exam-2-sol

# exam-2-sol - Haiman Math 1ACalculus Second Midterm Exam...

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Haiman Math 1A—Calculus Fall, 2006 Second Midterm Exam Solutions Name Student ID Discussion Section (Time and GSI) You may use one sheet of notes. No other notes, books or calculators allowed. There are 8 questions, on front and back. Write answers on the exam and turn in only this paper. Show enough work so that we can see how you arrived at your answers. 1. Find the equation of the tangent line to the hyperbola x 2 + 2 xy - y 2 + x = 9 at the point (2 , 1). Differentiate implicitly with respect to x , to get 2 x + 2 y + 2 xy - 2 yy + 1 = 0 . Set x = 2, y = 1, and solve for y = - 7 / 2. The equation of the tangent line is then y = ( - 7 / 2)( x - 2) + 1 = 8 - 7 x/ 2 . 2. Differentiate the function (ln x ) cos x . Using logarithmic differentiation, f ( x ) = f ( x ) d dx ln f ( x ) = (ln x ) cos x ( cos x x ln x - (sin x )(ln ln x ) ) 3. The surface area of a melting spherical ball of ice decreases at a rate of 1 cm 2 / min. How fast is the volume decreasing when the radius of the ball is 10 cm? For your information, the surface area and volume of a sphere of radius r are given by A = 4 πr 2 and V = 4

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