Zero to Calculus
Calculus rests on three ideas: the limit, the derivative, and the integral.
Calculus classes spend much more time teaching you how to use these three ideas than on what they actually are. And while knowing how to use them is important, especially for AP tests, that’s much easier to learn once you wrap your head around the three core ideas.
This slideshow of Khan Academy videos will let you understand those three ideas at your own pace.
Understanding them takes an understanding of functions, which takes an understanding of certain parts of algebra, which takes an understanding of arithmetic.
The first slide actually starts near the end (derivatives) but for every math term that’s mentioned, there’s a link to another slide explaining it in simpler terms, going all the way down to basic arithmetic. This means there’s no need to worry about any terms or concepts you haven’t covered or don’t remember. See the next slide for clearer instructions on how it works.
1. Start by watching the video on the next slide. Don’t use the “Present” button, just click on the play button. If the narrator (his name’s Sal) says, draws, or writes anything that you’re not familiar with, pause the video.
2. To the right of the video, find the link or note with the video time (for example, 1:36) that looks closest to the point in the video where you paused. If there’s a link, click it and it will take you to a new slide.
3. Keep repeating steps 1-2 as necessary! When you’ve finished and understood a full video, click on the link below it that takes you to the next video in the sequence and keep going.
Once you’ve understood all the videos in slides 3-13, you’re done!
How it works
Finished?
Move on to Derivatives 2
0:23 - What are “coordinate axes?”
0:36 - What is “slope?” or “change in y” or “change in x?”
1:08 - What does ∆ mean?
2:04 - What is y=mx+b?
2:15 - How would I graph the equation y=x2?
3:01 - What’s a tangent line?
4:02 - What’s a limit?
Hold that thought! We need to talk about functions first (5:04). We’ll get to limits at 8:22 in this video.
4:17 - What’s a quadrant?
One of the four sections of the coordinate plane.
5:04 - What do f(a) and f(a+h) mean?
5:14 - What’s a secant line?
8:22 - What’s a limit?
Finished?
Move on to Derivatives 3
Finished?
Move on to Indefinite Integrals
Finished?
Move on to Definite Integrals 1
Finished?
Move on to Definite Integrals 2
Finished?
Move on to Definite Integrals 3
Finished?
Move on to Definite Integrals 4
Finished?
Move on to Fundamental Theorem of Calculus 1
Finished?
Move on to Fundamental Theorem of Calculus 2
Finished?
Move on to Definite Integrals 5
Finished?
Awesome. You know calculus!
For further studying, try Paul’s Online Math Notes or the rest of the Khan Academy calculus series.
1:12 - What’s algebra? Math with amounts that change, or are unknown. We call these amounts “variables” and usually show them as letters.
1:20 - What’s geometry? Math with figures - points, lines, angles, shapes, stuff like that - and the relationships between them.
1:27 - What’s an equation? A statement in math which uses = to say two expressions are equal. What are expressions?
1:28 - What are symbols? In this case, he just means “variables.”
1:31 - What are values? Specific, countable amounts.
1:33 - 2x? 2 times x.
Times? Multiplication? What are those?
1:58 - Negative 2? What’s a negative?
2:03 - 2(-2): 2 times negative 2.
How do you multiply negative numbers?
Why does he do 2x first, not x-1 first?
2:43 - What’s a square root?
“Five-halves”? “Six sevenths”? What do those mean?
4:32 - What’s radii? The plural of “radius.”
4:52 - What’s a plane? An imaginary flat surface that extends forever in all directions.
Back to Derivatives 1
0:02 - What’s arithmetic?
The simplest operations you can do on numbers, like putting them together (addition) or taking them away from each other (subtraction).
0:06 - 23+5?
0:12 - 2x7?
0:17 - 3 ÷ 4? (3 divided by 4)
0:29 - Why do we even have variables?
0:34 - What are values? “Amounts,” or “quantities.” (These three words are interchangeable.)
What’s an expression? Anything that can have a specific value, even if that value is changing or unknown. They can look like “3+5” or “x-7+y.” Expressions can have as many numbers and variables in them as you want. (He explains this more at 1:47.) An equation, on the other hand, is a statement setting two expressions equal to one another.
1:12 - Negative 7? What’s a negative?
2:07 - What does evaluate mean?
“Calculate,” or “find out the exact value of.”
2:11 - y+z? How do you deal with two variables?
3:11 - What does constrain mean? ”Narrow down.”
3:48 - Does that say “y=3+z=2”? Nope, just “y=3 & z=2"
4:30 - What does he mean by “context”? The circumstances of the problem. What is it asking me to find? How is it asking me to do it?
4:43 - x to the y power?
5:42 - Negative 2 times negative 2? How do I multiply negative numbers?
6:00 - What does √ mean?
6:00 - For “square root of x+y-x”,why do we add first, then take the square root, and then subtract?
6:48 - Principal root?
Another term for “square root.”
Back to Coordinate axes
Back to subtraction
Back to multiplication
Back to Coordinate axes
0:49 - 3+3?
1:49 - 7.1?
Back to Coordinate axes
5:56 - 3 minus 4?
Back to Negative numbers
0:15 - 4+3?
Note: if you haven’t learned what the order of operations is, stop watching the video at 3:15. We’ll get to that soon!
Back to Coordinate axes
Back to Dealing with 2 variables
Back to Coordinate axes
Back to Coordinate axes
Back to Coordinate axes
Back to Decimals
Back to Why we use variables
1:21 - 1/10? What’s that?
0:02 - What’s a graph?
A way of visually describing a mathematical relationship, usually using a coordinate plane like the one in this video.
0:17 - Change in y? Change in x?
To find the total change in something, we just subtract the initial amount we’re dealing with from the final amount we’re dealing with.
0:20 - What does one number “over” another number mean?
0.36 - What does ∆ mean?
The ∆ symbol is the Greek letter “delta,” and usually means “the difference in” or “the change in” some quantity.
In other words, if we say t is time, then ∆t is change in time (ending time minus starting time).
1:11 - What’s an integer?
Back to Derivatives 1
Back to Slope
Back to Slope
2:13 - What’s pi?
Back to Derivatives 1
0:04 - One-third x? What’s one-third?
How do I multiply numbers by fractions?
0:30 - What’s a y-intercept?
1:22 - Solving for y? How do you do that?
Back to Intercepts
Back to solving for a variable
Back to Derivatives 1
“Tangent” is a term from geometry.
A tangent line is a straight line that touches the edge of a circle (or any other curve, really) at exactly one point.
This means that the slope of a curve at a certain point equals the slope of the line that’s tangent to the curve at that exact point.
If this line was tilted either way even a little bit, it would cross the curve, and have to touch it at two points.
Back to Derivatives 1
1:18 - What’s the difference between a function and an equation? (Or, what’s the difference between f(x) and y?)
Back to Derivatives 1
Back to Functions
A secant line, unlike a tangent line, crosses a circle or curve at exactly two points.
The slope of the secant line is the average slope of that curve between those points.
As you bring P2 reeeeally close to P1, the secant line between them is almost exactly the same as what the tangent line would be at P1.
This fact is really important for calculus. You’ll see why very soon.
Back to Derivatives 1
1:17 - Why is dividing by zero undefined? Why wouldn’t it just be infinity?
5:06 - { }? What kind of a function is that??
A “piecewise” function. Most functions are the same expression no matter what input you plug in, but you can also define a function as one expression for certain x-values, and a different expression for other x-values. This breaks the function into pieces, making it “piecewise.”
Back to Derivatives 1
Back to Limits
Back to Derivatives 2
What’s a binomial?
Any expression with two terms, where both terms are constants or variables raised to constant powers (like 2x+5 or a3-b; no fancy things like 4x)
Back to Definite Integrals 2
2:38 - What’s pi?
Back to Sigma notation
Back to Number types
2:50 - Epsilon-delta definition?
The true mathematical definition of a limit uses the Greek letters epsilon and delta. It’s no longer tested by the AP Calculus exam, but if you’re curious about it, it’s explained here.