#1: *
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#2: *
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#3 Find the horizontal asymptote of the following function:: *
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#4 Which of the following statements must be correct?: *
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| #5 Describe the limit of f(x) as x approaches 4.: * |
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Questions 6 - 7: Consider a function f(x), differentiable on [-1, 5]. Suppose f(1) = f(4) = 0 and f(3) = 6.
#6 Which of the following is not necessarily true? : * |
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| #7 What is the average rate of change on [1, 3]?: * |
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#8 Which is an example of a function that is continuous at x = 5 but not differentiable at x = 5?: *
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#9 What value of c will make f(x) continuous at x = 3?: *
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| #10 Find the derivative of g(x) = (x3 – 5)(x + 4).: * |
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Questions 11 - 13: Suppose f(x) and g(x) are differentiable functions on [-5, 5] and have values given by the following table:: *
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: *
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| #13 Find (f o g) ’(-2), where o denotes usual functional composition.: * |
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#14: *
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| #15 If f(x) = sin2(e3x), find f ’ (x): * |
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#16 Find the derivative y’ of the equation x + y = xy.: *
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| #17 Which of the following functions satisfy f(ab) = f(a) + f(b) for all positive real values a and b?: * |
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| #18 Which of the following are the domain and range, respectively, of f(x) = Arctan(x) where Arctan denotes the principal inverse tangent function?: * |
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Questions 19 - 21: Consider the following graph of the derivative f‘(x) of a three times differentiable function f(x), defined on the open interval (-3, 3) (thus f ’, f ’’, and f ’’’ all exist). Do not consider any behavior outside of this interval in answering.: *
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| #20 At what value(s) of x, if any, does f(x) have a local maximum?: * |
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| #21 On which of the following intervals is f ’(x) concave up?: * |
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#22: *
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#23: *
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#24: *
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#25: *
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#26: *
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For 27-28, consider the graphs y = 2x and y = 2x2 on [0, 1].
#27 Let S be the region enclosed by those graphs. What is the volume of the solid generated when S is revolved about the line y = 3? : *
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#28 The area bounded by the two curves on [0, 1] is given by: *
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#29: *
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#30: *
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| #31 Find the maximum value of f(x) = 18x – 6x3 on [-2, 5].: * |
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Question 32 uses the following slope field for a particular differential equation.: *
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For questions 33-34, assume f(x) is twice differentiable on [1, 7], f(1) = 0, f(7) = 6, f(3) = 4, f ’(3) = 0, and f ’’ (3) = -2.
#33 Which of the following must be true? : * |
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| #34 Where does f(x) assume its absolute maximum on [1, 7]?: * |
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#35 Two people are 50 feet apart. One of them starts walking north at a rate so that the angle θ between the two people’s paths is changing at a constant rate of 0.01 radians per minutes. At what rate is the distance between the two people changing when the θ = 0.5 radians?: *
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| #36 Suppose the length of a diagonal of a square increases from 16 cm to 16.1 cm. What would the change in the area of the square be?: * |
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#37: *
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#38: *
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#39: *
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| #40 Suppose f(x) is twice differentiable on [0, 7]. If f(3) = 5, f ’(3) = 2, and f ’’(3) = -1, which of the following statements is/are correct?: * |
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| #41 Let x = t5 - 4t3 and y = t2 be parametric equations. Find the first derivative for this set of parametric equations at t = 1.: * |
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| #42 Let f(3) = 5, f(5) = 2, f ’ (3) = 4, and f ’(5) = 9. Let g(x) be the inverse of f(x) and assuming both f(x) and g(x) are differentiable for all real numbers. Find g’(5).: * |
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#43 Find the Maclaurin series expansion of f(x) = e2x.: *
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| #44 Let Sk = 3 - (k - 1) be a sequence starting at k = 1. To what value does the sequence Sk converge?: * |
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#45: *
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#46 Which of the following series converge?: *
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#47: *
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#48: *
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Questions 49 - 51: Define the two vectors a = <1, 3, -4> and b = <0, 5, 1>
#49 Find the dot product. : * |
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| #50 Find the cross product.: * |
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#51 The angle x between vectors <-1, 1, 3> and <2, 0, -2> is given by solving which equation?: *
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| #52 Consider the function y = x2 + 4. Compute the left endpoint, right endpoint, and midpoint approximations to the area under the curve on the interval [0, 2] with four subdivisions. Which of the following statements is correct?: * |
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#53 Find the arc length of y = sin(√x) on [a, b] where a and b are positive.: *
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Questions 54 - 56: A particle is moving with the velocity function v(t) = t2 – 14t + 48.
#54 At what point(s) does the particle have zero acceleration? : * |
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| #55 Where is the particle moving backwards?: * |
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| #56 Let s(t) represent the position of the particle at time t and suppose the initial position is 10. Find s(2) rounded to the nearest integer.: * |
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#57: *
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#58 Which integral can be used to calculate the area enclosed by the smaller loop of the graph of the polar equation r = 1 + 2 sin θ?: *
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| #59 Which of the following is not necessarily true about a real-valued function f(x)?: * |
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#60 All of the following statements are always correct for functions f(x) and g(x) except:: *
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