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

• Can you double check the question? I suspect (a) is supposed to be $f'(x)$ evaluated at one of the listed $x$ values. Commented Jan 13, 2013 at 7:04

For part(a) recall the standard limit definition of derivative to come up with very rough estimates for the derivative with your limited samples.

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 $\int_a^b f^{\prime}(x) \mathrm{d}x = f(b) - f(a)$.

• The standard limit definition might be overkill, especially since I suspect this is a question for an AP Calculus course. But the idea of linear approximation to a function is definitely appropriate; I suspect the answer has to do with finding the slopes of the secants for these points. (Not sure yet what twice differentiability has to do with it.) Commented Jan 10, 2013 at 6:30
• @proximal - I was trying to not suggest evaluate the limit, but to see f'(x) = lim h->0 (f(x+h) - f(x-h))/(2h) we can approximate f'(x) ~ (f(x+h) - f(x-h))/(2h) at several points, exactly as you said. Just was trying to leave it as "hints". Commented Jan 10, 2013 at 7:00
• I didn't think you were suggesting that either. I still wonder where the second derivative comes in. Commented Jan 10, 2013 at 7:17
• 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
Commented Jan 10, 2013 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
Commented Jan 10, 2013 at 7:35