Step by step
The question
Question: find the tangent line to y = x³ − 3x at x = 2.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the point
The line touches the curve at x = 2.
Its y is f(2): put 2 in.
We get 2.
So the point is (2, 2).
2 Find the slope
The slope of the curve is the derivative.
Use the power rule on each term.
Now we have f′(x) = 3x² − 3.
At x = 2: put 2 in.
The slope is m = 9.
Careful!
The slope of a line is one number.
f′(x) changes with x, so we put x = 2 in.
3 Use point-slope form
Point-slope form: the line through the point (x₁, y₁) with slope m is y − y₁ = m(x − x₁).
Put in the point (2, 2) and m = 9: y − 2 = 9(x − 2).
4 Write as y = mx + b
We also write it as y = mx + b.
Then we can read the slope and where the line crosses the y-axis.
The 9 in front of the bracket multiplies every part inside.
So we write 9 times each part.
Work out each multiplication.
9 times 2 is 18.
Plus times minus is minus.
Now we have y − 2 = 9x − 18.
Is y alone now?
Not yet: − 2 is still with y.
How do we remove it?
We want y by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 cancel each other.
They make zero.
−18 plus 2 is −16.
So y = 9x − 16.
5 Check
Let's check on a picture.
The curve y = x³ − 3x.
Our line touches it at (2, 2), with the same steepness.
Right!
6 Pick the choice
Our answer is y = 9x − 16.
That is choice B.
7 Why not the others
A is y = 9x.
Wrong: left out the point (2, 2).
C is y = 3x − 4.
Wrong: used the slope 3: y′ = 3x² − 3 = 9 at x = 2.
D is y = 9x + 20.
Wrong: used + 18: y − 2 = 9(x − 2) gives − 18.
Let's look back.
First: find the point.
Then: find the slope.
Then: use point-slope form.
Then: write as y = mx + b.
Then: check.
Then: pick the choice.
Last: why not the others.