Prerequisites: Slope, Function Notation

Goal: To understand the definition of the derivative.


The blue curve represents the graph of a function y=f(x).
  1. What do the top and bottom sliders do?
  2. How is the yellow line generated?
  3. Can you move the sliders so that the yellow line is horizontal? Why or why not? What do you notice about the two points on the graph when the line is horizontal?
  4. Can you move the sliders so that the yellow line is vertical? Why or why not? How close to vertical can you get?
  5. What about the two points on the curve does the yellow line describe?
  6. What is the largest number of times you can make the yellow line touch the blue curve?
  7. Can you make the yellow line touch the curve at only one point? What is a line called that touches a curve at only one point?
  8. Using the lower slider, place the red point anywhere other than x=-3 or x=2 (at this time, farther away is better). Now use the upper slider to move the blue point closer to the red point. As it moves closer, whhat do you notice about the relationship of the yellow line and the blue curve near the red point?
  9. The expression at the bottom of the window gives the slope of the line between the two points on the curve. What happens to the slope line as the points get closer and closer together?
  10. How would you approximate the slope of a line tangent to the curve?
  11. How would you make that approximation better?
  12. How would you find the actual slope of the tangent line?
  13. Use limit notation to write an expression for the slope of the tangent line to the blue curve at the point x=-2. (Hint: what is the algebraic expression for the blue curve?)
  14. Use limit notation to write an expression for the slope of the tangent line to the blue curve at the point x=a.
  15. Since the "steepness" of the tangent line shows how fast f(x) is changing with respect to changes in x, the tangent line also shows the rate of change of the graph. What is the relationship between the equation for the slope of the tangent line you found above and the instantaneous rate of change?
  16. The instantaneous rate of change is called the ________?
  17. Write the definition of the derivative using the slope of the tangent line.
  18. Where on the graph (there are two points) is the tangent line horizontal? What is the derivative at each point? What do you notice about the shape of the graph at each point?
  19. Place the red point at x=-3.
    1. What happens to the slope of the yellow line when the blue point approaches x=-3 from the left?
    2. What happens to the slope of the yellow line when the blue point approaches x=-3 from the right?
    3. What can you say about the derivative at a corner? Why?
  20. Place the red point at x=2.
    1. What happens to the slope of the yellow line when the blue point approaches x=2 from the left?
    2. What happens to the slope of the yellow line when the blue point approaches x=2 from the right?
    3. What can you say about the derivative at a discontinuity? Why?