University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2019
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495

Pages:

13
Author:

InterstellarInnovator

Sample Examination II 1. ∫(2x³ + 3) dx = (2x⁴/4 + 3x)|₀² = (x⁴/2 + 3x)|₀² = (8 + 6) - (0 + 0) = 14 The correct choice is (C). 2. The average value of f' is given by 1/(2-(-2)) ∫₋₂² f'(x) dx. Using the Fundamental Theorem of Calculus, 1/(2-(-2)) ∫₋₂² f'(x) dx = (f(2) - f(-2))/4 = (e - a)/4 The correct choice is (B). 3. You need to find t such that dy/dt = 4. dy/dt = (2t - 3)/(4t + 1) Solving 2t - 3/(4t + 1) = 4, you get 4(4t + 1) = 2t - 3 or 16t + 4 = 2t - 3, and t = -1/2. The correct choice is (A). 4. f(x) = (2 + 3x)⁴ f'(x) = 4(2 + 3x)³(3) f''(x) = 3 * 4(2 + 3x)²(3)² f'''(x) = 2 * 3 * 4(2 + 3x)(3)³ f⁴(x) = 1 * 2 * 3 * 4 * (3)⁴ = 4!(3⁴) If y = (a + bx)ⁿ, then the nth derivative, dⁿy/dxⁿ = n!(bⁿ) The correct choice is (C). 22. Sample Examination II 5. f(x) = x⁴ - 8x² f'(x) = 4x³ - 16x = 4x(x² - 4) f'(x) = 0 when x = 0, -2, and 2. x | -∞ < x < -2 | -2 < x < 0 | 0 < x < 2 | x > 2 ---+--------------+------------+------------+------ f'(x) | - | + | - | + f(x) has a relative minimum where f'(x) switches sign from - to +. Therefore, f(x) has a relative minimum at x = 2 and x = -2 only. The correct choice is (D). 6. ∫√(x+2) dx = ∫(x+2)^(1/2) dx = (2/3)(x+2)^(3/2) + C = 2/5 x^(3/2) + 4/3 x^(3/2) + C The correct choice is (D). 7. A rational function may have only one horizontal asymptote. If it does, then as x → ±∞, the function must approach this value. This is expressed by the limit in (B). For example, the graph of f(x) = 7x²/(x² + 3) is such a function. Its graph and asymptote are shown below: The correct choice is (B). 37. The derivative may be approximated by calculating the slope from any two points near x = 4, for example, f'(4) ≈ (1.16016 - 1.15782) / (4.00100 - 4.00000) = 0.00234 / 0.001 = 2.340 Using other points gives comparable results. The correct choice is (D). The length of the path = ∫(√(dx/dt)² + (dy/dt)²) dt dx/dt = cos t and dy/dt = -2 sin(2t) Length = ∫(√(cos²t + 4sin²(2t))) dt Use a calculator to find the length to be 9.294. The correct choice is (B). Graph the two functions y = cos 2x and y = sin 3x in the interval 0 ≤ x ≤ 5. Choices (A) through (D) represent the areas of the regions enclosed by the graphs from left to right. It is easy to see that choice (D) is the largest. Choice (E) has the wrong function "on top" and therefore represents a negative number. The correct choice is (D). First graph the function f(x) in the window [0, 1.4] by [-10, 10]. Identify the zero at x = 1.042 (which is ln(π)/ln(3)). Then use the derivative function to find the derivative; and, the derivative at the point (x = 1.042) is 3.451. Do not be misled by the vertical asymptote near x = 0.411. The correct choice is (C). 32. Sample Examination II 41. The interval [0,6] is divided into three subintervals [0,2], [2,4], and [4,6]. The integral is approximated using the function's value at the midpoint of each interval multiplied by the width of the interval: (0.25)(2) + (0.68)(2) + (0.95)(2) = 3.76 For additional practice: (A) is the left Riemann sum, (E) is the right Riemann sum and (B) is the trapezoidal approximation, all with three subintervals. The correct choice is (D). 42. By the Fundamental Theorem of Calculus (version II), d/dx ∫[0,x] sin(t²) dt = 3x²sin(x⁶) - sin(x²) The correct choice is (C). 43. The amount of water, A, after x hours, is given by the accumulation function A(x) = ∫[0,x] 300√t dt After 4 hours, there are ∫[0,4] 300√t dt number of gallons. ∫[0,4] 300√t dt = 300 ∫[0,4] √t dt = 300 * (2/3) * t^(3/2) |[0,4] = 200(4^(3/2) - 0) = 1600 gallons The correct choice is (D). 44. Solve 2 sin x = x to obtain x = 1.89549. Let A = 1.89549. Using a graphing calculator, evaluate V = π ∫[0,A] (2 sin x)² - x² dx ≈ 6.678. The correct choice is (D).

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