184midtermreviewsolns

# 184midtermreviewsolns - Math 184 Midterm Review Solutions 1...

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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 left-hand 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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184midtermreviewsolns - Math 184 Midterm Review Solutions 1...

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