Extending Beyond Quadratics

Tangent lines are lines that are tangent to a curve or graph at a tangent point. There must be a point on the curve. At that tangent point, the curve or graph is behaving in some manner, which the tangent line is modeling.

Best job means the line shares the tangent point with the curve and the slope of the line is the same as the “slope” of the curve at the tangent point.

This is possible even at an endpoint on a graph.

Endpoints

This situation might occur if we had a restricted domain for a function.

This point \((4, -3)\) is an endpoint on the graph.

However, there still might be a line that models one side of the graph.

A tangent line, well, a one-sided tangent line.

We have a tangent line. It is a one-sided tangent line.

Secant lines on the left approach the tangent line.

\(\blacktriangleright \) desmos graph

The secant lines on the left smoothly turn into the tangent line as you move to the right.

There are no secant lines on the right, so we don’t have to worry about them.

In this case, we have a one-side tangent line.

We have a one-side derivative.

For our curve above, we have a left tangent line or left tangent ray.

For the function, we have a left derivative.

Similarly, we might have a one-side tangent line on the right.

We would have a one-side derivative.

We would have a right derivative.

The Derivative (Two-Sided)

Note: The term “derivative” implies a two-sided derivative. Otherwise, we would say ‘right’ or ‘left’.

The Left or Right Derivative (One-Sided)

So, we have two separate situations.

In the two-sided situaiton, we have a domain number inside an open interval in the domain, so that there are two sides. Since, it is a domain number, we know there is a point on the graph of the function and the graph is on both sides. We can examine if there is a tangent line or not and if it has a slope or not.

A different situation involves a domain number that is the end of an interval in the domain. The domain number only has one side in the domain. This automatically disqualifies the function from having a derivative value at this domain number. The domain number is automatically a critical number. However, this is still a nice situation, which we would like to describe.

We would still like to describe one-sided situations.

There can still be a tangent line, a one-sided tangent line. That tangent line can still have a slope.

So, we also have a one-sided derivative, which is like half of a derivative.

Note: It is possible that we have a domain number where there is both a left and right derivative value and they are different values. We saw this with an absolute value function. In this case, there would be no value of the derivative. If there is a derivative value, then the left and right sides must match.

No Side

If the domain number is isolated, meaning no domain numbers immediately to the left or right, then there just is no derivative there.

Isolated domain numbers are automatically critical numbers.

The term derivative implies that there is some sort of approaching.

We are just beginning to think about these types of mental exercises. We want to think about how to think about them. Calculus will officially take over this story and give all of the details.