AP Calculus AB Medium practice

Apply the squeeze theorem to evaluate $\underset{x\to 0}{\lim }{x}^2\sin \left(\frac{1}{x}\right)$.

Find the volume of the solid with square cross-sections perpendicular to the $x$-axis, where the base is the region bounded by $y=\sqrt{x}$ and $y=0$ on $[0,4]$.

If $f(x)=\frac{x^2-4}{x-2}$ for $x\ne 2$ and $f(2)=5$, then $f$ is:

The volume of a cube is decreasing at $12$ cm³/s. How fast is the side length decreasing when the side is $2$ cm?

Newton's law of cooling gives $\frac{dT}{dt}=-k(T-T_s)$, where $T_s=20$°C is room temperature. If an object cools from $100$°C to $60$°C in $10$ minutes, what is the temperature at $t=20$ minutes?

If $y={\cos }^3(x^2)$, find $\frac{dy}{dx}$.

The graph of $f^{\prime }$ is shown (not provided here — described): $f^{\prime }(x)>0$ for $x<2$, $f^{\prime }(2)=0$, and $f^{\prime }(x)<0$ for $x>2$. What can be concluded about $f$ at $x=2$?

Differentiate $y={\tan }^{-1}\left(\frac{x}{3}\right)$ with respect to $x$.

Solve $\frac{dy}{dx}=-2x{y}^2$ with $y(0)=1$.

Find $\frac{d}{dx}\left[\frac{x^2}{x+1}\right]$.

Use the limit definition of the derivative to find $f^{\prime }(2)$ for $f(x)=x^2+1$.

Find the area between $y=\sin x$ and $y=\cos x$ on $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$.

Use implicit differentiation to find $\frac{dy}{dx}$ for $x^2+y^2=25$.

Let $f(x)=x^3-3x^2-9x+5$. On what interval(s) is $f$ increasing?

Find the absolute maximum of $f(x)=-x^2+4x+1$ on the interval $[0,5]$.