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Sagot :
Answer:
Step-by-step explanation:
∫41 xh''(x)dx=41[x∫h''(x)dx-∫{1∫h''(x)dx}dx]+c
The value of [tex]\int\limits^{4}_{1} {x\cdot h''(x)} \, dx[/tex] is 8.5. The choice that represent the best approximation is A.
How to determine the result of a definite integral based on a formula and a table
This integral can be approximated by the following Riemann sum:
[tex]A = \Sigma\limits_{i=0}^{2} \left\{(x_{i+1}-x_{i})\cdot x_{i}\cdot h''(x_{i})+\frac{1}{2}\cdot (x_{i+1}-x_{i})\cdot [x_{i+1}\cdot h''(x_{i+1})-x_{i}\cdot h''(x_{i})] \right\}[/tex]
[tex]A = \frac{1}{2} \cdot \Sigma\limits_{i=0}^{2} \left\{(x_{i+1}-x_{i})\cdot [x_{i+1}\cdot h''(x_{i+1})+x_{i}\cdot h''(x_{i})] \right\}[/tex]
Then, the approximate value of the integral is:
[tex]A = \frac{1}{2}\cdot \{(2-1)\cdot [(2)\cdot 2+(1)\cdot (-5)]+(3-2)\cdot [(3)\cdot 1+(2)\cdot 2]+(4-1)\cdot [(4)\cdot 2+(3)\cdot 1]\}[/tex]
[tex]A = 8.5[/tex]
The value of [tex]\int\limits^{4}_{1} {x\cdot h''(x)} \, dx[/tex] is 8.5. The choice that represent the best approximation is A. [tex]\blacksquare[/tex]
Remark
The statement is incomplete and poorly formatted and table is missing. Complete statement is:
Selected values of the twice-differentiable function and its first and second derivatives are:
Function
[tex]h(1) = 3[/tex], [tex]h(2) = 6[/tex], [tex]h(3) = 2[/tex], [tex]h(4) = 10[/tex]
First derivative
[tex]h'(1) = 4[/tex], [tex]h'(2) = -4[/tex], [tex]h'(3) = 3[/tex], [tex]h'(4) = 5[/tex]
Second derivative
[tex]h''(1) = -5[/tex], [tex]h''(2) = 2[/tex], [tex]h''(3) = 1[/tex], [tex]h''(4) = 2[/tex]
Selected values of the twice-differentiable function [tex]h[/tex] and its first and second derivatives are given in the table above. What is the value of [tex]\int\limits^{4}_{1} {x\cdot h''(x)} \, dx[/tex]?
A. 9, B. 13, C. 23, D. 38
To learn more on Riemann sums, we kindly invite to check this verified question: https://brainly.com/question/21847158
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