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Unformatted text preview: Math 221, Fall 2006, Lecture 7, Solutions to Midterm 2. Try to do first the problems that look simplest to you. You dont have to follow the order. Avoid spending too much time on one question with others not done yet. Good luck! 1. Define what is the inverse function and what is the condition for its existence. State the formula for the derivative of inverse function. Evaluate cos(sin 1 (1 / 2) cos 1 (1 / 2)) . The inverse function f 1 exists if and only if f is one to one. If b is in the range of f and f ( a ) = b, then f 1 ( b ) = a. Recall that cos(  ) = cos cos  sin sin . Also, cos(sin 1 1 / 2) = q 1 (sin(sin 1 1 / 2)) 2 = p 1 (1 / 2) 2 = 3 / 2 . Similarly, sin(cos 1 1 / 2) = 3 / 2 . Then cos(sin 1 (1 / 2) cos 1 (1 / 2)) = cos(sin 1 (1 / 2))cos(cos 1 (1 / 2))+ sin(sin 1 (1 / 2))sin(cos 1 (1 / 2)) = ( 3 / 2)(1 / 2) + (1 / 2)( 3 / 2) = ( 3 / 2) . 2. A particle moves along the hyperbola y = 1 /x in the first quadrant in such a way that its xcoordinate (measured in meters) increases at a steady 3m/sec. How fast is the angle of inclination of the line joining the particle to the origin changing when...
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 Spring '06
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 Math

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