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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