How can you estimate f'(4). The graph of f could look like almost anything on the interval (3, 5), so any estimate is plausible.
For the second question how can you evaluate an integral involving f'(x) when you don't know what f'(x) is.
$\endgroup$1 Answer
$\begingroup$(a) You can approximate a derivative by a so-called finite difference: $$f'(4)\approx \dfrac{f(5)-f(3)}{5-3}=\dfrac{-2-4}{5-3}=-3.$$ You do not know what happens inside the interval, so this is the most useful thing you can say about the derivative at 4. If you would know function values closer to 4, the estimate of $f'(4)$ would be better.
(b) The integral of a derived function is the original function. $$\int_2^{13}(3-5f'(x))\mathrm{d}x=\int_2^{13}3\mathrm{d}x-5\int_2^{13}f'(x)\mathrm{d}x=3[x]_2^{13}-5[f(x)]_2^{13}\\=3(13-2)-5(f(13)-f(2))=33-5(6-1)=8.$$
$\endgroup$
