History UsefulNotes / MathCurriculum

** Use the regular function (sine, cosine and tangent) if you have the &#920 (degrees or radians) already.
** Use the arc function (sin[[superscript:-1]], cos[[superscript:-1]], tan[[superscript:-1]]) when you're dividing a side of a right triangle by another side to ''find'' the &#920 (degrees or radians).

** Use the regular function (sine, cosine and tangent) if you have the degrees or radians already.
** Use the arc function (sin[[superscript:-1]], cos[[superscript:-1]], tan[[superscript:-1]]) when you're dividing a side of a right triangle by another side to ''find'' degrees or radians.

** LOOKING AHEAD: The derivative equation in calculus - where you'll actually start to "do calculus stuff" in calculus - requires composing functions.

*** LOOKING AHEAD: The derivative equation in calculus - where you'll actually start to "do calculus stuff" in calculus - is a slope equation.
** Curves have a constantly changing slope -- you'll learn how to find those in calculus.

** All the work with lines and parabolas in algebra were functions. From here on out you stop using a "y" and instead use '''function notation''': a fancy letter f with an x in parenthesis " f(x) " (The parenthesis isn't multiplication here - you draw the "f" bigger or use a small "(x)")

** If you ever hear someone say or mumble to themselves "sohcahtoa"[[note]]"so-ka-toe-ah"[[/note]], it's a mnemonic to help remember which trig functions go with which sides of a right triangle. "soa" means "'''S'''ine (equals) '''O'''pposite side (Over) '''H'''ypotenuse" and so on.)

** All the work with lines and parabolas in algebra were functions. From here on out you stop using a "y" and instead use '''function notation''': a fancy letter f with an x in parenthesis " f(x) "
*** f(x)=x+1 means "You can plug number x into function f". You're about to do operations on functions themselves, so you're allowed to use other letters for the functions now, too. g(x), h(x), x(x). So... here we go...

** All the work with lines and parabolas in algebra were functions. From here on out you stop using a "y" and instead use a fancy letter f with an x in parenthesis " f(x) "
*** f(x)=x+1 means "You can plug number x into function f". Your'e about to do operations on fuctions themselves, so you're allowed to use other letters for the functions now, too. g(x), h(x), x(x). So... here we go...

** A simple function is y = x+1 . You plug different numbers into x (like "4") and get an answer (here it's "5"). As functions get more complicated, this is still the basic point. You can make a chart of every single possible answer to y = x+1. It's silly with such a simple equation (it's just a diagonal line) but when you get to complex equations dealing with science and finance you can look and see the trends.

* '''Vectors''' look like what were called "rays" back in geometry. A vector isn't merely an arrow, it has a magnitude ("strength", and is always positive) and a direction (usually in degrees).

-->-- '''E.T.Bell'''

Generally given to teenagers, algebra helps us look at math problems in a new light, showing that some back door work in unexpected directions can give an answer which seemed originally impossible to know. For teenagers it's broken into two sections: Algebra 1 is order of operations and computation, Algebra 2 is lines and parabolas (functions). If you take it in college you'll get that all in one class.

After that you'll learn about how to plot all the possible answers for an equation on a graph, leading you to the equations of lines and curves -- which are the first steps to understanding equations called functions.

Generally given to teenagers, algebra helps us look at math problems in a new light, showing that some back door work in unexpected directions can give an answer which seemed originally impossible to know.

*** f(x)=x+1 means "You can plug number x into function f". Your'e about to do operations on fuctions themselves, so you're allowed to use other letters for the functions now, too. g(x), h(x), X(x). So... here we go...

** g(x), h(x), X(x) simply clarify which function is which. You're still going to have to write them all out longhand and do all the work on paper. Sorry.

*** The more complicated side is when they were found by doing research and then using math to see if there's a pattern connecting them. This is a computation-heavy subject in statistics called ''regression analysis'' where you average out the data points you found into the closest function you can create.

*** The more complicated side is when they were found by doing research and then using math to see if there's a pattern connecting them. This is a computation-heavy subject in statistics called ''regression analysis'' where you average out the data points you find into the closest function you can find.

** Note that "10" isn't one of those, though it's usually counted. 0-9 are the only numbers allowed in the ones place. "10" is the "zero" in the tens place.

** You'll also have to show what set of numbers the inequality refers to, using a "set".

* Using the integral and the derivative on a function is very handy in pretty much everything. When you apply the derivative or the integral to a function the new function ''crosses the x axis exactly where the precise top or bottom of the "hump" was in the old equation.
** Simple example -- if you take the derivative of a parabola, the line that results crosses the x-axis at the same x value the parabola was highest or lowest.

*** You've probably come across the meme on the internet that tosses an equation at you and then the comments section falls into arguments about order of operations, such as :"8 ÷ 2(2 + 2)=?". These are thrown out there [[ForTheEvulz for people who like to watch the arguing]]. Fact is, there is no single correct answer without any context. Without any explanation on what you're doing (you're obviously not using this equation for engineering, after all), you technically CAN do the equation however ''you'' want. But don't bother commenting unless you want a fight -- just keep scrolling past it, or [[TakeAThirdOption use set notation (below) to list every possible answer]] and annoy ''everyone''.

*** You've probably come across the meme on the internet that tosses an equation at you and then the comments section falls into arguments about order of operations, such as :"8÷2(2+2)=?". These are thrown out there [[ForTheEvulz for people who like to watch the arguing]]. Fact is, there is no single correct answer without any context. Without any explanation on what you're doing (you're obviously not using this equation for engineering, after all), you technically CAN do the equation however ''you'' want. But don't bother commenting unless you want a fight -- just keep scrolling past it, or [[TakeAThirdOption use set notation (below) to list every possible answer]] and annoy ''everyone''.

* '''Functions''' are equations likened to machines where you put something in [[note]]an x value "domain" on the aforementioned cartesian plane[[/note]] and when you do the algebra it spits out an answer [[note]]as an y value "range" on the cartesian plane[[/note]].

** Warning: '''there's no official derivative notation'''. There are quite a few different ways people use. Just use what your teacher tells you to, then go your own way. In the picture the derivative is shown by dx/dy and an apostrophe by the ''f'' in the derivative equation: ''f'' '(x)

** You're going to learn rules for manipulating limits that won't really come up much in most lines of work. Humor your calculus teacher or professor anyway and do them.

-->--E.T.Bell

->''He say; one and one and one is three''
-->--Music/TheBeatles "Come Together"

-->--Jordan Ellenberg

->''Dear algebra, [[NotGoodWithRejection stop asking us to find your X, they're not coming back]].''
-->--Unknown

-->--Unknown

->''A generating function is a clothesline on which we hang up a sequence of numbers for display.''
-->--Herbert Wilf

->''Secant, tangent, cosine, sine!''
->''Three-point-one-four-one-five-nine!''
-->--GeniusBonus sports chant from a few of the nerdier colleges such as MIT and RPI.

->''If you're going through Hell, keep going.''
-->--Winston Churchill

->''Mathematics as we know it ... could never have come into being without some disregard for the dangers of the infinite.''
-->--David Bressoud

** You're going to learn rules for manipulating limits that won't really come up much in most lines of work. Humor your calculus professor anyway and do them.
* '''Differential calculus''' runs whole functions through a formula called the '''derivative equation''', which is the "rise over run" slope fraction from algebra on steroids - see the fraction in the picture?

** To '''integrate''' is to do the antiderivative - the opposite of a derivative.

->''...in mathematics, you don't understand things. You just get used to them.''
-->--John Von Neumann