Step by step
The question
Question: find f(x), if f′(x) = 8x³ + 6x and f(−1) = 2.
f′(x) = 8x3 + 6x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 8 × x44 + 6 × x22 + C
Now we have f(x) = 2x⁴ + 3x² + C.
= 2x4 + 3x2 + 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) = 2(−1)4 + 3(−1)2 + C
Work it out: 5 + C.
= 5 + C
It must equal 2.
5 + C = 2
Take away 5 on both sides: C = −3.
C = 2 − 5 = −3
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) = 2x⁴ + 3x² − 3.
f(x) = 2x4 + 3x2 − 3
4 Check
Let's check: its derivative is 8x³ + 6x, the slope we were given.
f′(x) = 8x3 + 6x
And at x = −1 it gives 2.
Right!
x = −1: 5 − 3 = 2
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = 2x4 + 3x2 − 3