AP®︎/College Calculus AB
Course: AP®︎/College Calculus AB > Unit 6Lesson 7: The fundamental theorem of calculus and definite integrals
Antiderivatives and indefinite integrals
What's the opposite of a derivative? It's something called the "indefinite integral". Created by Sal Khan.
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- So, is it all about writing "possible functions" from its derivatives?(27 votes)
- Well, essentially.
In application, you'll have additional constraints, which will narrow down the possibilities.(27 votes)
- How can we use derivatives, integrals and anti-derivatives in real life situations or in an occupation like engineering? I'm kinda confused here, if anyone can give an example and a detailed explanation I would really appreciate it. I really want to understand this intuitively. Thanks in advance!(16 votes)
- Okay after studying so far I understand that:
derivatives are used to find the minimum or maximum of something to optimize something.(5 votes)
- Are their any good websites that provide integration problems?(5 votes)
- go to http://www.math.ucdavis.edu/~kouba/ProblemsList.html
and scroll down to beginning integral calculus.
Hope this helped =).(31 votes)
- I'm currently in Grade 10, and our math subjects are mainly on trigonometry, quadratic equations, and general algebra 2 stuff. I can safely say that I've learnt all Grade 10, 11 and even 12 math level subjects with Khan Academy since I am so fascinated by math. I was wondering whether I should start learning calculus since it's so profound and deep. Should I just wait it out till college or can I start learning this by myself?(9 votes)
- Start now!
Start here at the beginning of the Pre Calculus Track: https://www.khanacademy.org/math/precalculus
Do the work in the order presented - if you don't get one or two things, ask a question.
If you are not getting a lot of things, best do some appropriate review.
Have Fun!(8 votes)
- cant the anti derivative of 2x be x^2+c1+c2+...+cn ?
is it so that all the constants are summed and they become one constant?
Which gives us just one constant notation c?
Have I understood correctly?(4 votes)
- Yes, that is correct. That will be a useful understanding when you are solving differential equations, which will depend heavily on those arbitrary constant.(6 votes)
- Will there be videos about solving rational,irrational and hyperbolic function integrals?(12 votes)
- What is the range of points that you need an area for?(0 votes)
- In the integral notation, why is there a dx at the end? I've looked everywhere for an explanation, but I just can't find one. I would really appreciate an answer.(3 votes)
- The symbol dx has different interpretations depending on the theory being used. In Leibniz's notation, dx is interpreted as an infinitesimal change in x and his integration notation is the most common one in use today. If the underlying theory of integration is not important, dx can be seen as strictly a notation indicating that x is a dummy variable of integration; if the integral is seen as a Riemann integral, dx indicates that the sum is over subintervals in the domain of x; in a Riemann–Stieltjes integral, it indicates the weight applied to a subinterval in the sum; in Lebesgue integration and its extensions, dx is a measure, a type of function which assigns sizes to sets; in non-standard analysis, it is an infinitesimal; and in the theory of differentiable manifolds, it is often a differential form, a quantity which assigns numbers to tangent vectors. Depending on the situation, the notation may vary slightly to capture the important features of the situation. For instance, when integrating a variable x with respect to a measure μ, the notation dμ(x) is sometimes used to emphasize the dependence on x. Source: http://en.wikipedia.org/wiki/Integral#Terminology_and_notation(7 votes)
- is this calculus 2 material?(3 votes)
- Calc 1 should include at the very least a brief lesson on this. Calc 2 goes much farther in-depth with integrals.(5 votes)
- What will be the derivative of :
Squareroot of ax.
a being a constant does it effect the derivation.
naive in maths(3 votes)
- Power Rule states ax^1/2 = (1/2)ax^((1/2)-1) = (1/2)ax^(-1/2) = a/2x^1/2(1 vote)
- If a Derivative shows the rate of change of a curve & if an Integral shows the area under the curve.
Then what is an Antiderivative? What is it used for?(3 votes)
- At first, mathematicians studied three (or four if you count limits) areas of calculus. Those would be derivatives, definite integrals, and antiderivatives (now also called indefinite integrals).
When you learn about the fundamental theorem of calculus, you will learn that the antiderivative has a very, very important property. There is a reason why it is also called the indefinite integral.
I won't spoil it for you because it really is incredible!(4 votes)
We know how to take derivatives of functions. If I apply the derivative operator to x squared, I get 2x. Now, if I also apply the derivative operator to x squared plus 1, I also get 2x. If I apply the derivative operator to x squared plus pi, I also get 2x. The derivative of x squared is 2x. Derivative, with respect to x of pi of a constant, is just 0. Derivative, with respect to x of 1, is just a constant, is just 0. So once again, this is just going to be equal to 2x. In general, the derivative, with respect to x of x squared plus any constant, is going to be equal to 2x. The derivative of x squared, with respect to x, is 2x. Derivative of a constant, with respect to x, a constant does not change with respect to x, so it's just equal to 0. So you have-- You apply the derivative operator to any of these expressions and you get 2x. Now, let's go the other way around. Let's think about the antiderivative. And one way to think about it is we're doing the opposite of the derivative operator. The derivative operator, you get an expression and you find it's derivative. Now, what we want to do, is given some expression, we want to find what it could be the derivative of. So if someone were to tell-- or give you 2x-- if someone were to say 2x-- let me write this. So if someone were to ask you what is 2x the derivative of? They're essentially asking you for the antiderivative. And so you could say, well, 2x is the derivative of x squared. But you could also say 2x is the derivative of x squared plus 1. You could also say that 2x is the derivative of x squared plus pi, I think you get the general idea. So if you wanted to write it in the most general sense, you would write that 2x is the derivative of x squared plus some constant. So this is what you would consider the antiderivative of 2x. Now, that's all nice, but this is kind of clumsy to have to write a sentence like this, so let's come up with some notation for the antiderivative. And the convention here is to use kind of a strange looking notation, is to use a big elongated s looking thing like that, and a dx around the function that you're trying to take the antiderivative of. So in this case, it would look something like this. This is just saying this is equal to the antiderivative of 2x, and the antiderivative of 2x, we have already seen, is x squared plus c. Now, you might be saying, why do we use this type of crazy notation. It'll become more obvious when we study the definite integral and areas under curves and taking sums of rectangles in order to approximate the area of the curve. Here, it really should just be viewed as a notation for antiderivative. And this notation right over here, this whole expression, is called the indefinite integral of 2x, which is another way of just saying the antiderivative of 2x.