Math Tips

Y-intercept and X-intercept

Modified: 2006-08-31 14:56:10

I'd forgotten how to easily calculate the x or y intercept.

It's actually very simple.

To get the y-intercept get the value of y when x=0. To get the x-intercept, just make y=0 and see what the value of x is.

Learn more on the Y-intercept page in Wikipedia.

An expression with an exponent to a power is that expression to the exponent times the power

Modified: 2006-11-21 18:23:38

I.E. (t^3)^2 = t^6 But remember if you are just multiplying two expressions with powers times eachother, you just add the powers together: I.E. (t^3)*(t^4) = t^7

Limits and division by zero

Modified: 2006-10-28 14:10:40

It is expected that you know that for "the function y = 1/x. As x approaches 0, y approaches infinity (and vice versa)." See Limits and division by zero in Wikipedia.

OpenOffice.org has a nice Mathematical Formula Editor

Modified: 2007-03-25 01:11:23

Just go to "Insert -> Object -> Formula". Now you can easily put integrals and stuff in your documents. :)

Enter the Natural Log, e in your TI-89 Calculator

Modified: 2006-09-17 20:46:56

1. Press "Diamond"
2. Presss "X"
3. Now erase "^(" or type "1)"
4. If you hit "Enter" your calculator should display something like this:

   e                        2.7182818

(Thanks to Eric my roomate for helping me find this out!)

The longer way (what I used before):

1. Press "2ND"
2. Press "CHAR" (the "+" key)
3. Press "2" for "Math"
4. Press "5" for e

Domain and Range

Modified: 2006-09-21 15:53:09

From Section P-3: Functions and their Graphs

"Domain is the set of x-values which can be used and range is the set of values which y actually uses. Remember that range depends on domain"

A nice description of the derivative

Modified: 2007-03-13 23:33:42

"The value of a nonconstant function may change every time x changes. The derivative of a function is another function that equals the ratio of the change in the value of the function to the change in x when x is changed a very, very small amount. Thus the derivative of a function equals the rate of change of the function and the value of the derivative for any value of x is the value of the slope of the tangent line to the graph of the function at that value of x."¹

1 Saxon, John and Frank Wang. Calculus with Trigonometry and Analytic Geometry (Norman, Oklahoma: Saxon Publishers, Inc., 1988), 146

Easy way to find Greatest Common Divisors

Modified: 2007-03-04 17:56:39

Use The Euclidean Algorithm.

How to get a series to have just odd exponents

Modified: 2007-04-22 17:27:45

You know how the Taylor series of sin(x) has just odd exponents?

If you want just odd exponents, use this:

`x^(2n + 1)`

Shortcut for entering "limit("

Modified: 2006-09-11 12:55:42

Instead of using the catalog you can press the following keys to get "limit(":

1. Press "F3" (Calc)
2. Pess "3" limit(

3. Now your calculator should show the following:

   limit(

Open VS Closed Intervals

Modified: 2006-09-21 16:00:01

From Section P-3: Functions and their Graphs:

"An open interval is one in which the endpoints are not included. This equats to the > or < signs of algebra and the open circles on a graph. Thuis the open interval (2,10) is equivalent to the ineqaulity 2 < x < 10"

"A closed interval will include the endpoints and the notation uses brackets: [4,20] includes both endpoints. You can also have half-open intervals where you combine the two symbols, just as you can in algebra. [2,∞) is equivalent to the algebraic notation 2 ≤ x < ∞"

Tangent Lines

Modified: 2007-03-13 23:33:45

The Derivate is the formula which give the slope of the tangent line at any point for f(x). I found a nice graphical description of tangent lines.

What is the remainder of a small number divided by a larger one?

Modified: 2007-03-04 18:00:05

Answer: The small one. Since it could not be divided into a larger piece, the small number itself is the remainder.

Therefore:

1 mod 5 = 1

because 1 could not be divided into 5 pieces.

In the same way:

2 mod 5 = 2

because 2 could not be divided into 5 pieces. 2 is smaller than 5, so it is the remainder.

So:

2 mod 5 = 2

12 mod 5 = 2

because 2 is the remainder when 12 is divided by 5.

How to get a series to be negative on the odd terms

Modified: 2007-04-22 17:47:25

Just use this:

(-1)^n

Shortcut for entering "solve("

Modified: 2006-09-11 12:55:34

Instead of using the catalog you can press the following keys to get "solve(":

1. Press "F2" (Algebra)
2. Pess "1" solve(

3. Now your calculator should show the following:

   solve(

Range of sine and cosine

Modified: 2006-09-30 13:48:34

From: SparkNotes: SAT Math Level 1: Graphing in the Entire Coordinate Plane.

"The range of sine and cosine, as you can see in its graph or by analyzing the unit circle, is –1 ≤ y ≤ 1. The graphs of these two functions never rise above 1 or fall below –1, and every point on the unit circle has an x and y value between –1 and 1. Occasionally, you may see a question in which the answer choices are possible values of sine or cosine. If any of them are greater than 1 or less than –1, you can eliminate them."

Taking the derivative of a natural exponential

Modified: 2007-03-14 00:39:22

What is a natural exponential?

Say you have a function like:

y=e^x

The variable x in the equation is called the 'exponent', the function is called the exponential function.

If you have an equation:

With the Euler number e as base of the power, the function is the natural exponential function exp(x), sometimes abbreviated as exponential function

(quotes from: exponential function)

If you are still fuzzy on exponential functions, you may want to see the nice Visual Calculus Flash walk-thru on Exponential Functions.

Accoring to PurpleMath.com's page titled Exponential Functions: The "Natural" Exponential "e":

(The number "e" is the "natural" exponential, because it arises naturally in math and the physical sciences (that is, in "real life" situations), just as pi arises naturally in geometry. This number was discovered by a guy named Euler (pronounced "OY-ler"; I think he was Swiss), who described the number and named the number "e", and then swore that this stood for "exponential", and not for his own name.)

How do you take a derivative of a natural exponential?

If you just have the following equation:

y=e^x

amazingly enough, the derivative is simply:

e^x

I just take go ahead and agree to the following formula:

`d/dx(e^x) = e^x`

work in progress..............

Use the TI-89 to factor and expand expressions

Modified: 2006-09-11 13:09:01

See the third paragraph under TI-89 series User features in Wikipedia for some examples of how it works.

Remebering what the Graphs of Sin and Cosine Look Like

Modified: 2007-03-03 16:42:53

I just thought of a way that I think will finally allow me to easily remember what the graphs of the sine and cosine functions look like.

I'll just try to remember this:

Sin Starts at y axis
Cos Crosses y axis.

:)

Properties of Relations: Reflexive, Symmetric, Anti-Symmetric, and Transitive

Modified: 2007-03-24 16:33:51

Reflexive

(a,a) ∈ R for all a

Example:

If you have a relation on a set {1,2,3} then your relation will need to contain (1,1), (2,2),(3,3)

Symmetric

if (a,b) ∈ R, then (b,a) ∈ R

Example:

If you have (2,1) in your relation, you also need to have (1,2) for the relation to be symmetric.

Anti-Symmetric

if (a,b) ∈ R and (b,a) ∈ R then a=b

In other words, for a relation to be anti-symmetric, there can not be any pairs like (2,1) and (1,2) in the relation, but it can still have pairs like (1,1) and (2,2) and be anti-symmetric.

It is possible for a relation to be both symmetric and anti-symmetric. For example, a relation containing just (1,1) and (2,2) would be both symmetric and anti-symmetric.

Transitive

if (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R

Example:

If you have (1,2) and (2,3) in your relation, you need to have (1,3) for the relation to be transitive.

More information

Wikipedia has some Really nice diagrams and more explanations for properties of relations.

Derivative of the inverse tangent

Modified: 2007-08-22 18:10:33

Integral of 1 / (x^2 +1) is inverse tangent.

Exact/Approx Answers on TI-89

Modified: 2006-09-12 22:12:52

See Exact/Approx Answers on TI-89 on the Physics Forums

Complement and Inverse of Relations

Modified: 2007-04-12 23:12:38

Complement of a Relation

The complement of a relation is represented by:

R

To find it you simply change all the 0's to 1's and the 1's to 0's (i.e. flip all the bits).

Inverse of a Relation

The inverse of a relation is represented by:

R^-1

To find it you take the transpose of the matrix represented by:

R^T

The transpose just means making all rows into columns (The TI-89 can do the transpose of a matrix, just go to catalog and hit T).

When to use long division

Modified: 2007-08-25 10:31:46

If degree of numerator is greator than the degree of denominator, then do long division.

Get rid of @n1 in solves involving cos() and sin()

Modified: 2007-03-08 18:03:08

Add limits to your solve like so:

solve(0=cos(x))| x >_ and x <_ PI

Update: My Calc teacher said that the @n1 (or @n2 etc) variable is a counter which you can let equal zero, then 1, then 2, etc.

Volume of Disk:
V = Bh = PI*r^2*h
Washer:
PI (TOP Curve^2 - Bottom curve^2) * dx
Shell:
2PI*r*h
Volume: 2*PI*r*f(x)*DELTA X

Area of triangle: .5*b*h
Area of circle: PI*r^2