Step by step
The question
Question: find dy for y = x² when x = 3 and dx = 0.1, and compare it with Δy.
y = x2
1 The rule
dx is a small change in x.
dy is the change the tangent line gives: the slope times dx.
dy = f′(x) dx
dy = f′(x) dx
2 Find f′(x)
Use the power rule on each term.
f′(x) = 2x^(2 − 1)
Now we have f′(x) = 2x.
= 2x
3 Put the numbers in
The slope at x = 3 is 6.
f′(3) = 2(3) = 6
So dy = 0.6.
dy = 6 × 0.1 = 0.6
Careful!
dy is the slope times dx.
Do not forget the times 0.1.
Careful: not dy = f′(3) = 6 ✗
4 The real change
Δy is the real change: f at 3.1, minus f at 3.
Δy = f(3.1) − f(3)
A calculator gives f of 3.1: 9.61.
And f of 3 is 9.
= 9.61 − 9
So Δy = 0.61.
Very close to dy, 0.6.
= 0.61
5 Check
Let's check: dy and delta y differ by only 0.01.
A tiny gap, because dx is small.
Right!
dy − Δy = −0.01: tiny
Let's look back.
First: the rule.
Then: find f′(x).
Then: put the numbers in.
Then: the real change.
Last: check.
Answer
Careful: not dy = f′(3) = 6 ✗