Step by step
The question
Question: find dy for y = x² − 5x when x = 3 and dx = -0.1.
y = x2 − 5x
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) − 5 × 1
Now we have f′(x) = 2x − 5.
= 2x − 5
3 Put the numbers in
The slope at x = 3 is 1.
f′(3) = 2(3) − 5 = 6 − 5 = 1
So dy = −0.1.
dy = 1 × (-0.1) = −0.1
Careful!
dy is the slope times dx.
Do not forget the times -0.1.
Careful: not dy = f′(3) = 1 ✗
4 Check
Let's check with the real change: f at 2.9, minus f at 3, is −0.09 on a calculator.
Close to −0.1.
Right!
Δy = f(2.9) − f(3) = −0.09
Let's look back.
First: the rule.
Then: find f′(x).
Then: put the numbers in.
Last: check.
Answer
Careful: not dy = f′(3) = 1 ✗