AP Calculus BC Medium practice

For the logistic model $dP/dt=0.2P(1-P/500)$ with $P(0)=50$, at what population is the growth rate $dP/dt$ maximized?

Evaluate ${\int }_0^{\pi }{\sin }^{2}xdx$.

A particle's position is $s(t)=2\sin (\pi t)$ for $t\ge 0$. What is the particle's acceleration at $t=1$?

A particle moves with position $\mathbf{r}(t)=⟨t^2-1,2t⟩$. Find the speed at $t=2$.

Verify that $y=C{e}^{2x}+3$ is the general solution to $\frac{dy}{dx}=2(y-3)$. What is the particular solution satisfying $y(0)=7$?

Use L'Hôpital's Rule to evaluate ${\lim }_{x\to 1}\frac{x^3-3x+2}{x^2-2x+1}$.

Find the slope of the tangent line to the polar curve $r=1+\sin \theta$ at $\theta =\pi /2$.

Evaluate $\int x\cos xdx$ using integration by parts.

For $p(x)=\frac{\sin x}{x(x-\pi )}$, identify and classify all discontinuities on $(-\infty ,\infty )$.

Find $\frac{d}{dx}[\sinh x]$ using the definition $\sinh x=\frac{e^x-e^{-x}}{2}$.

If $g(x)=f^{-1}(x)$ and $f(3)=7$, $f^{\prime }(3)=4$, find $(g{)}^{\prime }(7)$.

Find the Lagrange error bound for approximating $\sin (0.1)$ using the third-degree Taylor polynomial about $x=0$.

Find the arc length of $y=\frac{2}{3}{x}^{3/2}$ from $x=0$ to $x=3$.

Find the volume of the solid generated by revolving $y=e^x$ on $[0,1]$ around the $x$-axis.

A particle's position is $s(t)=2\sin (\pi t)$ for $t\ge 0$. What is the particle's acceleration at $t=1/2$?