HW10-solutions

# HW10-solutions - husain(aih243 – HW10 – Gilbert...

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Unformatted text preview: husain (aih243) – HW10 – Gilbert – (56215) 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points The graph of f is shown in the figure 2 4 6 8 10 2 4 6 2 If the function g is defined by g ( x ) = integraldisplay x 3 f ( t ) dt, for what value of x does g ( x ) have a maxi- mum? 1. x = 7 correct 2. x = 9 3. not enough information given 4. x = 3 5. x = 4 . 5 6. x = 8 Explanation: By the Fundamental theorem of calculus, if g ( x ) = integraldisplay x 3 f ( t ) dt, then g ′ ( x ) = f ( x ). Thus the critical points of g occur at the zeros of f , i.e. , at the x- intercepts of the graph of f . To determine which of these gives a local maximum of g we use the sign chart g ′ + − 3 7 9 for g ′ . This shows that the maximum value of g occurs at x = 7 since the sign of g ′ changes from positive to negative at x = 7. 002 10.0 points If f is a linear function whose graph has slope m and y-intercept b , evaluate the inte- gral I = integraldisplay 2 f ( x ) dx . 1. I = 4 m + 2 b 2. I = 4 m + b 3. I = 2 m + 2 b correct 4. I = 2 m 5. I = 4 m 6. I = 2 m + b Explanation: Since the graph of f has slope m and y- intercept b , f ( x ) = mx + b . But then by the Fundamental Theorem of Calculus, integraldisplay 2 f ( x ) dx = integraldisplay 2 ( mx + b ) dx = bracketleftBig 1 2 mx 2 + bx bracketrightBig 2 . Consequently, I = 2 m + 2 b . husain (aih243) – HW10 – Gilbert – (56215) 2 003 10.0 points Determine g ′ ( x ) when g ( x ) = integraldisplay x radicalbig 5 − 3 t 2 dt . 1. g ′ ( x ) = − 3 x radicalbig 5 − 3 x 2 2. g ′ ( x ) = x √ 5 − 3 x 2 3. g ′ ( x ) = radicalbig 5 − 3 x 2 correct 4. g ′ ( x ) = − x √ 5 − 3 x 2 5. g ′ ( x ) = − radicalbig 5 − 3 x 2 6. g ′ ( x ) = 3 x √ 5 − 3 x 2 Explanation: By the Fundamental Theorem of Calculus, d dx parenleftBig integraldisplay x a f ( t ) dt parenrightBig = f ( x ) . When g ( x ) = integraldisplay x f ( t ) dt , f ( t ) = radicalbig 5 − 3 t 2 , therefore, g ′ ( x ) = radicalbig 5 − 3 x 2 ....
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## This note was uploaded on 11/30/2010 for the course M 408c taught by Professor Mcadam during the Spring '06 term at University of Texas.

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HW10-solutions - husain(aih243 – HW10 – Gilbert...

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