express it as an inequality involving x. 29. Express the graph shown using interval notation. Also express it as an inequality involving x. 30. Express the graph shown using interval notation. Also express it as an inequality involving x. 4

5 31. Simplify the expression. Assume that all variables are positive when they appear. 32. Simplify the expression. Assume that all variables are positive when they appear. 33. Simplify the expression. Assume that all variables are positive when they appear Simplify the expression. Assume that all variables are positive when they appear Simplify the expression. Assume that all variables are positive when they appear. ( + )( - ) 36. Simplify the expression. Assume that all variables are positive when they appear. 37. Simplify the expression. (-8) 4/3 38. Simplify the expression /3 39. Solve the equation. - x = 40. Solve the equation. x = 4 + x 5

6 41. Solve the equation. = 42. Solve the equation by factoring. x 2-4x - 32 = Solve the equation by factoring. x(x - 10) + 24 = Solve the equation by factoring. x(x + 4) = Solve the equation by factoring. 16x 2-40x + 25 = Solve the equation by completing the square. x 2 + x - = Find the real solutions, if any, of the equation. Use the quadratic formula. 5x 2 + x - 2 = Find the real solutions of the equation. 2(x + 1) (x + 1) + 12 = Find the real solutions of the equation. (x - 4) 2 + 3(x - 4) - 18 = Solve the equation. = Solve the equation. = 7 6

7 52. Find the value for the function. Find f(2x) when f(x) =. 53. Find the value for the function. Find f(x + h) when f(x) = -2x 2-5x Find the value for the function. Find f(x + h) when f(x) =. 55. Find and simplify the difference quotient of f,, for the function. f(x) = 6x Graph the function. f(x) = 7

8 57. Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. f(x) = (x - 7) Graph the function by starting with the graph of the basic function and then using the techniques of shifting, compressing, stretching, and/or reflecting. f(x) = x For the given functions f and g, find the requested composite function value. f(x) = 4x + 6, g(x) = 4x 2 + 1; Find (g f)(4). 60. For the given functions f and g, find the requested composite function. f(x) =, g(x) = 8x + 7; Find (g f)(x). 8

9 61. Graph the function. f(x) = e x 62. Solve the equation. 4 -x = 63. Solve the equation x = 64. Find the exact value of the logarithmic expression. log Find the exact value of the logarithmic expression. log 5 9

10 66. Graph the equation by plotting points. y = 2x Find the slope of the line containing the two points. (3, -4); (-2, 3) 68. Find the slope of the line containing the two points. (6, -6); (-7, -1) 69. Solve the equation. log 12 (x 2 - x) = Solve the equation. 6 ln 5x = Solve the equation. e 4x = Solve the equation. e x + 5 = Express as a single logarithm. 9ln (x - 3) - 11 ln x 10

11 74. Express as a single logarithm. 56 log 7 + log 7 (56x 4 ) - log Solve the equation. log 3 (x - 1) + log 3 (x - 7) = Find the exact value. Do not use a calculator. cos 77. Find the exact value. Do not use a calculator. tan (- ) 78. Find the exact value. Do not use a calculator. sin 79. Find the exact value. Do not use a calculator. cos Find the exact value. Do not use a calculator. tan Find the exact value. Do not use a calculator. tan 11

12 82. Graph the sinusoidal function using key points. y = 2 sin(3x) 83. Graph the sinusoidal function using key points. y = cos x Find the vertical asymptotes of the rational function. R(x) = 85. Find the vertical asymptotes of the rational function. F(x) = 12

13 86. Give the equation of the horizontal asymptote, if any, of the function. H(x) = 87. Give the equation of the horizontal asymptote, if any, of the function. F(x) = 88. Give the equation of the horizontal asymptote, if any, of the function. G(x) = 89. Give the equation of the horizontal asymptote, if any, of the function. R(x) = Essay 90. An electric company has the following rate schedule for electricity usage in single-family residences: Monthly service charge $4.93 Per kilowatt service charge 1st 300 kilowatts Over 300 kilowatts $ /kW $ /kW What is the charge for using 300 kilowatts in one month? What is the charge for using 375 kilowatts in one month? Construct a function that gives the monthly charge C for x kilowatts of electricity. 13

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