1. Overview. Suppose we have a line in the plane . What data do we need to specify a line?
We need a slope and a point on the line . Then we can specify completely by the equation
We can rewrite this as
Lets remember this.
Last time we discussed the derivative, and the derivative gives us the slope at a point . But look: we have the slope from the derivative.
What more information do we need to compute a line? We need a point on the curve! So if is a curve, and is a point on the curve, then we can compute the line tangent to at as
This uses the derivative as the slope, and the data .
(We should quickly prove to ourselves this passes through the desired point. It is a one minute check that . We also see it has the desired slope, and it is in fact a line.)
Lets now consider a few examples.
2. Basic Algorithm. So, what’s the basic algorithm when finding the tangent line of the curve at the point ?
Step 1. Find the derivative .
Step 2. Using Step 1’s results, evaluate the derivative at , i.e., evaluate .
Step 3. Using Step 2’s results, plug it into the formula .
We will do each step and draw the doodles as we go.
Example 1. Lets find the tangent line passing through the point for the curve . What to do?
Step 0: draw the curve. We will doodle the curve in blue, just so we understand which part of the graph is the curve.
Step 1: find the derivative . We see that the derivative for is, well, by using linearity
Then we invoke the power rule to find
Thus we find
This concludes step 1.
Step 2: Evaluate . OK, so we just found and we have . Thus we find
This concludes step 2.
Step 3: Construct Tangent Line. Well, we have the formula for the tangent line be
We have everything we need. We plug in and , and results from step 2, to get
But we may simplify things out:
Thus the tangent to at is
We can doodle this:
Note the curve is doodled in blue, the tangent point is emphasized, and the tangent line is doodled in pink.
We will conclude this post with motivation for the next:
Problem. We want to find some curve such that at any point on the curve its tangent line satisfies the property . What is ?
(The field called “Differential Equations” studies this general sort of problem “Find the curve whose derivative satisfies…”)