830
Answer Key
11 . 4 ∑ k =1 25
k 2 −100 ∑ k =1 25
k = 4(25)(26)(51) 6
− 50(25)(26) = −10, 400
13 . R 4 = -0.25 15 . R 6 =0.372 17 . L 4 =2.20 19 . L 8 =0.6875 21 . L 6 =9.000= R 6 . The graph of f is a triangle with area 9. 23 . L 6 = 13.12899 = R 6 . They are equal. 25 . L 10 = 4 10 ∑ i =1 10 4− ⎛ ⎝ −2+4 ( i −1) 10 ⎞ ⎠
e −1 100 ∑ i =1 100
⎛ ⎝ 1+( e −1) i
⎞ ⎠
27 . R 100 =
ln
100
29 . R 100 =0.33835, L 100 =0.32835. The plot shows that the left Riemann sum is an underestimate because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.
31 . L 100 =−0.02, R 100 =0.02. The left endpoint sum is an underestimate because the function is increasing. Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates.
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