TL7 Limits and Derivatives

Finally, we have all the machinery necessary to construct the tangent line at a fixed point $P$. With this construction, we will also have a precise definition for the derivative of a function.

Core Concepts¶

Constructing a Tangent Line¶

The difficulty in constructing the tangent line precisely is that it takes 2 points to define a line, but we want our tangent line to only touch the circle or curve at 1 point.

As we are going to see, the construction technique involves secant lines and limits:

1. Pick a point $Q$ on the curve which is close to $P$.

2. Draw a secant line connecting points $P$ and $Q$.

3. Calculate the limit as point $Q$ approaches point $P$.

Secant

Right Side Limit

Left Side Limit

Observations

Notice that as this point $Q$ moves closer and closer to point $P$, from either the right or the left side, the secant line looks more and more like what we would want for the tangent line.

Once $Q$ is very, very close to $P$ the secant line and the tangent line are indistinguishable. Both lines include point $P$ and have a very similar slope (essentially they are pointing in almost the same direction).

Limit Definition¶

Primary Examples¶

Homework¶

Write out each question as well as your solution on the homework page template.

• Make sure you leave a good amount of blank space between each question.

• Write down the question as well as your solution.