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Suppose that you don't know an algebraic expression for the function $g$, but you do know that $g(3)=5$ and that $g'(x)=\frac{1}{x^2+16}$; $\forall x \in \mathbb{R}$. Use linear approximation to estimate the values of $g(2.96)$ and of $g(3.05)$.

The only solution I see is to calculate the anti-derivative in other words its integral and calculate the linear approximation with that result, but the course I'm taking is differential calculus and the teacher has not mention derivatives more than 3 times in the whole semester. Is there another way to solve the problem or another way to calculate $g(x)$

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  • $\begingroup$ If you were to integrate and then do a linear approximation on the integral, you'd need to take the derivative of the antiderivative of $g'$, which is just $g'$ itself.... $\endgroup$
    – Andrew D. Hwang
    Commented May 28, 2017 at 1:34

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From Taylor's theorem $$g(x+\delta x)\approx g(x)+\delta x\cdot g'(x)$$ This can be applied to your example with $\delta x=-0.04, 0.05$, $x=3$.

Also, as a note, your integral is not actually too difficult to compute - it is $$g(x)=k+\frac14\arctan\frac x4.$$ However this is not the intended method of the question, due to the wording including the word approximate.

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