1. 2.4 Rates of Change and Tangent Lines
Objectives
Students will be able to:
Directly apply the definition of the slope of a curve in order to calculate slopes
Find the equations of the tangent line and normal line to a curve at a given point
Find the average rate of change of a function

2. Average Rates of Change
The average rate of change of a function over an interval is the amount of change (y) divided by the length of the interval (x).

4. A line through two points on a curve is a secant to the curve.
The average rate of change of a function is the slope of the secant line through the function.

5. Tangent to a Curve
To find the rate of change at one particular point on a curve, we find the slope of the tangent line to the curve at that point. This represents the rate at which the value of the function is changing with respect to x at any particular value x=a.
The problem we run into is that we are trying to find the slope of a line only being given one point on that line. How do we do that? We have always needed two points on a line to calculate its’ slope.

6. What we do is start off by finding the average change between an x value “a” and another x value “h” units away from “a.” This is essentially the equivalent of finding the slope of the secant line between the ordered pairs:
Then we find the limit of the slope as the second point approaches the first point (as h approaches 0).

7. In short, the slope of a curve y=f(x) at the point P(a,f(a)) is the number:
This is provided that the limit exists.
The tangent line to the curve at point P is the line through P with this slope.

8. It is important to realize that these are all the same:
The slope of y=f(x) at x=a
The slope of the tangent to y=f(x) at x=a
The (instantaneous) rate of change of f(x) with respect to x at x=a

11. Here is what we now know:

15. Now we have the slope of the tangent line
Now we have the slope of the tangent line. We still need to find the equation of the tangent line. Let’s now find the point at which the curve intersects the tangent line, or in other terms, find f(2).

16. Finally, write the equation of the line passing through (2,6) with a slope of 11

18. Normal to a Curve
A normal line to a curve at a point is the line perpendicular to the tangent at that point.
Remember that perpendicular lines have opposite sign, reciprocal slopes.

19. Function: blue
Tangent: red
Normal: purple