quiz4-sol - AMS 151 (Fall, 2009) Joe Mitchell Applied...

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AMS 151 (Fall, 2009) Joe Mitchell Applied Calculus I Quiz # 4, Thursday, October 22 – Solution Notes 1. Find the equation of the tangent line to the curve y = 2 x + 1 ( x 0) at the point (4,5). Since the slope is given by the derivative function, y ( x ) = 2 · (1 / 2) x 1 / 2 = 1 x , we know that the slope of the tangent line at (4,5) is y (4) = 1 / 2. Thus, the equation of the tangent line at (4,5) has the form y = (1 / 2) x + b , and we can ±nd b using the fact that the line must pass through the point (4,5): 5 = (1 / 2) · 4 + b , implying that b = 3. Thus, the equation of the tangent line at (4,5) is y = (1 / 2) x + 3. 2. If h ( b ) = xb 3 + b 2 - x 2 b , ±nd h (1). Also ±nd h ′′ ( x ). h ( b ) = x · 3 b 2 + 2 b - x 2 · ( - 1) b 2 = 3 xb 2 + 2 b + x 2 b 2 . Taking the second derivative, we get h ′′ ( b ) = 6 xb + 2 + x 2 · ( - 2) b 3 = 6 xb + 2 - 2 x 2 b 3 . Thus,
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This note was uploaded on 02/07/2010 for the course AMS 151 taught by Professor Zhang during the Fall '08 term at SUNY Stony Brook.

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