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sample midterm 2000

# sample midterm 2000 - the surface area of a sphere of...

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Your Name Section Number Your TA’s Name Section Time I agree to abide by the Honor Code. Signature Math 20 Winter 2000 Prof. Moore Midterm 1 Directions: You may use a graphing calculator to compute and graph things during the exam, but you may not use any notes or books, or receive help from any person. Show all of your work, explaining as much as possible. 1. Find each of the following limits: a) (4 points) lim x 0 1 - cos x e x b) (4 points) lim x →∞ ln(4 + e x ) 2 x

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1. c) (6 points) lim x →∞ ( x + 5) 2000 (2 x 2 - 3) 2 (5 - x ) 1996 d) (6 points) lim x 0 (1 + sin x ) 1 /x
2. (12 points) Find the absolute maximum and minimum values of the function f ( x ) = 2 x - x 3 / 2 on the interval [ 1 4 , 3].

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3. (15 points) Two ﬂies are sitting on a spherical balloon while it is being inﬂated at a constant rate of 5 cubic centimeters per second. If one ﬂy is sitting at the north pole and the other is on the equator, how quickly is the distance between them (measured along the balloon) changing when the balloon has a radius of 3 cm? Deﬁne each variable you use, and simplify your answer. (Note:

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Unformatted text preview: the surface area of a sphere of radius r is 4 πr 2 , and its volume is 4 3 πr 3 .) 4. Consider the function f ( x ) = x 2 ln x . a) (2 points) Give the interval of x-values for which the function is deﬁned. b) (10 points) Give a mathematical justiﬁcation of the behavior of the function at each end of this interval. 4. c) (12 points) Find all relative maxima and minima of f , giving full mathematical justiﬁcation. 4. d) (8 points) Find any inﬂection points of f , giving full mathematical justiﬁcation. e) (6 points) Sketch a graph of y = f ( x ) that includes all interesting features of the curve. Include the scales used on your axes. 5. (15 points) One hallway (which is 4 feet wide) meets another hallway (which is 8 feet wide) in a right-angled corner. What is the length of the longest ladder which can be carried horizontally around the corner? Give an exact answer, assuming the ladder has no width....
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sample midterm 2000 - the surface area of a sphere of...

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