# How to solve this calculus problem?

$$\begin{array}{c|ccccc} x & 2 & 3 & 5 & 8 & 13 \\ f\left(x\right) & 1 & 4 & -2 & 3 & 6 \\ \end{array}$$

Let $f$ be a function that is twice differentiable for all real numbers. The table above gives values of $f$ for selected points in the colsed interval $2\leq x\leq 13$.

(a) Estimate $f^{\prime}\left(x\right)$. Show the work that leads to your answer.

(b) Evaluate $\int_2^{13} \left(3-5f^{\prime}\left(x\right)\right) \,\mathrm{d}x$

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Can you double check the question? I suspect (a) is supposed to be $f'(x)$ evaluated at one of the listed $x$ values. – JohnD Jan 13 '13 at 7:04

Part (b) is straightforward; you should know how to split up the integral into multiple integrals using the property of linearity; specifically $\int a f(x) + b g(x) \mathrm{d}x = a \int f(x) \mathrm{d}x + b \int g(x) \mathrm{d}x$ where $a$ and $b$ are constants.
To evaluate the second integral note recall the fundamental theorem of calculus $\int_a^b f^{\prime}(x) \mathrm{d}x = f(b) - f(a)$.
I know the formal definiation of a derivative is $\lim_{x\to\infty} \frac{f\left(x+h\right)-f\left(x\right)}{h}$ I understand how to get $f^{\prime}$ between the data points, but the over all derivative, I am a bit confused on. I doubt that I take a sample from each interval and then find the mean of that sample and that will be my estimated derivative for the complete interval. – yiyi Jan 10 '13 at 7:33
would I use something along the lines of $f\left(a+h\right)\approx f\left(a\right) + f^{\prime}\left(a\right)h$? – yiyi Jan 10 '13 at 7:35