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

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
up vote 3 down vote accepted

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

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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.) – Gyu Eun Lee Jan 10 '13 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". – dr jimbob Jan 10 '13 at 7:00
I didn't think you were suggesting that either. I still wonder where the second derivative comes in. – Gyu Eun Lee Jan 10 '13 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 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

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