Step by step
The question
Question: find dy for y = 2x² − x when x = 4 and dx = 0.05.
y = 2x2 − x
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) = 2 × 2x^(2 − 1) − 1
Now we have f′(x) = 4x − 1.
= 4x − 1
3 Put the numbers in
The slope at x = 4 is 15.
f′(4) = 4(4) − 1 = 16 − 1 = 15
So dy = 0.75.
dy = 15 × 0.05 = 0.75
Careful!
dy is the slope times dx.
Do not forget the times 0.05.
Careful: not dy = f′(4) = 15 ✗
4 Check
Let's check with the real change: f at 4.05, minus f at 4, is 0.755 on a calculator.
Close to 0.75.
Right!
Δy = f(4.05) − f(4) = 0.755
Let's look back.
First: the rule.
Then: find f′(x).
Then: put the numbers in.
Last: check.
Answer
Careful: not dy = f′(4) = 15 ✗