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HW03-solutions

# HW03-solutions - nunez(djn358 – HW03 – Radin –(54915...

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Unformatted text preview: nunez (djn358) – 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 + 6 n parenrightBig 2 + 2 n parenleftBig 4 + 12 n parenrightBig 2 + . . . + 2 n parenleftBig 4 + 6 n n parenrightBig 2 using sigma notation. 1. n summationdisplay i = 1 6 i n parenleftBig 4 + 2 i n parenrightBig 2 2. n summationdisplay i = 1 2 n parenleftBig 4 + 6 i n parenrightBig 2 correct 3. n summationdisplay i = 1 6 n parenleftBig 4 i + 2 i n parenrightBig 2 4. n summationdisplay i = 1 2 i n parenleftBig 4 + 6 i n parenrightBig 2 5. n summationdisplay i = 1 6 n parenleftBig 4 + 2 i n parenrightBig 2 6. n summationdisplay i = 1 2 n parenleftBig 4 i + 6 i n parenrightBig 2 Explanation: The terms are of the form 2 n parenleftBig 4 + 6 i n parenrightBig 2 , with i = 1 , 2 , . . . , n . Consequently in sigma notation the sum becomes n summationdisplay i = 1 2 n parenleftBig 4 + 6 i n parenrightBig 2 . 002 10.0 points The graph of a function f on the interval [0 , 10] is shown in 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 ≈ 52 2. area ≈ 53 correct 3. area ≈ 56 4. area ≈ 55 5. area ≈ 54 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 ≈ 53 , 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 3 n cos iπ 3 n nunez (djn358) – HW03 – Radin – (54915) 2 without evaluating the limit. 1. braceleftBig ( x, y ) : 0 ≤ y ≤ cos 3 x, ≤ x ≤ π 6 bracerightBig 2. braceleftBig ( x, y ) : 0 ≤ y ≤ cos x, ≤ x ≤ π 3 bracerightBig correct 3. braceleftBig ( x, y ) : 0 ≤ y ≤ cos 2 x, ≤ x ≤ π 3 bracerightBig 4. braceleftBig ( x, y ) : 0 ≤ y ≤ cos x, ≤ x ≤ π 6 bracerightBig 5. braceleftBig ( x, y ) : 0 ≤ y ≤ cos 4 x, ≤ x ≤ π 6 bracerightBig 6. braceleftBig ( x, y ) : 0 ≤ y ≤ cos 3 x, ≤ x ≤ π 3 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 )Δ 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 3 n cos iπ 3 n , therefore, we see that f ( x i ) = cos iπ 3 n , Δ x = π 3 n , where in this case x i = iπ 3 n , f ( x ) = cos x, [ a, b ] = bracketleftBig , π 3 bracketrightBig ....
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HW03-solutions - nunez(djn358 – HW03 – Radin –(54915...

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