It is required to construct a piecewise linear interpolant for the function: $$f(x)=(e)^{2x}$$ provided that the values of the function at the nodes are known: $$x=0,x=0.05,x=0.1$$ Using the obtained interpolant, it is required to find an approximation of the value of the integral$$\int_{0}^{0.1} f(x) dx$$ and compare with the exact value of the integral.
As I far as I understand, interpolant is going to consist of 2 straight line equations. Something like: $$P(0) = a_1x+b_1$$, $$P(1)=a_2x+b_2$$ So how I can obtain the value of the integral with such interpolant? Or maybe I have wrong thoughts about representation of it?
I suspect that $f(0)=0$ for your function. For the first half of the integral, you approximate $$\int_0^{0.05}f(x)\,\mathrm dx\approx \int_0^{0.05}g(x)^{\frac1{2x}}\,\mathrm dx$$ where $g$ is linear, i.e., $g(x)=ax+b$, with coefficients determined by $$ g(0)=\lim_{x\to0^+}f(x)^{2x},\quad g(0.05)=f(0.05)^{0.1}.$$
Do the same (without the problems of indeterminancy of the form $0^0$) for the second half $[0,05,0.1]$ with different $g$ of course.