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School started recently, and with it, my AP Calculus BC class. Seen as a lot of you will be going into higher math classes when you get to high school, here’s a beginning guide on some of pre-calculus! (I dunno why I wanted to make this. I hate math lol.)
** Every image here was screenshotted from Desmos, an online graphing calculator.
If you’re in junior high or above, you’ve probably already seen basic equations and their graphs. Stuff like the following:
f(x) = 1 (red)
f(x) = x (blue)
f(x) = x^2 (green)
f(x) = 1/x (purple)
These are pretty easy graphs. In general, we take the x-coordinate input, use math to solve the equation, and get a y-coordinate (or f(x)) output. But calculus doesn’t just take inputs and outputs.
For equations like f(x) = 1, there is a slope of zero. For equations like f(x) = x, there is a slope of one. However, for equations like f(x) = x^2, it can be hard to determine slope.
Of course, there is the average slope between two x-coordinates. We can take f(2) = 4 and f(4) = 16, subtract 4 from 16, then divide from the distance between the x-coordinates. Calculus, though, takes advantage of instantaneous slopes, not average slopes. And exponential graphs like f(x) = x^2 has an infinite amount of them.
At f(2), the instantaneous slope is 4, and seen in the tangent line above. But before you get to instantaneous slopes, you have to understand limits.
** Tangent lines usually only cross a graph once with simpler equations. A tangent line, however, is defined as “a line that touches a curve at a point, matching the curve’s slope there.”
Calculus has everything to do with limits. A limit is where a graph may appear to reach a certain coordinate, though that said x-value may not be defined by a y-value. As my teacher says, “It’s about the journey, not the destination.”
For example, f(x) = x^2 is defined on all x-values. Thus, the limit of f(x) as x approaches any x-value from both sides will always be the output of the original equation. On the other hand, f(x) = 1/x is NOT defined on all x-values. A number can never be divided by zero, so f(0) when f(x) = 1/x is labeled as UNDEFINED. This does not mean f(x) = 1/x does not have any limits whatsoever.
I like to say that there are three types of basic limits:
- The limit of f(x) as x approaches __ from the left
- The limit of f(x) as x approaches __ from the right
- The limit of f(x) as x approaches __ from both sides
In the case of f(x) = 1/x, the limit of f(x) as x approaches zero from the left is negative infinity, as identified visually on its graph. Also, the limit of f(x) as x approaches zero from the right is positive infinity.
HOWEVER, because the limits from right and left are different, there is no limit of f(x) as x approaches zero from both sides.
** Infinity, both negative and positive, is not a number, but a concept. Therefore, we cannot say that f(0) equals positive or negative infinity.
Calculus has a lot to do with limits. In a way, it’s like taking two super close limits like that of f(0.999) and f(1) then taking the average slope between them… except you’re not taking an average. At all. You’re finding an exact answer.
I’ll be honest with you: when -- or if -- you get to learning calculus, the actual calculus part isn’t hard. It’s pretty easy to understand. Trust me, this is coming from a person who had Cs in her algebra classes yet passed with an 89% in basic calculus. The hardest part is utilizing the chain rule, meaning that a seemingly simple equation can take up half a page when doing derivatives… then somehow not messing up on any of the basic math.
** Derivatives are how you find instantaneous slope for the majority of your time in calculus. You can get derivatives analytically with limits in order to prove the derivative rules.
I hope this helped a little, and if not, at least you have an idea for what’s ahead! See you guys later!
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