fall-04-sol2 - Math 1A, Calculus 1. Find d2 (sec x). dx2...

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Math 1A, Calculus Second Midterm Exam Solutions Haiman, Fall 2004 1. Find d 2 dx 2 (sec x ). sec 3 x + sec x tan 2 x 2. Differentiate x ( e x ) . x ( e x ) e x (ln x + 1 x ) 3. If h ( x ) = f ( g ( x )) and f (0) = 0, g (0) = 1, f 0 (0) = 2, g 0 (0) = 3, f 0 (1) = 4, g 0 (1) = 5, find h 0 (0). h 0 (0) = f 0 ( g (0)) g 0 (0) = f 0 (1) g 0 (0) = 4 · 3 = 12 4. If x 2 + y 3 = 17 and dx/dt = 10, find dy/dt when x = 3. Differentiating gives 2 x dx/dt + 3 y 2 dy/dt = 0. Solve the original equation for y , getting y = 2 at x = 3. Then 60 + 12 dy/dt = 0, so dy/dt = - 5. 5. A cube is measured to be 6 cm on each side, with a possible error of ± . 5 cm. Use a linear approximation or differentials to estimate the error in computing the volume of the cube. V = a 3 , dV = 3 a 2 da . With a = 6 and da = . 5, get dV = 54 cm 3 (giving the answer as ± 54 cm 3 is OK too). 6. Find all local and absolute minima and maxima of the function f ( x ) = x 2 ( x + 6) on the interval [ - 5 , 3]. Include local minima and maxima at the endpoints if there are any.
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fall-04-sol2 - Math 1A, Calculus 1. Find d2 (sec x). dx2...

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