TL2 Slope of a Curve

Core Concepts¶

Tangent Lines and Circles¶

We begin by considering the case of the tangent line at a point $P$ on a circle.

1. The tangent line just barely touches the circle at this one point.

2. Near point $P$, the circle and the tangent line look very similar.

We then define the slope of the circle at point $P$ to be the slope of the tangent line at $P$. We can then extend this definition to general curves $y=f(x)$.

Graphical Estimates¶

How do we use the graph of a function to estimate its slope at particular points?

1. Find the particular point on the graph.

2. Draw a little tangent line at the point, so that the line and the curve are both pointing in the same direction.

3. Extend the tangent line in both directions.

4. Use two points on the tangent line to calculate its slope.

Slope Formula¶

It would be helpful if there was a nice, easy to use formula that told us the slope of the curve at any $x$-value, rather than us having to draw a new tangent line at each point.

As we are going to see in the next section, such a formula does exist, but it depends on the formula of the original function. As an example:

• Original function: $f(x)=x^2$

• Curve-slope formula: $2x$

Important Equations¶

There are no important equations to know for this section.

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.