Notice to visitors

I just want you to know that comments will not be noticed, because the class this was written during is over and my schedule is rather tight.

Feel free to read this and send links to it to anyone who could benefit from it.

Thank you,

David Nemati

Last Minute Advice

The class is over, and the final is the day after tomorrow (Friday the 5th, 7:30 to 10 AM).

If you have not finished (or perhaps not started!) your cheat sheet, hurry up and make it now. Remember, it can be both sides of one sheet or one side each of two sheets. Write the most important formulas and rules, especially those which you have difficulty remembering or remembering the names of. 

I wrote the name of each rule above its definition, and would advise you to do the same.

I also used the strategy of going through all of the book we covered in the order we read it in (almost) so that I did not miss any sections.

You may also want to write rules from the Appendices - the Double-Angle Formulas, for example.

God bless you all. I hope everyone passes the test with flying colors.

The Chapter 6 Test has been Moved

By popular demand in class today, Prof. Newberry moved the Ch. 6 Test from Thursday to Friday.

One more day to study! Don't waste it!

When to use which method of finding volumes

When it's clear that, when using the disk/washer method, you will be able to define the function variable in terms of the differentiation variable, you can use that method.

Illustration: the way to calculate volume with the disk method is pi∫(A to B)[F(x)^2-f(x)^2]dy , requiring a translation of [F(x)^2 - f(x)^2] into terms of y; finding F inverse and f inverse.

Otherwise, you should use the cylindrical shells, which have integration in terms of the same variable as the function inside.

Click on one of the "Math Notes" pictures, and then click "View All" after it sends you to Picasa, and you will see that one of the pictures shows the table that Prof. Newberry set up, showing when to use the disk/washer method and when to use the cylindrical shell method.

The idea of the disk method is ∫pi(radius squared)(thickness), or pi ∫ f(x)^2 dx .

When there is a hole in the middle, the washer method is used: 

∫ [pi (big radius squared) - pi (little radius squared)] (thickness) , 

or pi ∫ [F(x)^2 - f(x)^2] dx .

The cylindrical shell method fully takes care of whether and where there is a hole using the limits of the integration, which show where the outermost and innermost radii end. 

The integral is written: 

∫ (circumference)(height)(thickness) , 

or ∫ (2 pi radius)(F(x) - f(x)) dx , 

or 2 pi ∫ ( x - [axis] )( F(x) - f(x) ) dx

Wondering what "The Survey" is?

Prof. Newberry has been announcing and asking for "The Survey" in class.

If you are doing your Honors Project with the IAC (Instructional Assistance Center), then there's a survey that you are expected to fill out. At least, I think it's the IAC - it might be one of the other campus bureaucracies. Anyway, it's something about the Honors Project, if you're doing it with one of the offices on campus.

Any Test Problems from 5.4?

Does anyone remember whether the professor promised any problems on the test from 5.4?

I remember when he wrote what's in the picture below on the board, he said he would show us which types of problems from 5.4 he would give us on the test, but I do not remember whether he followed through with that. 

Anyone remember? Please comment

In Case You Haven't Yet Memorized the Trig Function Derivatives

The way I remember them is:

1. sine and cosine are mutual derivatives (almost; see rule 2)
2. every co- function has a minus sign in its derivative
3. when working with secant or tangent, you get the derivative of the one by multiplying the other by secant
4. when working with cosecant or cotangent, you get the derivative of one by multiplying the other by cosecant (very similar to rule 3)

If anyone else has good, easy to remember rules, or improvements on these, please comment.

Advice on how to do well in class

This post will be added on to by all authors who are willing, and if you aren't an author and have a tip, feel free to make a comment.

1) Read every section the day before the class. This will help you be more on top of the lesson in class. It doesn't matter whether you're not doing well or if you're a prodigy, this will help you.

The Substitution Rule

The Substitution Rule is basically the reverse of the Chain Rule.

You find a sub-function within the function you are integrating, and represent it with a new variable. (The book, and Prof. Newberry too, uses u to represent the function).

Before you rewrite $g(f(x))dx  as  $u du , * make sure you go through and evaluate du:

du/dx = _    =>    du = _dx

If you're lucky, the value multiplied by dx when evaluating du will account for some of the things the function represented by u was multiplied by. Whether it does or doesn't, you will need to divide the remaining parts of the original function by the right side of the equation evaluating du to cancel out the multiplication by du.

To make this a bit clearer, I'll do Section 5.5 Example 2:

Find   Int = $ sqrt(2x+1) dx 

For u = 2x +1 ,

du/dx = 2   =>   du = 2 dx  => dx = du/2

Int = $ sqrt(u) du/2  =  1/2 $ u^(1/2) du

= (1/2) * u^(3/2) / (3/2)  + C

= (1/3) * u^(3/2)  + C

= (1/3) * (2x+1)^(3/2)  + C

If you took the trouble to read the above, you will have seen that in the first step of solving the integral using u, du was divided by 2 to correct for the fact that du/dx = 2 . 

I hope this helps.

*I don't have an integral symbol on my keyboard (who does?), so I'm using $ instead of the vertically stretched S.

Helpful Notes on Anti-Deriving

When taking a derivative of an ax^n equation, you first multiply the coefficient by the power, and then you subtract 1 from the exponent.

With integrals, it's the reverse. First you add 1 to the exponent, and then you divide the coefficient by the new exponent.

I have found this a little hard to remember in my math autopilot, but if this sequence can be memorized, it should be easier.

Keep in mind: write the x and its new exponent first, then figure out the coefficient.

God bless!

Slant Asymptotes

Ok, I have a question.

How do you figure out slant asymptotes?

I just was working on S 4.5 #21, use the guidelines to sketch y=(x^2+x-2)^(1/2)

It looked like there might be, there oughta be, some slant asymptotes, but the guidelines given in the book only work for rational functions, not radical functions.

Either I'm wrong that this has slant asymptotes, or there is some other way besides long division to figure out the equation of a slant asymptote. Or both.

Curve Sketching

This might be the hardest thing we've done yet. (Or maybe I'm just getting lazy all of a sudden. [lol])

So, the book says:

1st figure out the domain, the values of x for which the equation can be used.

2nd find the x- and y-intercepts

3rd figure out whether the function is even (symmetric about the y-axis), odd (rotationally symmetric about the  origin) or neither

4th a) find horizontal asymptotes; if the numerator is an nth degree polynomial and the denominator is an mth  degree polynomial, with leading coefficients A in the numerator and B in the denominator, then:

Case 1: m>n  ->  horizontal asymptote at y = 0

Case 2: m=n  ->  horizontal asymptote at y = A/B

Case 3: m  no horizontal asymptote*

b) find vertical asymptotes; if the denominator reaches zero at a value of x, there is a vertical asymptote there.

c) if in Case 3, n=m+1, there will be a diagonal asymptote with slope A/B. Not all slant asymptotes go through the origin; it will be necessary to use long division to find exactly what the asymptote's equation is. (Have questions on how? Leave a comment.)

5th Find f prime, find where it is equal to zero, where it is positive, and where it is negative.

6th Find extrema, by finding critical numbers, places where f prime equals zero or does not exist, and then  checking either the f prime values between critical numbers or the f double prime values at the critical numbers.  The second method will not always work; if f(x)=x^4 , then f'(x)=4x^3 and f''(x)=12x^2 . In this case, the origin  is an extreme, but it is also a critical number when looking for inflection points; the second derivative does not  show whether the curve is concave up or down at that point.

7th Find possible inflection points, values of x where f''=0, and find the concavity between those points.

8th Sketch the curve

Some rules of thumb to keep in mind: find the zeroes of the function and both derivatives; find asymptotes

What exactly is an inflection point?

(An example of the types of questions you could ask here)

If you have an equation such as f(x)=x^4 , the double derivative will be f^(2)(x) = 12x^2 .

This double derivative will equal zero at x = 0 , but f(x) will not change concavity.

An inflection point is defined as a point where the curve changes concavity in the book, but then it is defined as that before double derivatives are brought into the picture; I think it's possible that an inflection point might be defined as simply a point where f''=0 . But then it might not be.

I will try to ask this in class tomorrow, but if I don't, feel free to post your opinion as a comment.

A Simple Request

Dear Classmates,

The purpose of this blog is to help the students of this class. I was hoping to do that by making (nearly) everyone in our class an author of this blog - someone who can make posts.

If you are able to make a post, then whenever you have a question that Prof. Newberry can't answer, or one which you thing the answer to would be helpful to others, you could post your question on the blog, and at least I will be quick to try and answer it. 

(I took calculus before, got rusty on it, and am taking this class to warm up again. I would like to put my knowledge to good use immediately, and so I made this blog.)

If you feel independent, but would like to help others, there is no reason (except shortness of time) why you can't start making posts that make statements about the math - clarifying things you think others might have trouble with.

If any of you would like to make posts, please give me your name and email address in class, and I will make sure that you are sent an email inviting you to become an author of this blog.

Thank you,

David Nemati

Advice on the Word Problems

1. It might help to glance at the text of Section 3.7 to get a bit of an idea of which rates correspond to each other.

2. Make sure you read the text of 3.8

3. Pay attention to the warning on page 185 - substituting given information for variables should only be done after all the differentiation is done.

4. Remember how Prof. Newberry mentioned that differentiating equations with x and y with respect to t would be important? That will be a big thing in this section.

Implicit Differentiation

Implicit differentiation is a valuable tool. 

When y can't be isolated without turning one equation into multiple equations, you can just take the derivative of both sides, using all the rules (Chain Rule, Product Rule, Quotient Rule, etc), without worrying about isolating y, and then isolate y'. 

It will generally be easier to isolate y' than y when implicit differentiation is necessary because y prime always appears in the numerator, and to the first power. 

The cases when implicit differentiation is necessary are generally when y is put to some power other than 1, and especially (not exclusively!) when it is locked inside a root after being added to something else.

The Chain Rule

I think the Chain Rule is just so cool. 

The Leibniz notation for it is

dy/dx = dy/du du/dx

In the book it says that these shouldn't be thought of as real quotients, but that doesn't quite make sense, because dy/dx means something like "the infinitesimal change in y divided by the corrresponding infinitesimal change in x".

Any thoughts?

Welcome to Our Class's New Blog!

Hey, Classmates,

The whim popped into my head of creating a blog for our class.

It seemed good because we would be able to share questions, and help each other with difficulties.

If you want to be able to make posts, please put your name and email address as a comment on this post, and as soon as I read your comment, I'll send you an email inviting you to be an author, and then delete your comment as a measure of safety - in case some spammers stumble across our site. Or if you don't want to risk that, hand me your email address and name in class.

To become an author, someone who is able to make posts, it will be necessary to have a Google account, but if you don't have one, don't worry, because easy instructions for how to create one are included in the link in the invitation email.

I hope this works well.

- David Nemati