PM2bSolns - MATH 31A (Butler) Practice for Midterm IIa...

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MATH 31A (Butler) Practice for Midterm IIa (Solutions) 1. (a) Use linearization to give an estimate for 3 1017. We first note that 3 1017 is close to 3 1000 = 10 and so we can use lineariza- tion for the function f ( x ) = 3 x = x 1 / 3 at x = 1000. Before we start we note that f 0 ( x ) = 1 3 x - 2 / 3 = 1 3( 3 x ) 2 . So the linearization is L ( x ) = f ( a ) + f 0 ( a )( x - a ) = f (1000) + f 0 (1000)( x - 1000) = 3 1000 + 1 3( 3 1000) 2 ( x - 1000) = 10 + 1 300 ( x - 1000) . So we have that 3 1017 L (1017) = 10 + 1 300 (1017 - 1000) = 10 + 17 300 . (b) Is the estimate given in part (a) too large or too small? Explain. We note that f 00 ( x ) = - 2 9 x - 5 / 3 . In particular, at x = 1000 we see that f 00 ( x ) is negative and so the curve is concave down, i.e., it is bending down. So we would expect the actual function to be under the tangent line so that the answer in part (a) would be an overestimate. [Note: 3 1017 10 . 056348 while 10 + 17 300 10 . 056666. So an overestimate as predicted, but still a good guess.]
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2. What is the area of the largest rectangle that you can make where the bottom edge is on the x -axis and the top two vertices lie on the parabola y = 12 - x 2 ? For this problem it is good to draw a picture (I recommend you draw one
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PM2bSolns - MATH 31A (Butler) Practice for Midterm IIa...

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