Calculus Exam
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What is lim→∞ (x² - 4)/(2 + x - 4x²) ? A. -2 B. -1/4 C. 1/2 D. 1 E. The limit does not exist.
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B
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Suppose the second derivative of a function equals x² - 2x - 3. On what intervals would the function be concave up? A. (-∞ , ∞) B. (-∞ , -1) and (3 , ∞) C. (-∞ , -1) D. (1 , 3) E. (1 , ∞)
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D
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∫ (x - 1)√[x] dx A. (2/5)x⁵/² - (2/3)x³/² + C B. (1/2)x² + 2x¹/² + C C. (2/3)x³/² + 2x¹/² + C D. (1/2)x² - x + C E. (3/2)x¹/² - x⁻¹/² + C
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A
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Let f and g be functions defined by f (x) = sin x and g (x) = cos x. For which of the following values of a is the line tangent to the graph of f at x = a parallel to the line tangent to the graph of g at x = a? A. 0 B. π/4 C. π/2 D. 3π/ 4 E. π
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D
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The acceleration at time t, of a particle moving along the x - axis is given by the equation a (t) = 20t³ + 6. At time t = 0, the velocity of the particle is 0 and the position of the particle is 7. What is the position of the particle at time t? A. 120t + 7 B. 60t² + 7t C. 5t⁴ + 6t +7 D. t⁵ + 3t² + 7 E. t⁵ + 3t² + 7t
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D
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If f (x) = (sin x) / (2x), then f '(x) = A. (cos x) / 2 B. (x cos x - sin x) / (2x²) C. (x cos x - sin x) / (4x²) D.(sin x - x cos x) / (2x²) E.(sin x - x cos x) / (4x²)
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B
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↓x..↓f(x)↓f'(x)↓g(x)↓g'x↓ ↓10↓35..↓15...↓6.....↓4...↓ ↓20↓8....↓5.....↓12...↓10.↓ ↓30↓24..↓25...↓20..↓10.↓ Selected values of the functions f and g and their derivatives, f' and g', are given in the table above. If h(x) = f(g(x)), what is the value of h'(30)? A. 5 B. 15 C. 35 D. 50 E. 250
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D
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What is lim(h→0) {cos(π/2 + h) - cos π/2}/h ? A. -∞ B. -1 C. 0 D. 1 E. ∞
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B
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If x² + y³ = x³y², then dy/dx = A. (2x + 3y² - 3x²y²) / (2x³y) B. (2x³y + 3x²y² - 2x) / (3y²) C. (3x²y² - 2x) / (3y² - 2x³y) D. (3y² - 2x³y) / (3x²y² - 2x) E. (6x²y - 2x) / (3y²)
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C
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For which of the following functions does d³y/dn³ = dy/dn ? I. y = eⁿ II. y = e⁻ⁿ III. y = sin(n) A. I only B. II only C. III only D. I and II E. II and III
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D
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The vertical height, in feet, of a ball thrown upward from a cliff is given by s(t) = -16t² + 64t + 200, where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero?
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264
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Which of the following statements about the curve y = x⁴ - 2x³ is true? A. The curve has no relative minimum B. The curve has one point of inflection and two relative extrema. C. The curve has two points of inflection and one relative extrema. D. The curve has two points of inflection and two relative extrema. E. The curve has two points of inflection and three relative extrema.
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C
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d/dx (sin(cos x) = A. cos(cos x) B. sin(-sin x) C. (sin(-sin x))cos x D. -(cos(cos x))sin x E. -(sin(cos x))sin x
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D
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Let f be the function defined by f(x) ={ (x² - 25) / (x - 5) for x ≠ 5 ____ {0 ___________ for x = 5 Which of the following statements are true? I. The lim(x→5) f(x) exists. II. f(5) exists III. f(x) is continuous at x = 5 A. None B. I only C. II only D. I and II only E. I, II, and III
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D
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What is the average rate of change of the function f defined by f(n) = 100*2ⁿ on the interval [0,4] ? A. 100 B. 375 C. 400 D. 1,500 E. 1,600
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B
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If the functions f and g are defined for all real numbers and f is a antiderivative of g, which of the following statements is NOT necessarily true? A. If g(x) > 0 for all x, then f is increasing. B. If g(a) = 0, then the graph of f has a horizontal tangent at x = a. C. If f(x) = 0 for all x, then g(x) = 0 for all x. D. If g(x) = 0 for all x, then f(x) = 0 for all x. E. f is continuous for all x.
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D
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If f(x) = arctan(πx), then f'(0) = A. -π B. -1 C. 0 D. 1 E. π
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E
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A rectangle with one side on the x-axis and one side on the line x = 2 has its upper left vertex on the graph of y = x². For what value of x does the area of the rectangle attain its maximum value? A. 2 B. 4/3 C. 1 D. 3/4 E. 2/3
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B
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Let f(x) = x³ + x. If h is the inverse function of f, the h'(2) = A. 1/13 B. 1/4 C. 1 D. 4 E. 13
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B
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Let F be the number of trees in a forest at time t, in years. If F is decreasing at a rate given by the equation dF/dn = -2F and if F(0) = 5000, then F(n) = A. 5000n⁻² B. 5000e⁻²ⁿ C. 5000 - 2n D. 5000 + n⁻² E. 5000 + e⁻²ⁿ
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B
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The function f is given by f(x) = sin(12x). Which of the following is the local linear approximation for f at x = 0. A. y = 12x B. y = -12x C. y = 1 + 12x D. y = 1 - 12x E. y = -1 + 12x
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A
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What is the area of the region in the first quadrant that is bounded by the line y = 6x and the parabola y = 3x ?
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4
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Let f be a differentiable function defined on the closed interval [a,b] and let c be a point in the open interval (a,b) such that f'(c) = 0, f'(x) > 0 when a ≤ x < c, and f'(x) < 0 when c < x ≤ b. Which of the following statements must be true? A. f(c) = 0 B. f''(c) = 0 C. f(c) is an absolute maximum value of f on [a,b]. D. f(c) is an absolute minimum value of f on [a,b]. E. The graph of f has a point of inflection at x = c.
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C
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The function f is continuous on the open interval (-π,π). If f(x) = {cos(x - 1)} / xsinx for x ≠ 0, what is the value of f(0)? A. -1 B. -1/2 C. 0 D. 1/2 E. 1
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B
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g(20) = 0 g'(t) > 0 for all values of t The function g is differentiable and satisfies the conditions above. Let F be the function given by F(x) (subscript 0, superscript x) ∫ g(t) dt. Which of the following must be true? A. F has a local minimum at x = 20. B. F has a local maximum at x = 20. C. The graph of F has a point of inflection at x = 20. D. F has no local minima or local maxima on the interval 0 ≤ x < ∞. E. F'(20) does not exist.
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A
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The Riemann sum (i = 1,50) ∑(i/50)²1/50 on the closed interval [0,1] is an approximation for which of the following definite integrals? A. (0,1) ∫ x² dx B. (0,50) ∫ x² dx C.(0,1) ∫ (x/50)² dx D.(0,50) ∫ (x/50)² dx E.(0,1) ∫ x²/50³ dx
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A
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(-3,3) ∫ |x +2| dx = A. 0 B. 9 C. 12 D. 13 E. 14
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D
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A particle moves along the x-axis, and its velocity at time t is given by v(t) = t³ - 3t² + 12t + 8. What is the maximum acceleration of the particle on the interval 0 ≤ t ≤ 3? A. 0 B. 9 C. 12 D. 13 E. 14
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D
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(5,10) ∫ ln(10x)/x dx = A. 1.282 B. 2.952 C. 5.904 D. 6.797 E. 37.500
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B
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f(x) = {x - 3 for when x 2 _____{2___for when x = 2 What is the lim(→2) f(x) ? A. -1 B. 0 C. 1 D. 2 E. The limit does not exist.
answer
E
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If the function f is continuous for all real numbers and the lim(h→0) {f(a + h) - f(a)} / h = 7, then which of the following statements must be true? A. f(a) = 7 B. f is differentiable at x = a C. f is differentiable for all real numbers. D. f is increasing for x > 0 E. f is increasing for all real numbers.
answer
B
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Let f be a function with a second derivative given by f''(x) = sin(2x) - cos(4x). How many points of inflection does the graph of f have on the interval [0,10] ? A. 6 B. 7 C. 8 D. 10 E. 13
answer
B
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The area of the region in the first quadrant between the graph of y = x√[4 -x²] and the x-axis is A. 2/3√[2] B. 8/3 C. 2√[2] D. 2√[3] E. 16/3
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B
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The function f is given by f(x) = 3x² + 1. What is the average value of f over the closed interval [1,3] ?
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14
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Starting at t = 0, a particle moves along the x-axis so that its position at time t is given by x(t) = t⁴ -5t² + 2t. What are all the values of t for which the particle is moving to the left? A. 0 < t < 0.913 B. 0.203 < t < 1.470 C. 0.414 < t < 0.913 D. 0.414 < t < 2.000 E. There are no values of t for which the particle is moving to the left.
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B
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The function f has a relative maximum value of 3 at x = 1, as shown in the figure above. If h(x) = x² f(x), the h'(1) = A. -6 B. -3 C. 0 D. 3 E. 6
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E
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∫ cos²xsinx dx = A. -cos³x/3 + C B. -cos³xsin²x/6 + C C. sin²x/2 + C D. cos³x/3 + C E. cos³xsin²x/6 + C
answer
A
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f'(n) = 1 - (n² + n)e⁻ⁿ The first derivative of the function f is given above. At what value of x does the function f attain its minimum value on the closed interval [-5,5] ? A. -5.000 B. -1.235 C. -0.618 D. 0.160 E. 1.618
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B
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The function f is differentiable on [a,b] and a < c < b. Which of the following is NOT necessarily true? A. (a,b) ∫ f(x) dx + (b,c) ∫ f(x) dx B. There exists a point d in the open interval (a,b) such that f'(d) = {f(b) - f(a)} / (b - a). C. (a,b) ∫ f(x) dx ≥ 0 D. lim(→c) f(x) = f(c) E. If k is a real number, then (a,b) ∫ f(x) dx = k(a,b) ∫ f(x) dx.
answer
C
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g'(x) = tan[2/(1+x²)] Let g be the function with first derivative given above and g(1) = 5. If f is the function defined by f(x) = ln(g(x)), what is the value of f'(1) ? A. 0.311 B. 0.443 C. 0.642 D. 0.968 E. 3.210
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A
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Let r(t) be a differentiable function that is positive and increasing. The rate of increase of r³ is equal to 12 times the rate of increase of r when r(t) = A. ³√[4] B. 2 C. ³√[12] D. 2√[3] E. 6
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B
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Analyze the graph of f(x) = (1/3)x³ - x² + 5. Suppose that A is where the function equals zero, B is where the function is x has a negative value, C is the relative maximum, D is where x has a positive value, and E is the relative minimum; at which of the points could the derivative of f be equal to the average rate of change of f over the closed interval [-2,4]? A. A B. B C. C D. D E. E
answer
B
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d/dx (1,x) ∫ t² dt = A. 2x B. x² - 1 C. x² D. x³/3 - 1 E. x³/3 + C
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C
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A college is planning to construct a new parking lot. The parking lot must be rectangular and enclose 6000 square meters of land. A fence will surround the parking lot, and another fence parallel to one of the sides will divide the parking lot into two sections. What are the dimensions, in meters, of the rectangular lot that will use the least amount of fencing? A. 1000 by 1500 B. 20√[5] by 60√[5] C. 20√[10] by 30√[10] D. 20√[15] by 20√[15] E. 20√[15] by 40√[15]
answer
C
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↓..x..↓1..↓2...↓3↓4↓5↓ ↓f(x)↓15↓10↓9↓6↓5↓ The function f is continuous on the closed interval [1,5] and has values that are given in the table above. If two subintervals of equal length are used, what is the midpoint Riemann sum approximation of (1,5) ∫ f(x) dx? A. -3 B. 9 C. 14 D. 32 E. 35
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D
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If f is continuous for all x, which of the following integrals necessarily have the same value? I. (a,b) ∫ f(x) dx II. (0,b - a) ∫ f(x + a) dx III. (a + c,b + a) ∫ f(x + c) dx A. I and II only B. I and III only C. II and III only D. I, II, and III E. No two necessarily have the same value?
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A
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A spherical balloon is being inflated at a constant rate of 25 cm³/sec. At what rate, in cm/sec, is the radius of the balloon changing when the radius is 2 cm.(The volume of a sphere with radius r is V = 4/3πr³). A. 25/16π B. 25/8π C. 75/16π D. 32π/25 E. 32π/3
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A
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R is the region below the curve y = x and above the x-axis from x = 0 to x = b, where b is a positive constant. S is the region below the curve y = cosx and above the x-axis from x = 0 to x = b. For what value of b is the area of R equal to the area of S? A. 0.739 B. 0.877 C. 0.986 D. 1.404 E. 4.712
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D
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Let f be defined by f(n) = e³ⁿ, and let g be the function defined by f(n) = n³. At what value of x do the graphs of f and g have parallel tangent lines? A. -0.657 B. -0.526 C. -0.484 D. -0.344 E. -0.261
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C
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d/dx (sin⁻¹(5x)) = A. cos⁻¹(5x) B. 5cos⁻¹(5x) C. 1/√[1 - 5x²] D. 5/√[1 - 25x²] E. 5/(1 + 25x²)
answer
D
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The population P of bacteria in an experiment grows according to the equation dP/dt = kP, where k is a constant and t is measured in hours. If the population of bacteria doubles every 24 hours, what is the value of k? A. 0.029 B. 0.279 C. 0.693 D. 2.485 E. 3.178
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A
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