AP Precalculus SL Fall 2024 Exam Review PDF

# AP Precalculus – SL (Fall 2024) ## Exam Review 1. A wind turbine uses the power of wind to generate electricity. The blades of the turbine make a noise that can be heard at a distance from the turbine. At a distance of d = 0 meters from the turbine, the noise level is 105 decibels. At a distance...

# AP Precalculus – SL (Fall 2024) ## Exam Review 1. A wind turbine uses the power of wind to generate electricity. The blades of the turbine make a noise that can be heard at a distance from the turbine. At a distance of d = 0 meters from the turbine, the noise level is 105 decibels. At a distance of d = 100 meters from the turbine, the noise level is 49 decibels. The noise level can be modeled by the function S given by S(d) = *ab^d*, where S(d) is the noise level, in decibels, at a distance of d meters from the turbine. a. Use the given data to write two equations that can be used to find the values for constants *a* and *b* in the expression for S(d). * S(0) = 105 * S(100) = 49 * ab⁰ = 105 * ab¹⁰⁰ = 49 b. Find the values for *a* and *b*. * a = 105 * ab¹⁰⁰ = 49 * 105b¹⁰⁰ = 49 * b¹⁰⁰ = 49/ 105 * b = (49/105)^1/100 * b ≈ 0.992408 c. Use the given data to find the average rate of change of the noise level, in decibels per meter, from d = 0 to d = 100 meters. Express your answer as a decimal approximation. Show the computations that lead to your answer. * Average Rate of Change = (S(100) - S(0)) / (100 - 0) * = (49 - 105) / 100 * = -0.56 decibels per meter d. Interpret the meaning of your answer from (c) in the context of the problem. On average, as the distance from the turbine increases from 0 to 100 meters, the noise level decreases 0.56 decibels per meter. e. Use the average rate of change found in (c) to estimate the noise level, in decibels, at a distance of d = 120 meters. Show the work that leads to your answer. * y - y₁ = m(x - x₁) * y - 105 = -0.56(120 - 0) * y - 105 =- 67.2 * y = 37.8 The noise level at a distance of d = 120 meters is 37.8 decibels. f. Use the model S to find the value of m such that the noise level is 20 decibels at a distance of m meters from the turbine. * S(d) = *ab^d* * S(d) = 105(0.992408)^d * 20 = 105(0.992408)^d * d = log(20/105) / log(0.992408) * d ≈ 217.585 2. a. Use the following functions to solve the following problems. * g(x) = 4x - 1 * h(x) = 5 - 2x * j(x) = 1 - x² i. (g o h)(2) * (g o h)(x) = g(h(x)) * g(h(2)) = g(5 - 2 * 2) = g(1) * g(1) = 4(1) - 1 = 3 * (g o h)(2) = 3 ii. (j o h)(x) * (j o h)(x) = j(h(x)) * j(h(x)) = j(5 - 2x) = 1 - (5 - 2x)² * = 1 - (25 - 20x + 4x²) * = 1 - 25 + 20x - 4x² * = -4x² + 20x - 24 * = x² - 5x + 6 * = (x - 2)(x - 3) b. Find f^-1(x) when f(x) = (x - 4)² + 3. For what values of x is f^-1(f(x)) = x? For what values of x is f(f^-1(x)) = x? * y = (x - 4)² + 3 * x = (y - 4)² + 3 * x - 3 = (y - 4)² * √(x - 3) = y - 4 * y = √(x - 3) + 4 * f^-1(x) = √(x - 3) + 4; x ≥ 3 For values of x ≥ 3, f^-1(f(x)) = x and f(f^-1(x)) = x. 3. What exponential function goes through the following points a. (1, 8) and (2, 4) * f(x) = Ca^x * a = 4 / 8 = 1/2 * f(1) = Ca¹ = 8 * C(1/2) = 8 * C = 16 * f(x) = 16(1/2)^x b. (0, 4) and (1, 12) * f(x) = Ca^x * a = 12 / 4 = 3 * f(0) = Ca⁰ = 4 * C = 4 * f(x) = 4 * 3^x 4. a. Solve the exponential equation 9^-x+15 = 27^x for x. Express your answer as a decimal approximation. * 9^-x+15 = 27^x * 3^2(-x+15) = 3^3x * 2(-x + 15) = 3x * -2x + 30 = 3x * 5x = 30 * x = 6 b. If f(x) = 2x² + 5 and g(x) = 3x + a, find a so that the graph of f o g crosses the y-axis at 23. * (f o g)(x) = 2(3x + a)² + 5 * = 2(9x² + 6xa + a²) + 5 * = 18x² + 12xa + 2a² + 5 The graph crosses the y-axis when x = 0. * 18(0)² + 12(0)a + 2a² + 5 = 23 * 2a² + 5 = 23 * 2a² = 18 * a² = 9 * a = ±3