Step by step
The question
Question: find y, if dy/dx = 4x + 3 and y = 2 when x = 1.
dydx = 4x + 3
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 4 × x22 + 3x + C
Now we have y = 2x² + 3x + C.
= 2x2 + 3x + 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 and y = 2.
2 = 2(1)2 + 3(1) + 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 y value.
The other terms at x = 1 are not 0 here.
Careful: not C = 2 ✗
3 Write the function
So y = 2x² + 3x − 3.
y = 2x2 + 3x − 3
4 Check
Let's check: its derivative is 4x + 3, the slope we were given.
dy/dx = 4x + 3
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
y = 2x2 + 3x − 3