Step by step
The question
Question: find dy for y = x³ + x when x = 1 and dx = 0.2.
y = x3 + 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) = 3x^(3 − 1) + 1
Now we have f′(x) = 3x² + 1.
= 3x2 + 1
3 Put the numbers in
The slope at x = 1 is 4.
f′(1) = 3(1)2 + 1 = 3 + 1 = 4
So dy = 0.8.
dy = 4 × 0.2 = 0.8
Careful!
dy is the slope times dx.
Do not forget the times 0.2.
Careful: not dy = f′(1) = 4 ✗
4 Check
Let's check with the real change: f at 1.2, minus f at 1, is 0.928 on a calculator.
Close to 0.8.
Right!
Δy = f(1.2) − f(1) = 0.928
Let's look back.
First: the rule.
Then: find f′(x).
Then: put the numbers in.
Last: check.
Answer
Careful: not dy = f′(1) = 4 ✗