AP Calculus AB Hard practice

Use the washer method to find the volume when the region between $y=x$ and $y=x^2$ (on $[0,1]$) is rotated about the $x$-axis.

A particle's velocity is $v(t)=t^2-4t+3$ for $0\le t\le 4$. What is the total distance traveled?

The general solution to $\frac{dy}{dx}=ky$ is $y=C{e}^{kx}$. If $y(0)=5$ and $y(2)=20$, find $k$.

The function $f(x)=\{\begin{array}{ll}{x}^2& x<1\\ ax+b& x\ge 1\end{array}$ is continuous and differentiable at $x=1$. What are $a$ and $b$?

A 5-foot tall person walks away from a 20-foot lamppost at $3$ ft/s. How fast is their shadow lengthening?

Use L'Hôpital's rule to find $\underset{x\to 0^{+}}{\lim }x\ln x$.

For the initial value problem $\frac{dy}{dx}=x-y$, $y(0)=1$, use Euler's method with step size $h=1$ to estimate $y(2)$.

A function $f$ has $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)=0$. What can be concluded?

Find the volume of the solid formed when the region bounded by $y=e^x$, the $x$-axis, $x=0$, and $x=1$ is rotated about the $x$-axis.

A farmer wants to enclose a rectangular plot next to a river (no fence needed along the river) using $200$ m of fencing. What dimensions maximize the enclosed area?

A particle moves so that $x(t)=t^3-6t^2+9t$ for $t\ge 0$. On which interval(s) is the particle's speed increasing?

Water drains from a conical tank (vertex down) of height 10 m and radius 4 m at $2$ m³/min. How fast is the water level dropping when $h=5$ m? (Recall $V=\frac{1}{3}\pi r^{2}h$.)

Let $f(x)=\arcsin (x)$. Using the derivative of an inverse function, find $f^{\prime }(1/2)$.

A slope field for $\frac{dy}{dx}=\frac{x}{y}$ is sketched. Which statement best describes a solution curve passing through $(0,3)$?

Find $\frac{dy}{dx}$ by implicit differentiation: $x^3+y^3=6xy$.