The function $f$ is twice differentiable for all $x$ with $f(0) = 0$. Values of $f'$, the derivative of $f$, are given in the table for selected values of $x$.
| $x$ | 0 | $\pi$ | $2\pi$ |
| $f'(x)$ | 5 | 6 | 0 |
(a) For $x \geq 0$, the function $h$ is defined by $h(x) = \int_{0}^{x} \sqrt{1 + \left(f'(t)\right)^2}\, dt$. Find the value of $h'(\pi)$. Show the work that leads to your answer.
(b) What information does $\int_{0}^{\pi} \sqrt{1 + \left(f'(x)\right)^2}\, dx$ provide about the graph of $f$?
(c) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f(2\pi)$. Show the computations that lead to your answer.
(d) Find $\int (t + 5)\cos\left(\frac{t}{4}\right)\, dt$. Show the work that leads to your answer.