Step by step
The question
Question: find f(x), if f′(x) = 4x³ and f(1) = 5.
f′(x) = 4x3
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
f(x) = 4 × x44 + C
Now we have f(x) = x⁴ + C.
= x4 + 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) = 14 + C
Work it out: 1 + C.
= 1 + C
It must equal 5.
1 + C = 5
Take away 1 on both sides: C = 4.
C = 5 − 1 = 4
Careful!
C is not just the f value.
The other terms at x = 1 are not 0 here.
Careful: not C = 5 ✗
3 Write the function
So f(x) = x⁴ + 4.
f(x) = x4 + 4
4 Check
Let's check: its derivative is 4x³, the slope we were given.
f′(x) = 4x3
And at x = 1 it gives 5.
Right!
x = 1: 1 + 4 = 5
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
f(x) = x4 + 4