The test is on Friday!!!

This is the worksheet with answers. Please notify me if there are errors. (verzani@math.csi.cuny.edu).

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

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

ANS: without simplifying

$$\frac{((x+h)^2 - 2(x+h)) - (x^2 - 2x)}{h}$$

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

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

Ans. Without simplifying and specifying the rules you get:

$$10(2 x^1) - (-1x^{-2}),\quad (-16(2)x^1 - 32)(2x) + (-16x^2 - 32x)(2),\quad \frac{(-2x^1)(10x^5-5) - (12-x^2)(10(5x^4))}{(10x^5-5)^2}$$

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

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

Again, without simplfyind

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

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

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

It is easiest if we solve for $\frac{dy}{dx}$ as a number first, that means plug in $x=1,y=1$.

first: take the $x$ derivative of both sides

$$3x^2 - (4xy + 2x^2 \frac{dy}{dx}) + 2(3)y^2 \frac{dy}{dx} = 0$$

plug in $x=1,y=1$ to get

$$3 - 4 - 2 \frac{dy}{dx} +6\frac{dy}{dx} = 0$$

or

$$\frac{dy}{dx} = 1/4.$$

That is the slope at the point $(1,1)$ is 1/4. Now differentiate again to see what the second derivative is. To do this we take the $x$ derivative of the first line

$$3(2)x - 4(y+x \frac{dy}{dx}) - (4x \frac{dy}{dx} + 2x^2 \frac{d^2y}{dx^2}) + 6( 2y \frac{dy}{dx} + y^2 \frac{d^2y}{dx^2})= 0$$

Now plut in $x=1,y=1,\frac{dy}{dx}=1/4$ to get

$$6 - 4 - 4(1/4) - 4(1/4) - 2 \frac{d^2y}{dx^2} + 12 (1/4) + 6\frac{d^2y}{dx^2} = 0$$

SOlve for $\frac{d^2y}{dx^2} $ to get -3/4.

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?
ANS: did in class

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$^2$?
AND: did in class

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 $[-1,1]$.

ANS:

  1. First

    $$f'(x) = \frac{2x(x^2+3)-x^2(2x)}{(x^2+3)^2} = \frac{6x}{(x^2+3)^2}$$

    This is always defined and is zero at $x=0$. Thus $x=0$ is the lone critical point.

  2. We need to compare $f(1)= 1/4$,$f(0)=0$, $f(-1)=1/4$. We see that 0 is the minimum value on $[-1,1]$, and 1/4 the maximum value.

  1. Give an example of a function which does not have a limit at 0.
    ANS: try $\sin(1/x)$ or somthing with a jump at 0.

  2. Give an example of a function which has a limit at 0, but is not continuous.
    ANS: try $\sin(x)/x$

  3. Give an example of a function which is continuous at 0, but has no derivative at 0.
    ANS: try $|x|$.

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