Step by step
The question
Question: find f(x), if f′(x) = 9x² − 4 and f(1) = −2.
f′(x) = 9x2 − 4
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 9 × x33 − 4x + C
Now we have f(x) = 3x³ − 4x + C.
= 3x3 − 4x + 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 = 1.
f(1) = 3(1)3 − 4(1) + C
Work it out: −1 + C.
= −1 + C
It must equal −2.
−1 + C = −2
Add 1 on both sides: C = −1.
C = −2 + 1 = −1
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = −2 ✗
3 Write the function
So f(x) = 3x³ − 4x − 1.
f(x) = 3x3 − 4x − 1
4 Check
Let's check: its derivative is 9x² − 4, the slope we were given.
f′(x) = 9x2 − 4
And at x = 1 it gives −2.
Right!
x = 1: −1 − 1 = −2
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = 3x3 − 4x − 1