Question
Asked Nov 7, 2019
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What is an estimate of g(21) using the midpoint to get the most precise estimate?

Let g(x)
f(t dt, where fis the function whose graph is shown.
f
3-
18
12
t
(a) Evaluate g(x) for x 0, 3 , 6, 9, 12, 15, and 18
g(0) 0
g(3) 4.5
g(6)= 0
g(9)4.5
g(12) 0
g(15) = 13.5
g(18) 36
(b) Estimate g(21). (Use the midpoint to get the most precise estimate.)
g(21)
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Let g(x) f(t dt, where fis the function whose graph is shown. f 3- 18 12 t (a) Evaluate g(x) for x 0, 3 , 6, 9, 12, 15, and 18 g(0) 0 g(3) 4.5 g(6)= 0 g(9)4.5 g(12) 0 g(15) = 13.5 g(18) 36 (b) Estimate g(21). (Use the midpoint to get the most precise estimate.) g(21)

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Expert Answer

Step 1

We need to estimate g(21) by using the mid-point rule.

Note:

By looking at the part (a), it seems that the length of the sub-interval should be 3.

Step 2

We are given that

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g(x) f(t) dt. 12 We are finding g(12). That is, g(12) = f(t) dt. 21-0 3. (Ax = - Let n 7 so as to get Ax 7 15, Х6 18, x7 21 So, Xo — 0, х1 — 3, х2 — 6, х3 — 9, Хд — 12, х5 Thus, the interval becomes [0,3], [3,6], [6,9], [9,12], [12,15], [15,18], [18,21]. Let x be the midpoint of the i-th interval. So, xi 1.5, x = 4.5, x3 7.5,x4 = 10.5, x5 = 13.5, x = 16.5, x7 19.5.

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Step 3

Now,

...
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12 g(12) f(t) dt Ax f(xi)Ax f(x2) Ax f(x5) Ax f(x Ax f(x5) Ax f(x)Ax f(x7) ( f(1.5) + f(4.5) + f(7.5) + f(10.5) + f(13.5) * Δx f(16.5)f(19.5) (f(1.5) + f(4.5) + f(7.5) + f(10.5) f(13.5) * f(16.5)f(19.5) = 3

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Tagged in

Math

Calculus

Integration