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.