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Practice Exam 2

Practice Exam 2 - is that Martha an escaping prisoner is...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Fall 200 6

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± 18.01 Practice Exam 2 Problem 1. (20) Find the local maxima and minima and points of inﬂection of 2 x 3 + 3 x 2 12 x + 1 . Then use this data to sketch its graph on the given axes, showing also where it is convex (concave up) or concave (down). (Note that diﬀerent scales are used on the two axes.) Problem 2. (20) A new junk food NoKarb PopKorn is to be sold in large cylindrical metal cans with a removable plastic lid instead of a metal top. The metal side and bottom will be of uniform thickness, and the volume is ±xed to be 64 cubic inches. What base radius r and height h for the can will require the least amount of metal? Show work, and include an argument to show your values for r and h really give a minimum. Problem 3. (15) Evaluate the following inde±nite integrals: a) e 3 x dx b) cos 2 x sin x dx x dx c) 1 x 2 Problem 4. (15) A searchlight L is 100 meters from a prison wall. It is rotating at a constant rate of one revolution every 8 minutes. (How many radians/minute
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Unformatted text preview: is that?) Martha, an escaping prisoner, is running along the wall trying to keep just ahead of the beam of light. At the moment when the searchlight angle ± is 60 degrees, how fast does she have to run? Problem 5. (15: 5, 10) − x a) What value for the constant c will make the function e 1 + cx approximately constant, for values of x near 0? (Show work.) dx b) Find the solution x ( t ) to the diﬀerential equation = 2 t 1 − x 2 dt which also satis±es the condition x (0) = 1. Problem 6. (15: 8,7) a) Use the Mean-value Theorem to show that ln(1 + x ) < x, if x > 0. (You do not have to state the theorem.) b) Let c be any constant. Show that the function f ( x ) = x 3 + x + c cannot have two zeros. (Use the Mean-value theorem, or some other argument.)...
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Practice Exam 2 - is that Martha an escaping prisoner is...

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