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Question

Find the approximate value of : $$(3.97)^{4}$$


Solution

Let $$f(x)=x^{4}$$
Then $$f'(x)=\dfrac{d}{dx}(x^{4})=4x^{3}$$
Take $$a=4$$ and $$h=-0.03$$.
Then $$f(a)=f(4)=(4)^{4}=256$$ and 
$$f'(a)=f'(4)=4(4)^{3}=256$$
The formula for approximation is
$$f(a+h)\doteqdot f(a)+h.f'(a)$$
$$\therefore (3.97)^{4}=f(3.97)=f(4-00.03)$$
$$\doteqdot f(4)-(0.03)f'(4)$$
$$\doteqdot 256-0.03\times 256$$
$$\doteqdot 256-7.68$$
$$=248.32$$
$$\therefore (3.97)^{4}\doteqdot 248.32$$.

Mathematics

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