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HW03-solutions

# HW03-solutions - Munoz(gm7794 HW03 Radin(54915 1 10 This...

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Munoz (gm7794) – HW03 – Radin – (54915) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Rewrite the sum 2 n parenleftBig 4 + 5 n parenrightBig 2 + 2 n parenleftBig 4 + 10 n parenrightBig 2 + . . . + 2 n parenleftBig 4 + 5 n n parenrightBig 2 using sigma notation. 1. n summationdisplay i =1 2 n parenleftBig 4 i + 5 i n parenrightBig 2 2. n summationdisplay i =1 5 n parenleftBig 4 i + 2 i n parenrightBig 2 3. n summationdisplay i =1 2 n parenleftBig 4 + 5 i n parenrightBig 2 correct 4. n summationdisplay i =1 5 n parenleftBig 4 + 2 i n parenrightBig 2 5. n summationdisplay i =1 2 i n parenleftBig 4 + 5 i n parenrightBig 2 6. n summationdisplay i =1 5 i n parenleftBig 4 + 2 i n parenrightBig 2 Explanation: The terms are of the form 2 n parenleftBig 4 + 5 i n parenrightBig 2 , with i = 1 , 2 , . . . , n . Consequently in sigma notation the sum becomes n summationdisplay i =1 2 n parenleftBig 4 + 5 i n parenrightBig 2 . 002 10.0 points The graph of a function f on the interval [0 , 10] is shown in -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 6 8 Estimate the area under the graph of f by dividing [0 , 10] into 10 equal subintervals and using right endpoints as sample points. 1. area 55 2. area 56 3. area 53 4. area 54 5. area 52 correct Explanation: With 10 equal subintervals and right end- points as sample points, area braceleftBig f (1) + f (2) + . . . f (10) bracerightBig 1 , since x i = i . Consequently, area 52 , reading off the values of f (1) , f (2) , . . . , f (10) from the graph of f . 003 10.0 points Decide which of the following regions has area = lim n → ∞ n summationdisplay i =1 π 4 n sin 4 n

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Munoz (gm7794) – HW03 – Radin – (54915) 2 without evaluating the limit. 1. braceleftBig ( x, y ) : 0 y sin 2 x, 0 x π 4 bracerightBig 2. braceleftBig ( x, y ) : 0 y sin 4 x, 0 x π 8 bracerightBig 3. braceleftBig ( x, y ) : 0 y sin x, 0 x π 8 bracerightBig 4. braceleftBig ( x, y ) : 0 y sin x, 0 x π 4 bracerightBig correct 5. braceleftBig ( x, y ) : 0 y sin 3 x, 0 x π 4 bracerightBig 6. braceleftBig ( x, y ) : 0 y sin 3 x, 0 x π 8 bracerightBig Explanation: The area under the graph of y = f ( x ) on an interval [ a, b ] is given by the limit lim n → ∞ n summationdisplay i =1 f ( x i x when [ a, b ] is partitioned into n equal subin- tervals [ a, x 1 ] , [ x 1 , x 2 ] , . . ., [ x n 1 , x n ] each of length Δ x = ( b - a ) /n . When the area is given by A = lim n → ∞ n summationdisplay i =1 π 4 n sin 4 n , therefore, we see that f ( x i ) = sin 4 n , Δ x = π 4 n , where in this case x i = 4 n , f ( x ) = sin x, [ a, b ] = bracketleftBig 0 , π 4 bracketrightBig . Consequently, the area is that of the region under the graph of y = sin x on the interval [0 , π/ 4]. In set-builder notation this is the region braceleftBig ( x, y ) : 0 y sin x, 0 x π 4 bracerightBig .
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HW03-solutions - Munoz(gm7794 HW03 Radin(54915 1 10 This...

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