# Exploration of Secant and Tangent Lines to a Curve

- Author:
- Terri Rehner

- Topic:
- Calculus, Straight Lines

Worksheet
Secant / Tangent Activity
Part I
1. What is the average rate of change from A to B?
2. What is the instantaneous rate of change at point A?
3. As point B moves toward point A, describe 3 things that happen.
Move point A in order to answer the next two questions
4. When was the instantaneous rate of change the greatest?
5. When will the instantaneous rate of change be negative? Positive? 0?
Part II
Move point B so x = 2 and point A so x =1. At the bottom of the screen is an input line. Type (type abs for absolute value).
6. What is the average rate of change from A to B?
7. What is the instantaneous rate of change at point A?
8. As point B moves toward point A, describe what happens and explain why this is so.
9. Move point B back so x = 2 and point A so x = 3. Repeat question 8.
10. Move point B anywhere on either side of the vertex and point A so x = 2. What is the instantaneous rate of change? Why is this so?
11. How many tangent lines could you draw at the point (2, 4)?
12. When is there no tangent line?
Part III
13. Type in the input line: . Move point A. Is there any place where there is no tangent line defined?
14. Type in the input line: . Move point A. Is there any place where there is no tangent line defined?
15. Generalize what you have found out about the existence of tangent lines.