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Unformatted text preview: Math 184 Midterm Review Solutions 1. Use limits to find the derivative of the function f ( x ) = x 2 + 3. Solution: f ( x ) = lim h f ( x + h ) f ( x ) h = lim h ( x + h ) 2 + 3 ( x 2 + 3) h = lim h x 2 + 2 xh + h 2 + 3 x 2 3 h = lim h 2 xh h = lim h 2 x = 2 x. squaresolid 2. If the function f ( x ) = braceleftBigg ax 2 + ax 1 if x < 1 x 3 if x  1 is differentiable at x = 1, then what is the value of a ? Solution: In order for f ( x ) to be differentiable at x = 1, first the function needs to be continuous at x = 1. The limit of ax 2 + ax 1 as x  1 is a ( 1) 2 + a ( 1) 1 = 1, and the limit of x 3 as x  1 is ( 1) 3 = 1. Since these numbers are equal, no matter what value is chosen for a , lim x 1 f ( x ) = 1. Also, f ( 1) = ( 1) 3 = 1. So, since f ( 1) = lim x 1 f ( x ), f ( x ) is continuous at x = 1 no matter what value is chosen for a . In order for f ( x ) to also be differentiable at x = 1, the slopes of the two pieces of f ( x ) must match up at x = 1. The slope of the lefthand piece of f ( x ) is 2 ax + a , and at x = 1 this is 2 a ( 1) + a = 2 a + a = a . The slope of the righthand piece of f ( x ) is 3 x 2 , and at x = 1 this is 3( 1) 2 = 3. So, for f ( x ) to be differentiable at x = 1, you need a = 3, so a = 3. squaresolid 1 3. Suppose that the cost for producing x units of some product, measured in thousands of dollars, is given by C ( x ) = 300 200 x . Compute the cost C (10000) of producing 10000 units. Compute the marginal cost C (10000) of producing 10000 units. Use this to approximate the cost of producing 9998 units. Solution: C (10000) = 300 200 10000 = 300 200 100 = 3. Since C ( x ) is measured in thousands of dollars, the cost of producing 10000 units is $3000. C ( x ) = (300) (200 x 1 2 ) (300)(200 x 1 2 ) (200 x 1 2 ) 2 = 300( 1 2 x 1 2 ) (200 x 1 2 ) 2 = 150 x 1 2 (200 x 1 2 ) 2 . So, C (10000) = 150(10000) 1 2 (200 10000 1 2 ) 2 = 150 1 10000 (200 10000) 2 = 150 1 100 (200 100) 2 = 1 . 5 10000 = . 000015. The marginal cost of producing 10000 units is . 000015 thousand dollars per unit, or $ . 15 per unit. This means that at the point where 10000 units have been produced, each additional unit costs roughly $ . 15. So, producing 9998 units should cost ap proximately C (10000) 2($ . 15) = $3000 $ . 3 = $2999 . 70. squaresolid 2 4. Find the derivatives of the following functions. a. f ( x ) = ( x 2 1) 4 ( x 2 + 1) 5 Solution: f ( x ) = (( x 2 1) 4 ) ( x 2 + 1) 5 + ( x 2 1) 4 (( x 2 + 1) 5 ) = 4( x 2 1) 3 (2 x )( x 2 + 1) 5 + ( x 2 1) 4 5( x 2 + 1) 4 (2 x )....
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 Spring '09
 James

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