Step by step
The question
Question: find f(x), if f′(x) = 6x + 2 and f(0) = −3.
f′(x) = 6x + 2
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 6 × x22 + 2x + C
Now we have f(x) = 3x² + 2x + C.
= 3x2 + 2x + C
Every antiderivative has + C.
The point will tell us what C is.
+ C: we do not know it yet
2 Find C
Put in x = 0.
f(0) = 3(0)2 + 2(0) + C
The other terms are 0 at x = 0.
So C = −3.
C = −3
3 Write the function
So f(x) = 3x² + 2x − 3.
f(x) = 3x2 + 2x − 3
4 Check
Let's check: its derivative is 6x + 2, the slope we were given.
f′(x) = 6x + 2
And at x = 0 it gives −3.
Right!
x = 0: 0 − 3 = −3
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = 3x2 + 2x − 3