State the complete definition of Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?

Suppose $4 x^2+9 y^2=36$, where $x$ and $y$ are functions of $t$.

If $\frac{d x}{d t}=3$, find $\frac{d y}{d t}$ when $x=-2$ and $y=\frac{2}{3} \sqrt{5}$.

State the complete definition of First Derivative Test.

Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.

Suppose $4 x^2+9 y^2=36$, where $x$ and $y$ are functions of $t$.

If $\frac{d y}{d t}=\frac{1}{3}$, find $\frac{d x}{d t}$ when $x=2$ and $y=\frac{2}{3} \sqrt{5}$.

A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner’s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

Find the absolute maximum and absolute minimum values of $f(x)=5+54 x-2 x^3$ on the interval $[0,4]$.

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$$\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{b x}$$

Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?

Find the most general antiderivative of $f(x)=5 x^{1 / 4}-7 x^{3 / 4}$. (Check your answer by differentiation.)

Find the limit. Use l'Hospital's Rule where appropriate. If there is an applicable alternate method to l'Hospital's Rule, consider using it instead. If l'Hospital's Rule does not apply, explain why.

$$\displaystyle\lim _{x \rightarrow-2} \frac{x^3+8}{x+2}$$

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$$\lim _{x \rightarrow 0^{+}}(\tan 2 x)^x$$

Use Newton’s method to find the absolute maximum value of the function

$$f(t) = cos t + t - t^2$$

correct to eight decimal places.

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$$\lim _{x \rightarrow 0}(1-2 x)^{1 / x}$$