"What I am going to do... is make calculus look like one plus one."
"Everyone needs an understanding of basic calculus, whether they like it or not."
"Limits allow you to evaluate a function when x approaches a certain value; derivatives are functions that allow you to calculate the instantaneous rate of change... and integration... allows us to determine how much something accumulates over time."
"Integration takes you from the derivative to the function you had before you differentiated it."
"Implicit differentiation allows us to take a derivative without having Y explicitly solved for."
"We must treat Y as a function of X, even if we don't know exactly what it is."
"The derivative of X is 1, the derivative of 5x is going to be 5."
"The derivative of sine U is cosine of that U variable times the derivative of U."
"To find the derivative of a composite function, you use the chain rule."
"Both the derivative and the integral are only possible because of the limit, which is why this unit is so foundational for the rest of the course."
"Limits allow us to narrow this interval so that it becomes smaller and smaller... until eventually we have this tiny little interval and we're finding rate of change over that tiny little interval."
"If we want to find instantaneous rate of change, we apply the limits to those difference quotient formulas."
"Limits really are the basis of our calculus. It's how we do calculus."
"If you know how to do limits, you're going to be able to do calculus."
"This actually works itself out not too bad. The calculus is very easy."
"The limit of the sum is equal to the sum of the limits."
"Calculus pops up in several places in data science and machine learning."
"The Fundamental Theorem of Calculus: The most important thing in calculus."
"The hardest part of calculus is that we call it one variable calculus, but we're perfectly happy to deal with four variables at a time or five, or any number."
"All of this is only a high-level view: just a peek at some of the core ideas that emerge in calculus."
"At all points I want you to feel that you could have invented calculus yourself."
"For me, calculus is about the relation between two functions."
"Calculus has the job of given one of those functions, find the other one."
"Calculus is the mathematics that describes how a thing change instantaneously, whether that thing is velocity, acceleration, displacement, height, weight, volume, or whatever."
"The beauty of calculus is that it can solve just awesome real-world problems for us."
"The exercise that you could do to verify that a cycle like this is attracting, by the way, would be to compute the derivative of f(f(z)), and you check that at the input zero this derivative has a magnitude less than one."
"If you need to evaluate at a particular point all you do is take that point, plug it into the function you found, and that will tell you the value of that partial derivative at that particular point."
"This fairness of calculus is telling of how powerful relationships between ideas and people can be, even when they might not seem to work well together at first."
"The inflection point occurs when the slope stops increasing."
"If you have that A is equal to tanh of Z, then dA by dZ by propagating through tanh is just one minus A squared."
"It turns out that hard problems are hard to solve, and this is why we turn to calculus, which can help us to intelligently navigate to a solution."
"This instantaneous slope is also the derivative."
"Calculating derivatives using numerical differentiation would require multiple forward passes for a single weight or bias adjustment per sample."
"For multivariate functions, instead of doing something like multiple forward passes for our neural network, we can use partial derivatives."
"Just worry about the absolute value whenever you have definite integrals, right? So that's the usual deal."
"Here's the answer: one half secant theta times tangent theta plus one half ln absolute value secant theta plus tangent theta. I know it's a long one, yeah, I know."
"So the first derivative intuitively tells us how much f changes when we change x by a tiny amount."
"Calculus teaches us that if you zoom in on something enough, no matter how curvy it was, it begins to look straight."
"We can literally pretty much get the total area of this figure, and that is the main concept of integration in calculus."
"The process of finding the antiderivative is basically integration."
"We know that e to the z is its own derivative so if I were to zoom into a point z the transformation would look like being multiplied by e to the z."
"Calculus is hard because it is different. It introduces completely new concepts."
"Understanding this link will give us a better understanding of the motivation for calculus."
"Calculus is one of those things that once you see the world through that lens, you can't unsee it."
"Calculus is as close as we've come to discovering the secret of the universe."
"Calculus helped reshape civilization by allowing us to understand and manipulate the laws of nature."
"Calculus is one of those enriching, marvelous things that makes your life better outside of utilitarian need."
"You're using calculus all day long. You just don't know it."
"Calculus is not as bad as you might think."
"Calculus is definitely quite fascinating."
"Gradient descent is an amazing method from calculus that allows us to calculate the slope of our cost function and move in the negative direction of that slope."
"The chain rule allows us to calculate the effect of changing something downstream by multiplying together each gradient along the way."
"One of the reasons I love Vector calculus is that you can make these really General statements that are just true."
"Can you find the slope of a curve at a point? What was that, a derivative?"
"If we want to find the area under a certain function, we basically need to be able to undo this."
"The area of a curve is A of x. If I take the first derivative of the area, it will give you your original function back."
"If you want to find the area of the curve under the curve, what I'm going to do is undo the derivative. That gives me the area."
"If our function is X^2, and we know that a function is equal to the first derivative of its area function, then we say, 'All right, well if f(x) is X^2, then X^2 equals that.'"
"We have one, two, and we have an infinite number of possibilities for the area function that it could actually be."
"Algebra: Sometimes it's just a little thing, the little steps of your big calculus questions."
"Now that we get on to differentiation, I expect you to know what finding the derivative actually gives you."
"Since we know that this equals this and I want to find the integral do you remember anything about calculus you can integrate on both sides of an equation"
"And the truth is, if you want to integrate secant x, you can use this as your answer as well."
"The idea of the chain rule is that you're trying to figure out how the function f changes as the variable x changes but there's a function that's changing in between them."
"The chain rule can feel confusing because there are often four or more variables in the problem at the same time and so it is helpful to have a scheme to organize that information in a helpful way."
"Critical points for single variable calculus occur where the first derivative of the function is equal to 0 or is undefined."
"The derivative of x raised to the sine x is x sine x times (cosine x ln x + sine x over x)."
"In single-variable calculus, we were able to think of integration as area under a curve."
"Gauss's divergence theorem is an extremely powerful equality in vector calculus."
"The bottom times the derivative of the top minus the top times the derivative of the bottom."
"The method of Lagrange multipliers is, at least in my view, one of the great magic acts of multivariable calculus."
"So this turns into the integral from 0 to infinity of R e to the minus t over 1 minus R e to the minus t."
"We'd like to perhaps do a change of variables and the motivation here is that we see an e to the minus t in the denominator and its derivative is in the numerator another e to the minus t well that's off by a constant but that's like kind of good enough."
"The best way to explain how math has built structure to answer this question is calculus."
"...we use it extensively in calculus because very quickly the problems get much more complicated and you have to understand what a composite function is to do almost anything in calculus."
"That's the test of whether you really understand multi-variable calculus."
"this is a pretty solid book on calculus"
"The covariant derivative of a vector field on a curved surface is the ordinary derivative with the normal components subtracted."
"Without calculus, our modern-day world would not exist."
"Calculus effectively solves two problems for us."
"The first problem calculus solves for us is the area problem."
"The second problem calculus solves for us is something called the rate of change."
"Gradient of a scalar function is a vector that always points in the direction of greatest increase or decrease of the function at a point."
"He invented calculus. This is endless thanks to this man, genius."
"The good stuff that we're going to talk about for the rest of our section is parametric equations, the calculus of parametric equations, polar coordinates, and then the calculus of polar coordinates."
"The quantum integral of the quantum derivative from 0 to a is simply that function being evaluated at the two endpoints."
"...the limit does not exist as X approaches zero from either side."
"So let's find f prime, that's going to be three z squared times everything else."
"Antiderivative of 10^x is 10^x divided by natural log of 10."
"The derivative of the integral equals the original function."
"The main idea behind the chain rule: differentiate the outside, keep the inside, multiply by the derivative of the inside."
"We took the derivative algebraically and got a closed form solution without ever having to use the piecewise definition."
"And we found the derivative without referencing inverse functions or knowing anything about the derivative of e to the X itself."
"I'm very curious to hear from calc 1 teachers: Is this something that you would introduce in your class alongside the original definition, even if just used as a teaching tool for understanding the power rule intuitively? Let me know in the comments."
"There is a precise sense in which you think about derivatives in the sense of calculus, derivations, and these boundary operators."
"How to find average rate of change: Your rise over your run."
"At time equals 25 minutes, the rate at which hot chocolate is being removed is decreasing at 2.15 cups per minute per minute."
"Changing in a negative, I prefer to say 'decreasing', so the rate at which it's decreasing is 2.15 cups per minute per minute."
"Your job is to find all those derivative values like the f of 1, f prime of 1, f double prime."
Applications of derivatives: "Always need two pieces of information for a tangent line: slope and point."
Integration: "Remember, any area below the x-axis is negative."
Net change: "Rate in minus rate out, that's the rate of change of something."
Theorems: "Mean value theorem tells you that there's a point where the slope of the tangent line matches the slope of the secant line."
FRQ types: "You usually get a graph where they give you some graphs of f prime and you have to relate it to f of x."
"If the second derivative is positive, what kind of behavior does that tell me about the graph of my function? That means that the graph of the function is concave up."
"That's my justification; that's because the second derivative is positive."
"The calculus I want to talk about is made up from two seemingly unrelated strands now called differentiation and integration and is probably the most important mathematical technique ever devised."
"...it's such a cool result that it's called the fundamental theorem of the calculus."
"...Leibniz calculus originated in a different manner from that of Newton's been based on sums and differences rather than on speed and motion and in 1675 liveness introduced two symbols that would forever be used in the calculus."
"After you really know the arithmetic by all means go and start to study the calculus."
"It's the area under the curve that matters."
"Initial rates can be found using the gradient of a tangent."
"This is the beginning of calculus as almost as we would see it today."
"The second derivative is responsible for Movement and, therefore, life."
"The minimum occurs when the derivative is equal to zero."
"It's calculus with multiple variables essential skills workbook and again I think it's valuable because the examples that you find in this book you can't really find them in other books."
"The slope is the change in the error divided by the change in time."
"You can learn Calculus if you know a little bit of algebra."
"It's a great book if you want to learn calculus and you feel like you don't know enough math to do so."
"We all know that this stands for a certain limit, but the limit stands literally for the slope which is at that point."
"But most functions of course will change the slope, and for that reason there will be an error in the formula you get if you use this."
"If Δx is very, very small and it's your intention in the end to make very, very tiny Δxs then we say we will use this approximation."
"When dealing with an indefinite integral, you always need to add the constant of integration."
"It's not really too difficult finding the antiderivative of a function as you work out a few problems; it gets easier."
"The process by which we evaluate the definite integral comes from the fundamental theorem of calculus."
"The integral of \( f(x) \) DX, that is the indefinite integral, is equal to the antiderivative plus C."
"The antiderivative of \( e^u \) is \( e^u \) divided by \( u' \), if \( u \) is a linear function."
"U substitution is a common technique that you need to be familiar with when dealing with integration."
"The antiderivative of \( \frac{1}{x} \) is \( \ln(x) \), because the derivative of \( \ln(x) \) is \( \frac{1}{x} \)."
"The derivative of the inverse function with respect to x is one over the derivative."
"When it comes to differentiation, we are finding an expression that allows us to get the gradient at a point on a curve."
"Multiply the coefficient by the power and reduce the power by 1, that is pretty much all we're going to need to do."
"If we multiply the coefficient by the power of 3, 8, 16, 24, we get 24x, then reduce the power, that becomes x squared."
"We're going to have a look at obviously differentiating much harder graph equations than this one."
"It's all about getting into that straight line there and obviously just being very careful with your powers."
"To find a turning point, we have to do something called completing the square."
"The coordinates of our turning point are negative four and negative thirteen for the y coordinate."
"That'll be the reverse function there, the inverse function."
"Differentiation has key derivatives that you must make sure you know."
"The chain rule allows you to differentiate functions like y = f(g(x))."
"The product rule is for the product of functions."
"Integration by substitution requires you to get u equals right, and your first step is to find dy/dx."
"When you differentiate a function, what you actually get is something which is called its derivative."
"The derivative of a function tells you how it changes and it's often called its rate of change."
"DY by DX means the change in Y divided by the change in X."
"If a function is e to the x, then the first derivative of it is just going to be e to the x."
"We needed a type of calculus for this type of mathematics."
"We say that the limit as X approaches some number a of some function f of x is equal to some limit L."
"For every K between F of A and F of B, there exists a number C in A, B such that F of C is equal to K."
"The first derivative test is going to give you critical numbers and increasing and decreasing areas where the graph will increase or decrease."
"Tell me something, ladies and gentlemen, you need to know this: when X goes to Infinity, what's this do? What's this do? What's this do? Stays at one, that's 0 over one, that's zero."
"So where are our horizontal asymptotes? Zero, both left and right, both left and right. So horizontal asymptotes occur at y equals 0, both positive and negative Infinity."
"How many people feel okay with what we've done so far? Any calculus? Not really, just limits. Let's start doing our step number three, what step number three do? Our first test."
"Second derivative test was going to give you concavity."
"At the maximum value and at the minimum value, the slope of the horizontal tangent line is zero."
"To find the location of a maximum or a minimum, what you need to do is you need to take the first derivative of the function, set it equal to zero, and solve for x."
"The objective function is the function that we're going to find the first derivative of, set it equal to zero, and solve for either x or y."
"Euler's method works because it's based on the equation of the tangent line."
"That's the idea of the limit, right? Taking that width to zero."
"The range of this function is \( -\infty \) to \( -1 \), and \( 3 \) to \( +\infty \)."
"The integral sign means that we're going to add all of the areas of the cross sections from the top to the bottom."
"And the integral sign right here means that we add all of those volumes of our shells together."
"The volume formula will be V is equal to the integral from A to B of 2 times pi times R of Y times F of y dy."
"The derivative of natural log x is 1 over x."
"The derivative of sine is just cosine."
"We have to take the derivative of this thing, don't we? So I need to somehow get this area here all in terms of x or all in terms of y."
"The derivative of the integral of \( lnt \, dt \) is simply going to be \( ln(x) \)."
"The antiderivative from \( a \) to \( b \) of \( f(x) \, dx \) is simply \( F(b) - F(a) \)."
"The better you have the unit circle committed to memory, the easier calculus is going to be on you."
"We can say that the exact area underneath the curve y equals x squared on the interval from 0 to 1 is equal to 1/3."
"A partial differential equation or a PDE is basically an equation involving one or more derivatives of an unknown function of two or more independent variables."
"If I know my critical value evaluated in the second derivative comes out to be positive, that means my function is concave up there, so that critical value is a min."
"That's pretty nice when you realize that we were facing for the first time, more or less, the sort of tough problem of calculus."
"We now know the derivatives of two of the great functions of calculus."
"Imagine you have a curve like this... this will then become the exact area under the curve."
"Calculus is usually not the hardest part of the engineering classes themselves compared to coming up with an actual solution to a real-world engineering problem."
"These books will help you learn calculus."
"It talks about all of the things that are usually taught in a basic calculus one course in college."
"If you're trying to learn this on your own and you're struggling, don't worry, it's totally normal."
"If you're looking for a good calculus book where you can do practice problems from, I think any of these is an awesome choice."
"This book influenced the world as to how calculus books are written."
"It's the theory behind calculus in a way."
"We talk about what it really means to be a real number, what it really means to be a derivative, to be an integral."
"As long as we remember these two steps with chain rule, we'll be able to apply it to any function that's a composite function."
"The big takeaway from topic 4.1 is really just about being able to understand the relationship between the original function, its derivative, and its second derivative."
"The derivative of the product of two functions is the derivative of one of them multiplied by the other one plus the first one times the derivative of the other one."
"If you have two functions that are multiplied together, then the derivative is going to be the first times the derivative of the second plus the second times the derivative of the first."
"You must know your differentiation rules and when to use them."
"Once you're done with the practice questions per unit, jump into unit 3 where we'll dig into the chain rule."
"The limit exists at a point C if the limit from the left equals the limit as X approaches from the right."
"Horizontal asymptotes are when we take the limit of a function as X goes to positive or negative infinity."
"For a function to be continuous, the limit must equal the value of the function at a particular point."
"If you see 0 over 0 or infinity over infinity, that's an indeterminate form, then you apply L'Hôpital's."
"The derivative is defined as a limit, so the limit as H goes to zero of f of X plus H minus f of X all over H."
"If we've got a continuous function on some closed interval from A to B, then there is some X value between A and B such that we will hit the values in between."
"The integral from A to B is whatever that antiderivative is, that function evaluated at B minus the value evaluated at A."
"The integral can be written as an infinite sum of all of those rectangles of the infinitesimally wide rectangles, and you will get the exact area."
"Integration by parts rule is going to be the integral of u times v prime is going to be equal to uv minus the integral of v times u prime."
"Hello intrepid explorers of AP Calculus, I wish you good health, perseverance, and focus so that you may finish this year on a note that will make you proud."
"The moment I hear that phrase in calculus 'f is increasing', I just think f prime of x has to be positive."