Step by step
The question
Question: find y, if dy/dx = 5 − 2x and y = 7 when x = 2.
dydx = 5 − 2x
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 5x − 2 × x22 + C
Now we have y = 5x − x² + C.
= 5x − x2 + 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 and y = 7.
7 = 5(2) − 22 + C
Work it out: 6 + C.
= 6 + C
It must equal 7.
6 + C = 7
Take away 6 on both sides: C = 1.
C = 7 − 6 = 1
Careful!
C is not just the y value.
The other terms at x = 2 are not 0 here.
Careful: not C = 7 ✗
3 Write the function
So y = 5x − x² + 1.
y = 5x − x2 + 1
4 Check
Let's check: its derivative is 5 − 2x, the slope we were given.
dy/dx = 5 − 2x
And at x = 2 it gives 7.
Right!
x = 2: 6 + 1 = 7
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = 5x − x2 + 1