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Practice Final

Practice Final - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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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 Final Exam There are 19 problems, totaling 250 points. No books, notes, or calculators. This practice exam should take 3 hours. Generally useful trigonometry: sin 2 x = 1 cos 2 x ; cos 2 x = 1 + cos 2 x ; sec x = ln(sec x + tan x ) 2 2 1 sec x = ; sin 2 x + cos 2 x = 1; tan 2 x + 1 = sec 2 x cos x In a 30-60-90 right triangle, with hypotenuse 2, the legs are 1 and 3. 2 = 1 . 41 3 = 1 . 73 = 3 . 14 ln 2 = . 69 ln 10 = 2 . 3 Problem 1. (15) Evaluate each of the following: d ln x a) 2 ; simplify your answer. dx x b) d 3 sin 2 u + 2 c) d n e kx , k constant. du dx n x =0 Problem 2. (10) Find the equation of the line tangent to the graph of x 2 y 2 + y 3 = 2 at the point (1 , 1) on the graph. (Give the equation in the form y = mx + b .) Problem 3. (10) Using implicit differentiation, derive the formula for D cos 1 x by using the formula for D cos x . (Let y = cos 1 x .) x 2 + x + a, x 0 Problem 4. (10) Let f ( x ) = , a and b constants.
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