Step by step
The question
Question: find dy for y = x² + 3x when x = 1 and dx = 0.01.
y = x2 + 3x
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) + 3 × 1
Now we have f′(x) = 2x + 3.
= 2x + 3
3 Put the numbers in
The slope at x = 1 is 5.
f′(1) = 2(1) + 3 = 2 + 3 = 5
So dy = 0.05.
dy = 5 × 0.01 = 0.05
Careful!
dy is the slope times dx.
Do not forget the times 0.01.
Careful: not dy = f′(1) = 5 ✗
4 Check
Let's check with the real change: f at 1.01, minus f at 1, is 0.0501 on a calculator.
Close to 0.05.
Right!
Δy = f(1.01) − f(1) = 0.0501
Let's look back.
First: the rule.
Then: find f′(x).
Then: put the numbers in.
Last: check.
Answer
Careful: not dy = f′(1) = 5 ✗