1. Lesson 11.3 The Tangent Line Problem
Essential Question: How do you find the slope of a graph at any single point?

2. Tangent Line to a Graph
To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point. In simple terms, the tangent line to the graph of a function f at a point P (x1, y1) is the line that best approximates the slope of the graph at the point. Figure shows other examples of tangent lines.
Figure 11.19

3. Tangent Line to a Graph
From geometry, you know that a line is tangent to a circle when the line intersects the circle at only one point (see Figure 11.20). Tangent lines to noncircular graphs, however, can intersect the graph at more than one point.
For instance, in the first graph in Figure 11.19, if the tangent line were extended, then it would intersect the graph at a point other than the point of tangency.
Figure 11.20

4. The Tangent Line Problem
Given a function and a point on a graph, you will find the equation of the tangent line to a point.

5. The slope of the tangent line is said to be the limit of the slope of the secant line.
Because a tangent line approximates the slope of a graph at a point, the problem of finding the slope of a graph at a point is the same as finding the slope of the tangent line at the point.

6. Use the graph to approximate the slope of the graph of 𝑓 (𝑥)= 𝑥 2 at the point (1, 1).

7. Use the graph to approximate the slope of the graph at the point 𝑥,𝑦 .

8. Use the graph to approximate the slope of the graph at the point 𝑥,𝑦 .

9. Use the graph to approximate the slope of the graph at the point 𝑥,𝑦 .

10. Use the graph to approximate the slope of the graph at the point 𝑥,𝑦 .

11. The graph depicts the monthly normal temperatures (in degrees Fahrenheit) for Dallas, Texas. Approximate the slope of this graph at the indicated point and give an interpretation of the result.

12. Slope and the Limit Process
A more systematic method of approximating tangent lines makes use of a secant line through the point of tangency and a second point on the graph.

13. Slope and the Limit Process
If (𝑥, 𝑓 (𝑥)) is the point of tangency and
(𝑥 + ℎ, 𝑓 (𝑥 + ℎ)) is a second point on the graph of 𝑓, then the slope of the secant line through the two points is given by
𝑚 𝑠𝑒𝑐 = 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ
The right side of this equation is called the difference quotient. The denominator h is the change in x, and the numerator is the change in y.

14. Slope and the Limit Process
The beauty of this procedure is that you obtain more and more accurate approximations of the slope of the tangent line by choosing points closer and closer to the point of tangency.

15. Definition of the Slope of a Graph
The slope m of the graph of f at the point 𝑥,𝑓 𝑥 is equal to the slope of its tangent line at 𝑥,𝑓 𝑥 and is given by lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ provided the limit exists.

16. Find the slope of the graph of 𝑓 𝑥 = 𝑥 2 at the point −2,4

17. Find the slope of the graph of 𝑓 𝑥 = 𝑥 3 at the point 2,8

18. Find the slope of the graph of 𝑓 𝑥 =−2𝑥+4

19. Find the slope of the graph of 𝑓 𝑥 =−3𝑥+5

20. Find a formula for the slope of the graph of 𝑓 𝑥 = 𝑥 2 +1 What are the slopes at the points −1,2 and 2,5 ?

21. Find a formula for the slope of the graph of 𝑓 𝑥 = 𝑥 2 −2 What are the slopes at the points −3,7 and 1,−1 ?

22. The Tangent Line Problem
Finding the tangent line at point P comes down to finding the slope of the tangent line at point P.
To find the slope of the tangent line you have to find the derivative.

23. What is a Derivative? The derivative is…
The slope of the tangent line
A rate of change
Derivatives can be found using the limit process, analytically and implicitly.

24. Definition of the Derivative
The derivative of f at x is given by 𝑓 ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ provided the limit exists.

25. Find the derivative of 𝑓 𝑥 =3 𝑥 2 −2𝑥

26. Find the derivative of 𝑓 𝑥 =4 𝑥 2 −5𝑥

27. Find the derivative of 𝑓 𝑥 = 1 𝑥

28. Find the derivative of 𝑓 𝑥 = 𝑥 Then find the slope of the graph at the points 1,1 and 4,2

29. How do you find the slope of a graph at any single point?

30. Ticket Out the Door
Find the derivative of 𝑓 𝑥 = 𝑥 2 +𝑥−2