91Ó°ÊÓ

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{3}} \sin t^{2} d t $$

Find the area of the region. Use a graphing utility to verify your result. $$ \int_{\pi / 2}^{2 \pi / 3} \sec ^{2}\left(\frac{x}{2}\right) d x $$

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{2}} \sin \theta^{2} d \theta $$

Find the area of the region. Use a graphing utility to verify your result. $$ \int_{\pi / 12}^{\pi / 4} \csc 2 x \cot 2 x d x $$

Find a function \(f\) such that the graph of \(f\) has a horizontal tangent at \((2,0)\) and \(f^{\prime \prime}(x)=2 x\).

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{4} \frac{x}{\sqrt{2 x+1}} d x $$

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{2} x^{3} \sqrt{x+2} d x $$

If \(f^{\prime}(x)=\left\\{\begin{array}{cc}1, & 0 \leq x<2 \\ 3 x, & 2 \leq x \leq 5\end{array}, f\right.\) is continuous, and \(f(1)=3\), find \(f .\) Is \(f\) differentiable at \(x=2 ?\)

The total cost \(C\) (in dollars) of purchasing and maintaining a piece of equipment for \(x\) years is \(C(x)=5000\left(25+3 \int_{0}^{x} t^{1 / 4} d t\right)\) (a) Perform the integration to write \(C\) as a function of \(x\). (b) Find \(C(1), C(5)\), and \(C(10)\).

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{3}^{7} x \sqrt{x-3} d x $$