The test is on Friday!!!

The test will cover the material in chapter 2 and also section 3.1 on extrema. Make sure you know how to differentiate a function! There is no reason to suspect that this worksheet covers all the types of questions that will be asked on the exam!

Let $f(x) = x^2 - 2x$. Compute and simplify

$$\frac{f(x+h)-f(x)}{h}$$

Find the derivatives. Write down the rules you used for each.

$$10x^2 - 1/x,\quad (-16x^2 - 32x)(2x),\quad \frac{12 - x^2}{10x^5 - 5},\quad$$

Find the derivatives. Write down the rules you used for each.

$$\sqrt{x^2 - 3},\quad (\sin(x) + 1)^{-1/2},\quad \tan(\frac{x}{x+2}),\quad \sin^2(\cos^3(x)),\quad \sin(\cos(\sin(x)))$$

From the equation find both $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ at the point $x=1,y=1$.

$$x^3 - 2x^2y + 2y^3 = 1;$$

A bubble in the shape of a sphere is expanding at the rate of $5$in$^3$/sec. When the radius is $1$ inch what is the rate of change of the radius?

A circle is growing so that its area is increasing at a rate of 2 in$^2$/sec. What is the rate of change of the circumference when the area is 9$\pi$ in$^2$?

Let $f(x) = x^2(x^2+3)^{-1}$.

  1. Find the critical points of $f(x)$.

  2. Use your last answer to find the maximum and minimum values of $f(x)$ over the interval $[-1,1]$.

  1. Give an example of a function which does not have a limit at 0.

  2. Give an example of a function which has a limit at 0, but is not continuous.

  3. Give an example of a function which is continuous at 0, but has no derivative at 0.

  4. Which of the following functions have derivatives for all $x$?