Q1. Find an extremal for the functional \[ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t \] The boundary conditions are \( x(1)=4, x\left(t_{f}\right)=4 \) and \( t_{f}>1 \) is free.

Suppose \( \int_{2}^{4} f(x) d x=-8 \int_{2}^{7} f(x) d x=6 \), and \( \int_{2}^{7} g(x) d x=5 \). Evaluate the following integrals. \( \int_{2}^{7} 6 g(x) d x=30 \) \( \int_{2}^{7}[g(x)-f(x)] d x=-1 \) (Simplify your answer) (Simplify your answer) \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) (Simplify your answer.)

\( \int _ { 0 } ^ { 3 \pi / 2 } x \sin x d x = \square \)

\( \int_{0}^{5} f(x) d x \), where \( f(x)=\left\{\begin{array}{ll}5 & \text { if } x \leq 3 \\ 3 x-4 & \text { if } x>3\end{array}\right. \)

Complete the following steps for the given function interval, and value of \( n \). a. Sketch the graph of the function on the given interval. b. Calculate \( \Delta x \) and the grid points \( x_{0} x_{1} \quad x_{n} \) c. Illustrate the left and right Riemann sums, and determine which Riemann sum underestimates and which sum overestimates the area under the curve d. Calculate the left and right Riemann sums. \[ f(x)=\cos (x-5 \pi / 8) \text { on }[\pi / 2 \pi] n=4 \]

b) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \sin x-1}{2 \cos ^{2} x-1} \)