Step by step
The question
Question: find f(x), if f′(x) = 4x − 1 and f(2) = 9.
f′(x) = 4x − 1
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 4 × x22 − x + C
Now we have f(x) = 2x² − x + C.
= 2x2 − x + 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 = 2.
f(2) = 2(2)2 − 2 + C
Work it out: 6 + C.
= 6 + C
It must equal 9.
6 + C = 9
Take away 6 on both sides: C = 3.
C = 9 − 6 = 3
Careful!
C is not just the f value.
The other terms at x = 2 are not 0 here.
Careful: not C = 9 ✗
3 Write the function
So f(x) = 2x² − x + 3.
f(x) = 2x2 − x + 3
4 Check
Let's check: its derivative is 4x − 1, the slope we were given.
f′(x) = 4x − 1
And at x = 2 it gives 9.
Right!
x = 2: 6 + 3 = 9
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = 2x2 − x + 3