Please answer all of the following questions. The table gives the values of a fu

Question

Please answer all of the following questions.

The table gives the values of a function obtained from an experiment. Use them to estimate the integral below

using three equal subintervals with right endpoints, left endpoints, and midpoints.

(a) Estimate

using three equal subintervals with right endpoints.
R3 =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral? Choose from one of the following answers

-less than

-greater than

-one cannot say

(b) Estimate

using three equal subintervals with left endpoints.
L3 =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?

-less than

-greater than    

-one cannot say

(c) Estimate

using three equal subintervals with midpoints.
M3 =  

If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?

-less than

-greater than    

-one cannot say

Explanation / Answer

Sol:

In all cases, x = (9-3)/3 = 2

The subintervals are [3, 5], [5, 7], and [7, 9].

The right endpoints are 5, 7, 9.
The left endpoints are 3, 5, 7
The midpoints are 4, 6, 8

a) Right end

R3 (x)[f(5) + f(7) + f(9)]

= 2 [ -0.5 +0.9 + 1.9] = 4.6 >1 greater than 1

b) left end

I3 (x)[f(3) + f(5) + f(7)]

= 2 [ -3.6 +0.5 +0.9 ]

= -4.4 lessthan

c) mid point

M3 (x)[f(4) + f(6) + f(8)]

= 2 [- 2.3 +0.3 +1.4 ]

= -1.2 less than

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