Essential Knowledge 3.1A1

Essential Knowledge 3.1A1 Students will know that an antiderivative of a function f is a function g whose derivative is f.

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So, if I want to know whether or not function f is an antiderivative of function g, what can I do? Well think about this question.

Pretend you have no idea how to divide, but you do know division is the opposite of multiplication.

a) If I know 4 x 6 = 24, is it true that 24 ÷ 6 = 4?

Yes. Inverse operations will "undo" each other. So if multiplying by six brings 4 to 24, then dividing by 6 must bring 24 back to 4.

b) If I know 4 x 7 = 28, is it true that 28 ÷ 7 = 5?

No. Inverse operations will "undo" each other. So if multiplying by seven brings 4 to 28, then dividing by 7 must bring 28 back to 4.

We can use the same logic with derivatives and integrals

Example

Is g(x) = 2x³ + 5 an antiderivative of f(x) = 6x² ?

Well, even though we don't know how to take integrals yet, we do know they must be the opposite of a derivative. So, let's take a derivative of g(x) and see if we get f(x).

Sure enough we do. Therefore, yes, 2x³ + 5 is an antiderivative of 6x². Worth noting, there are other antiderivatives:

2x³ + 5

2x³ + 1

2x³ + -100

So in this way, it is a bit different than our multiplication/division analogy. Not a big deal. This is because constants become 0 when you take a derivative, so integrals can have any constant (more on this later).

Example

Is g(x) = cos(x) an antiderivative of f(x) = sin(x) ?

Well, let's see if g'(x) = f(x)

Close, but no cigar. That negative sign is not negligible. This is something you'll have to remember too, because many people forget that negative sign and get questions about sine and cosine wrong