AP Calculus BC Hard practice

For the function $f(x)=\frac{|x^2-4|}{x^2-4}$, evaluate ${\lim }_{x\to 2^{-}}f(x)$ and ${\lim }_{x\to 2^{+}}f(x)$, then classify the discontinuity at $x=2$.

Let $f(x)=x^3+a{x}^2+bx$ have a local maximum at $x=1$ and a local minimum at $x=3$. Find $a$ and $b$.

A point moves along the curve $y=x^2$. When $x=3$ the $x$-coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing at that moment?

Determine the radius of convergence of $\sum _{n=0}^{\infty }\frac{(x-3{)}^n}{4^n(n+1{)}^2}$.

Find the volume of the solid with a circular base of radius $2$ (centered at origin) and semicircular cross-sections perpendicular to the $x$-axis.

Find the area of the region inside $r=2$ and outside $r=2-2\cos \theta$.

Let $f(x)=x^x$ for $x>0$. Find $f^{\prime }(x)$.

A closed cylinder must hold volume $V=250\pi$ cm${}^3$. Find the radius $r$ that minimizes total surface area $S=2\pi r^2+2\pi rh$.

Two cars approach an intersection; one travels north at $30$ mph and the other travels east at $40$ mph. How fast is the distance between them decreasing when they are $5$ mi and $12$ mi from the intersection, respectively?

Find $\frac{d}{dx}\left[\arctan \left(\frac{y}{x}\right)\right]$ treating $y$ as a differentiable function of $x$.

Determine ${\lim }_{x\to 0^{+}}\frac{\ln x}{\cot x}$.

Find the arc length of the curve $y=\ln (\cos x)$ on $\left[0,\frac{\pi }{4}\right]$.

A point moves along the curve $y=x^2$. When $x=3$ the $x$-coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing?

Evaluate $\int x^2{e}^{x}dx$.

The Maclaurin series for $\frac{1}{1-x}$ is ${\sum }_{n=0}^{\infty }{x}^n$. Use this to find a series for $\frac{x}{(1-x{)}^2}$.