All of AP Calculus AB in 10 Minutes [Full Review]

追加: 2026-05-11

[00:00:00]The AP Calculus AB [music] exam is 8 months worth of information, all crammed into 3 hours and 15 minutes. Whether your test is months away or tomorrow morning, this video will speed run through [music] every major concept on the AP test. We're going to cover every single concept tested on the AP exam.

[00:00:17]That includes limits, [music] derivatives, integration, FTC, and all the applications.

[00:00:22]By the end of this, you'll have the full picture. Let's go.

[00:00:30]Limits are where all of calculus starts.

[00:00:32]A limit asks [music] one simple question. As X gets closer and closer to some value, what does the function [music] approach? Take this function on screen. It's undefined at X equals 1.

[00:00:44]There's a hole there.

[00:00:45]But as X creeps [music] toward 1 from both sides, the Y values get closer and closer to 2. That's the limit.

[00:00:51]>> [music] >> The limit is 2, even though the function never actually reaches it there.

[00:00:57]On the exam, limits will show up in three different ways. Direct substitution, factoring, and one-sided limits. Direct substitution is the easiest. You just plug in the value.

[00:01:09]Like if you have [music] the limit as X approaches 3 of X squared plus 2x, you just plug in 3, which gets you 9 plus 6, or 15.

[00:01:20]But sometimes when you [music] plug in, you get 0 over 0. That's called an indeterminate form, and you can't stop there. You need to factor and cancel.

[00:01:30]Take the limit as X approaches [music] 2 of X squared minus 4, all over X minus 2.

[00:01:37]Factor the [music] top into X plus 2 times X minus 2.

[00:01:41]The X minus [music] 2 cancels. Now you just plug in 2 and get 4.

[00:01:46]Sometimes [music] a function approaches a different value depending on which side you come from. If the left-hand limit and the right-hand [music] limit don't match, the overall limit does not exist. You'll write DNE. When X is flying toward [music] infinity, you only care about the biggest exponent.

[00:02:05]Take 3x ^ 2 + 1 all over [music] 5x ^ 2.

[00:02:10]Ignore everything except the 3x ^ 2 on top and the 5x ^ 2 on bottom.

[00:02:16]Those cancel down to just 3/5.

[00:02:19]That's your limit.

[00:02:21]The main part of calculus is the derivative. [music] This is the biggest section on the exam, but once you get what a derivative actually represents, the rules are just memorization.

[00:02:32]A derivative measures the instantaneous [music] rate of change of a function.

[00:02:36]On a graph, it's the slope [music] of the tangent line at any given point.

[00:02:41]As that tangent line slides along the curve, the slope keeps changing, [music] and that's what the derivative tracks.

[00:02:48]The derivative is able to be formally defined as a limit.

[00:02:52]The limit as h approaches zero of f of [music] x + h - f of x all over h.

[00:02:59]Okay, here is every derivative [music] rule.

[00:03:02]Power rule.

[00:03:03]This is the one you'll use constantly.

[00:03:06]Bring the exponent down front, [music] then subtract one from the exponent.

[00:03:11]So, x to the fifth becomes 5x to the fourth.

[00:03:15]Constant [music] rule. The derivative of any constant is just zero.

[00:03:20]The derivative of seven is zero.

[00:03:22]Sum and difference rule. If you have terms being added or subtracted, treat each term separately.

[00:03:28]Product rule.

[00:03:30]When two functions are multiplied together, it's the first function [music] times the derivative of the second plus the second function times the derivative of the first.

[00:03:39]Quotient rule. [music] Similar to the product rule, but flip the sign and square the denominator.

[00:03:45]Trig derivatives. [music] Just memorize these.

[00:03:48]Sine becomes cosine. Cosine becomes [music] negative sine. Tangent becomes secant squared. Chain rule.

[00:03:56]This is the [music] one that trips people up most. It's for functions inside other functions.

[00:04:02]You take the derivative of the outside, leave the inside alone, then multiply by the derivative of the inside.

[00:04:08]So, the derivative of sine of x squared >> [music] >> is cosine of x squared * 2x.

[00:04:15]Think of it like peeling an onion, starting with the outside first and then the inside. [music] Exponentials and logs.

[00:04:23]The derivative of e to the x is just e to the x, and the derivative of natural log of x is 1 over x.

[00:04:30]Implicit differentiation.

[00:04:33]This comes up when you can't isolate y.

[00:04:36]You differentiate [music] both sides with respect to x, and every time you hit a y term, you multiply by dy/dx.

[00:04:44]So, for x squared plus y squared equals 25, [music] you get 2x + 2y * dy/dx is equal to 0.

[00:04:51][music] Solve for dy/dx and you get -x over y.

[00:04:57]If derivatives [music] are about breaking things down, integration is about building them back up. [music] The integral finds the area under a curve between two points. So, something like the integral from 1 to [music] 3 of x squared, that's asking for the total area sitting underneath that curve.

[00:05:13][music] We can approximate that area using Riemann sums. You chop the region into a bunch of rectangles, [music] each with width delta x and height equal to the function value at that point.

[00:05:26]And you add them all up. As the rectangles get thinner [music] and thinner, the approximation gets more and more exact. And in the limit, that sum becomes the [music] integral. But we don't compute that limit every time. We use antiderivatives instead.

[00:05:41]Power rule for integrals.

[00:05:42]>> [music] >> Add one to the exponent, then divide by the new exponent. So, the integral of x cubed is x to the fourth over four.

[00:05:50]>> [music] >> If the integral has no bounds, do not forget the plus c at the end. The common integrals that you should memorize are the integral of e to the x is e to the x plus c.

[00:06:03]The integral of one over x [music] is the natural log of the absolute value of x plus c.

[00:06:09]The integral of cosine [music] is sine plus c. The integral of sine is negative cosine plus c. These are simply the reverses of their derivatives.

[00:06:22]Now, >> [music] >> u-substitution.

[00:06:24]This is the chain rule but in reverse.

[00:06:27]When you see a composite [music] function inside an integral, u-sub is your move.

[00:06:33]Take the integral [music] of 2x times cosine of x squared. Let u equal x squared, [music] then du is 2x dx. And look, that 2x dx is already sitting right there in the integral.

[00:06:46]Substitute it [music] all in and you get the integral of cosine of u, which is just sine of u plus c. Swap back and you get sine of x squared plus c.

[00:06:56]For integrals [music] with actual bounds, the answer is a number, not a function. Find the antiderivative, plug in the top bound, plug in the bottom bound, subtract.

[00:07:05]>> [music] >> So, the integral from one to three of x squared gives you x cubed over three evaluated from one to three.

[00:07:13]Plug in the bounds and solve.

[00:07:15]>> [music] >> Here it is 26 over three.

[00:07:19]The fundamental theorem of calculus.

[00:07:22]This is [music] the most important idea in the entire course. It's what connects derivatives and integrals, which on the surface look like completely separate things. Part one.

[00:07:32]>> [music] >> If you define a function as a definite integral with a variable upper bound, its derivative is just the integrand evaluated at that upper bound.

[00:07:42]So, if f of x equals the integral from a to x of f of t dt, then f prime of x is just f of [music] x.

[00:07:51]Part two, to evaluate a definite integral, find the antiderivative and subtract.

[00:07:57]The integral from [music] a to b of f of x equals f of b minus f of a.

[00:08:04]Applications. [music] The applications are where the exam gets hard, and no 10-minute video could cover all of them.

[00:08:10]Three big ones: [music] related rates, optimization, and area between curves.

[00:08:15]Related rates.

[00:08:17]These problems give you how fast one thing is changing and ask how fast something else is [music] changing.

[00:08:23]Both variables are functions of time.

[00:08:27]With optimization, you'll need to find the maximum or minimum of something, but you'll [music] always follow the same three steps.

[00:08:35]Step one, write the function you're trying to optimize. [music] Step two, take the derivative and set it equal to zero. Those are your critical points.

[00:08:46]Step three, [music] use the first or second derivative test to confirm whether it's a max or a min.

[00:08:53]If f double prime is negative [music] at that point, it's a max.

[00:08:57]If it's positive, it's a minimum. Area between curves.

[00:09:01]You've got two functions, one on top, one on bottom, and you want the area of the region between them.

[00:09:08]The formula is the integral from [music] a to b of the top function minus the bottom function.

[00:09:14]For example, if the top curve is x plus [music] two and the bottom is x squared, you integrate x plus two minus x squared between the [music] intersection points.

[00:09:25]That's all of AP Calc AB. For a deeper review, make sure to check out my other videos.