Calculus Blog

Friday, September 14, 2012

Limits Involving Infinity

It's hard to follow up the Sandwich Thm...but I'll try. :) HW help video at bottom!

Limits involving infinity really revolve around finding horizontal asymptotes and vertical asymptotes.

If you are trying to find a limit as "x" goes to infinity (or negative infinity), you are lookign for horizontal aysmpotes. The key? Look at the powers:
1. Degree on bottom is bigger: Limit = 0

Because the denominator's largest power is larger than the numerator's largest power, the denominator's highest powered term takes this expression over as x approaches inifinity. The denominator becomes larger much quicker than the denominator, therefore the limit of this expression must be zero. Because zero is a constant, this means that there is a horizontal asymptote at

2. Degree on top = degree on bottom: Limit = A/B (where A = leading coefficient on top, B = leading coefficient on bottom)

In this limit problem, the highest power term in the numerator is

and the highest power term in the denominator is

. Since these are the highest power terms, they dominate the limit problem, and we can ignore the other terms in determining the limit. We see that the limit is the quotient of the coefficients of the highest power terms.

3. Degree on top is bigger: Divide everything by biggest power on the bottom, and let x go to infinity of negative infinity and see where your function goes.

Because the numerator's largest power is larger than the denominator's largest power, the numerator's highest powered term takes this expression over as x approaches infinity. Therefore the limit of this type of expression must be positive infinity.

The Sandwich (or Squeeze) Theorem

The Sandwich Thm is a very creative way to find limits of functions that are down right impossible to solve using algebriac techniques. Many times, sandwich thm problems involve trig functions...although they don't have to. This is because we can easily "start" the problem be sandwiching sine or cosine between 1 and -1...as you no doubt recall from trig. :)

As a motivation let us consider the function

$\begin{displaymath}f(x) = x^2 \sin\left(\frac{1}{x}\right)\;\cdot\end{displaymath}$

When x get closer to 0, the function $\displaystyle g(x)=\sin\left(\frac{1}{x}\right)$ fails to have a limit. So we are not able to use the basic properties discussed in the previous pages. But we know that this function $\displaystyle g(x)=\sin\left(\frac{1}{x}\right)$ is bounded below by -1 and above by 1, i.e.

$\begin{displaymath}-1 \leq \sin\left(\frac{1}{x}\right) \leq 1\end{displaymath}$

for any real number x. Since $x^2 \geq 0$ , we get

$\begin{displaymath}- x^2 \leq x^2\;\sin\left(\frac{1}{x}\right) \leq x^2 \;. \end{displaymath}$

Hence when x get closer to 0, x² and -x² become very small in magnitude. Therefore any number in between will also be very small in magnitude. In other words, we have

$\begin{displaymath}\lim_{x \rightarrow 0}\quad x^2\;\sin\left(\frac{1}{x}\right) = 0\;.\end{displaymath}$

Enjoy!

Wednesday, August 29, 2012

Intro to Limits

Finally...a little CALCULUS! I know you're excited...

So what are limits? If you remember ONE thing from class today...I hope it's that limits are what functions are APPROACHING. NOT what they are equal to. Keep that in mind...especially as we start to explain the deeper meaning of limits during the next few classes.

I've included a few HW problems in the video below...(also use the HW hint sheet I gave you!)

Not that we would ever HAVE to do a limit problem like this...but the example below really illustrates the idea that we are interested in what number is being APPROACHED!

Numerical Approach to Limits

Example 1: Let f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. We first consider values of x approaching 1 from the left (x < 1).

x	f(x)
0.5	3
0.8	3.6
0.9	3.8
0.95	3.9
0.99	3.98
0.999	3.998
0.9999	3.9998
0.99999	3.99998

We now consider x approaching 1 from the right (x > 1).

x	f(x)
1.5	5
1.2	4.4
1.1	4.2
1.05	4.1
1.01	4.02
1.001	4.002
1.0001	4.0002
1.00001	4.00002

In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that lim_x→1 f(x) = 4.

NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined.

Tuesday, August 21, 2012

Are You Ready For Calculus? (Helpful Hints)

So your calculus teacher gave you a big packet...stuff you "should" know...but probably forgot. Here are a few helpful reminders:

I've included a few hints in the attached video below (you may have to pause the video...it goes a bit quick!)

You might find the notes below to be helpful as well.

Multiplying: take your time and multiply correctly, when taking a power to a power...you multiple the powers, you are NOT solving for for anything here

Factoring: recall how to factor, don't forget the diff. of two cubes and sum of two cubes formulas

a³ + b³ = (a + b)(a² – ab + b²) a³ – b³ = (a – b)(a² + ab + b²)

Tuesday, August 14, 2012

How To Ace McHale's Calculus Class

1. Sign up for calculus.

2. Pay attention. Easier said than done. I’ve spent years refining techniques to make your attention wander and your mind dip into numbness. (not true) If you are engaged during the 81 minutes of class, you will find yourself spending much less time out of class trying figure out what the heck I did in class.

3. Do your homework. Quite frankly, most of the learning takes place when you sit in the quiet of your bedroom, library, or jacuzzi, and do your HW problems. If you understand how to do your HW problems…life should be good for you in calculus land.

a. Do HW set assigned in chapter outline

b. Use examples we have done in class (there will be lots)

c. Feel free to use your book

4. Get help. When you are stuck on a problem, do not:

a. Bang your head against the wall

b. Decide that a couple hours of Playstation will help

c. Give up in disgust, and resign yourself to a career in Slurpee Sales

Instead, get help. From:

· Mr. McHale. That’s me.

· Your friends.

· Other math teachers.

· The book.

· Your notes.

· The world wide web

· Mr. McHale’s Calculus Blog (bhsmchalecalc.blogspot.com)

5. Know examples. Calculus is usually presented as a bunch of rules with occasional examples to illustrate them. In this class…we will do LOTS of examples. Pay attention to them. You may have the idea that these are randomly picked from an ocean of examples. However, the collection of examples is more like a small pond, so you should be particularly interested in the ones that I present to you. Mathematicians recycle good problems the same way comedians recycle jokes.

6. Study. Studying will be unattractive to some of you, unfamiliar to others, but the truth is that it cannot be avoided in the end. The key to success is to study effectively and in a way that is fun…do not avoid studying. If you cannot come to grips with this, you might consider a career in politics or some field that does not require mathematics or higher level thought.

7. Avoid the dark side. Almost without exception, cheating does not lead to higher grades. You might be able to squeak through one problem here or there, but pretty soon you will be in the middle of material that depends on material previously covered. Cheating is high risk and no reward. Don’t do it.

Monday, March 19, 2012

The Definite Integral

Remember the difference between the indefinite integral and the definite integral:
Indefinite

will have a "+ C"
answer will be a function
no upper/lower bound on integral symbol

Definite

no "+ C"
answer will be a NUMBER
it has upper/lower bounds on integral symbol

Steps to solving:

1. Find the integral

2. Plug in upper bound

3. Plug in lower bound

4. Subtract step #3 from step #2

Website with worked examples and good explanations!

Remember what we are REALLY doing...calculating definite integrals is the "easy" way to add up all those infinite rectangles and get the TRUE area under a curve.

Monday, March 12, 2012

Change of Variable...the Dirty Integral

As we attempt to undo chain rule-esque intergrals remember the following:

Pick a good "u". It's is usually the "inside stuff"...but doesn't have to be.
Find "du/dx"...take the derivative of u.
Your "dx" and anything else left over in the intergral should show up once you find "du/dx". Sub that into your intergral. You've now successfully changed your variables from "x" to "u". Hopefully, the new intergral is simplier. Take the integral.
Resub to get "x" in your answer.

Easy as pi... :)