Question

Find the first and second derivatives of the function tabulated below at the point
x= 1.5
x 1.5 2.0 2.5 3.0 3.5 4.0
f(x): 3.375 7.0 13.625 24.0 38.875 59.0​

Answer 1

Answer:

Done it

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Answer 2

Concept Introduction:-

Differentiation is a technique for determining a function's derivative.

Given Information:-

We have been given that

$x= 1.5\\x=1.5 2.0 2.5 3.0 3.5 4.0\\f(x): 3.375 7.0 13.625 24.0 38.875 59.0$

To Find:-

We have to find that the first and second derivatives of the function.

Solution:-

According to the problem

Here $\mathrm{x}_{0}=1.5, \mathrm{y}_{0}=3.375, \mathrm{~h}=0.5$

By Newton's Difference Formula, we have

${\left[\frac{d y}{d x}\right] \text { at } x \approx x_{0} \approx \frac{1}{h}\left[\Delta y_{0}-\frac{1}{2} \Delta^{2} y_{0}+\frac{1}{3} \Delta^{3} y_{0}-\frac{1}{4} \Delta^{4} y_{0}+\frac{1}{5} \Delta^{5} y_{0}-\ldots \ldots\right] }$

$\therefore & {\left[\frac{d y}{d x}\right] \text { at } x \approx 1.5 \approx \frac{1}{0.5}\left[3.625-\frac{1}{2}(3)+\frac{1}{3}(0.75)-\frac{1}{4}(0)+\frac{1}{5}(0)\right] }$

${\left[\frac{d y}{d x}\right] \text { at } x \approx 1.5 \approx \frac{1}{0.5}[3.625-1.5+0.25] }$

$\therefore & {\left[\frac{d y}{d x}\right] \text { at } x \approx 1.5 \approx \frac{2.375}{0.5} \approx 4.75 }$

And

${\left[\frac{d^{2} y}{d x^{2}}\right] \text { at } x \approx x_{0} \approx \frac{1}{h^{2}}\left[\Delta^{2} y_{0}-\Delta^{7} y_{0}+\frac{11}{12} \Delta^{4} y_{0}-\frac{5}{6} \Delta^{5} y_{0}+\ldots \ldots .\right] }$

$\therefore & {\left[\frac{d^{2} y}{d x^{2}}\right] \text { at } x \approx x_{0} \approx \frac{1}{h^{2}}\left[\Delta^{2} y_{0}-\Delta^{2} y_{0}+\frac{11}{12} \Delta^{4} y_{0}-\frac{5}{6} \Delta^{5} y_{0}+\ldots \ldots . .\right] }$

$\therefore & {\left[\frac{d^{2} y}{d x^{2}}\right] \text { at } x \approx 1.5 \approx \frac{1}{(0.5)^{2}}\left[3-0.75+\frac{11}{12}(0)-\frac{5}{6}(0)\right] }$

$\therefore & {\left[\frac{d^{2} y}{d x^{2}}\right] \text { at } x \approx 1.5 \approx \frac{2.25}{(0.25)} \approx 9$

Final Answer:-

The first and second derivatives of the function are $4.75$ and $9$ respectively.