Help me fast......I will give Upvote...... Answer the following questions:Q#1: Use the data in the following table to calculate the first derivative f'(6) (using forward, backward,and central schemes) and f"(10).f(x)13.406223423.550558593.932236857.4.251230896.4.5001799114.70053047134.86754123155.01063621u fur the ato niven in the followine

Answered: = Use the data in the following table…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

ChapterP: Prerequisites

SectionP.6: Analyzing Graphs Of Functions

Problem 6ECP: Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.

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Help me fast......I will give Upvote......

Expert Solution

Step 1

Note: Since you have posted a question that is time-consuming in computation and lengthy, we have solved the question only for forward and backward difference schemes. Kindly, post the question again mentioning the question to be solved for the central difference scheme and to calculate f''(10).

Given the table,

To calculate f'(6) using forward and backward difference schemes.

Let us compute the difference table,

Step 2

Forward difference formula for the 1st derivative,

${\frac{d y}{d x}}_{x = x_{0}} = \frac{1}{h} [∆ y_{0} + \frac{2 p - 1}{2!} ∆^{2} y_{0} + \frac{3 p^{2} - 6 p + 2}{3!} ∆^{3} y_{0} + \frac{4 p^{3} - 18 p^{2} + 22 p - 6}{4!} ∆^{4} y_{0} + \frac{5 p^{4} - 40 p^{3} + 105 p^{2} - 100 p + 24}{5!} ∆^{5} y_{0} + \cdot \cdot \cdot] w h e r e, p = \frac{x - x_{0}}{h}$

In our case,

$x = 6, x_{0} = 5, h = 2 \Rightarrow p = \frac{6 - 5}{2} = 0.5 And, ∆ y_{0} = 0.31899404 ∆^{2} y_{0} = - 0.07004503 ∆^{3} y_{0} = 0.02144659 ∆^{4} y_{0} = - 0.00618796 ∆^{5} y_{0} = 0.00035336 Now, \frac{2 p - 1}{2!} = 0 \frac{3 p^{2} - 6 p + 2}{3!} = - 0.041666667 \frac{4 p^{3} - 18 p^{2} + 22 p - 6}{4!} = 0.041666667 \frac{5 p^{4} - 40 p^{3} + 105 p^{2} - 100 p + 24}{5!} = - 0.036979167 Substituting, we get {\frac{d y}{d x}}_{x = 6} = 0.158914767$

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= Use the data in the following table to calculate the first derivative f'(6) (using forward, backward, entral schemes) and f"(10).