AB Calculus
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Transcript of AB Calculus
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AB Calculus
Midterm Review Problems
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A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of ¼ ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?
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A tank of water in the shape of a cone is leaking water at a constant rate of 2 ft3/hr. The base radius of the tank is 5 ft and the height of the tank is 14 ft. a) At what rate the depth of the water in the tank
changing when the depth of the water is 6 ft?
b) At what rate is the radius of the top of the water in the tank changing when the depth of the water is 6 ft?
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A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?
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If y = (x3 + 1)2, then dy/dx =
a) (3x2)2 b) 2(x3 + 1) c) 2(3x2 + 1)
d) 3x2 (x3 + 1) e) 6x2 (x3 + 1)
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The graph of a function f is shown above. At which value of x is f continuous, but not differentiable?
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If the line tangent to the graph of the function f at the point (1, 7) passes through the point (-2, -2), then f ‘(1) is
a) -5 b) 1 c) 3 d) 7 e) undefined
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Let f be the function given by f(x) = 2xex. The graph of f is concave down when
a) x < -2 b) x > -2 c) x < -1
d) x > -1 e) x < 0
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The derivative g’ of a function g is continuous and has exactly two zeros. Selected values of g’ are given in the table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the following intervals?
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A curve has a slope 2x + 3 at each point (x, y) on the curve. Which of the following in an equation for this curve if it passes through the point (1, 2)?
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The second derivative of the function f is given by f ‘’ (x) = x(x – a)(x – b)2. The graph of f ‘’ is shown below. For what values of x does the graph of f have a point of inflection?
a) 0 and a onlyb) 0 and m onlyc) b and j onlyd) 0, a, and be) b, j, and k
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Let f be the function defined by f(x) = 4x3 – 5x + 3. Which of the following is an equation of the line tangent to the graph of f at the point where x = -1?
a)y = 7x – 3b)y = 7x + 7c)y = 7x + 11d)y = -5x – 1e)y = -5x – 5
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A particle moves along the x-axis so that at time t ≥ 0 its position is given by x(t) = 2t3 – 21t2 + 72t – 53. At what time t is the particle at rest?
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What is the slope of the line tangent to the curve 3y2 – 2x2 = 6 – 2xy at the point (3, 2)?
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Let f be the function defined by f(x) = x3 + x. If g(x) = f -1 (x) and g(2) = 1, what is the value of g’(2)?
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A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by v(t) = 3 + 4.1cos(0.9t). What is the accelerations of the particle at time t = 4?
a)-2.016b)-0.677c)1.633d)1.814e)2.978
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The radius of a circle is increasing at a constant rate of 0.2 m/s. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20π m.
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The function f is continuous for -2 ≤ x ≤ 1 and differentiable for -2 < x < 1. If f(-2) = -5 and f(1) = 4, which of the following statements could be false?
a) There exists c, where -2 < c < 1, such that f(c) = 0.b)There exists c, where -2 < c < 1, such that f ‘ (c) = 0.c) There exists c, where -2 < c < 1, such that f(c) = 3.d)There exists c, where -2 < c < 1, such that f ‘ (c) = 3.e)There exists c, where -2 ≤ c ≤ 1, such that f(c) ≥ f(x)
for all x on the closed interval -2 ≤ x ≤ 1.
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For which of the following does exist?
A) I onlyB) II onlyC) III onlyD) I and II onlyE) I and III only
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Let f be the function with derivative given by f ‘ (x) = sin(x2 + 1). How many relative extrema does f have on the interval 2 < x < 4?
a) One b) two c) threed) four e) five
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The rate of change of the altitude of a hot-air balloon is given by for 0 ≤ t ≤ 8. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?
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The function f has the first derivative given by . What is the x-coordinate of the inflection point of the graph of f ?
a) 1.008b) 0.473c) 0d) -0.278e) the graph of f has no inflection point
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Let f be a differentiable function with f(2) = 3 and f ‘(2)= -5, and let g be the function defined by g(x) = x f(x). Which of the following is an equation of the line tangent to the graph of g at the point where x = 2?
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For all x in the closed interval [2, 5], the function f has a positive first derivative and a negative second derivative. Which of the following could be a table of values for f ?