Given that $f(0.8)=2.2255, f(1.0)=2.7183, \int_{0.8}^{1.0}f(x)e^{-x}dx=0.2$,
I need to find an approximation of $f'(1.0)$ using a method with error of degree $~2~$.
I'm studying for a test and I've seen many questions like this one. My guessing is that I should use Central Difference as this is the only method of error of degree $~2~$ that I studied, though I don't know how to to proceed really.
I'd like to get some intuition on how to approach such questions.
You have 3 pieces of data, thus you can compute an approximating function with 3 parameters, for ease of application take the product of a quadratic polynomial with the exponential function, $p(x)=e^x(ax^2+bx+c)$. Then you get the 3 equations \begin{align} e^{0.8}(a0.8^2+b0.8+c)&=2.2255\\ e^1(a+b+c)&=2.7183\\ \frac{a}3(1-0.8^3)+\frac{b}2(1-0.8^2)+c(1-0.8)&=0.2 \end{align} or \begin{align} 0.64a+0.8b+c&=1\\ a+b+c&=1\\ 0.81333a+0.9b+c&=1 \end{align}
Solve this and then compute $p'(1)=e^1(3a+2b+c)$.