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Math 180

Math 180. 2 .1 – Rates of Change and Tangents to Curves. Ex 1. Suppose you drop a calculus book from the top of building 61. What is the average speed of the book during the first 2 seconds of free fall?

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Math 180

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  1. Math 180 2.1 – Rates of Change and Tangents to Curves

  2. Ex 1. Suppose you drop a calculus book from the top of building 61. What is the average speed of the book during the first 2 seconds of free fall? What is the average speed of the book during the 1-second interval between second 1 and second 2?

  3. Ex 1. Suppose you drop a calculus book from the top of building 61. What is the average speed of the book during the first 2 seconds of free fall? What is the average speed of the book during the 1-second interval between second 1 and second 2?

  4. Ex 1 (continued). What is the speed of the book at second? (This is called the instantaneous speed at .) Instantaneous speed: _____________

  5. Ex 1 (continued). What is the speed of the book at second? (This is called the instantaneous speed at .) Instantaneous speed: _____________

  6. Ex 1 (continued). What is the speed of the book at second? (This is called the instantaneous speed at .) Instantaneous speed: _____________

  7. Ex 1 (continued). What is the speed of the book at second? (This is called the instantaneous speed at .) Instantaneous speed: _____________

  8. Ex 1 (continued). What is the speed of the book at second? (This is called the instantaneous speed at .) Instantaneous speed: _____________

  9. Ex 1 (continued). We can graph the distance the book has traveled with respect to time.

  10. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Graphically, an average speed is the ______of a ___________.

  11. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Graphically, an average speed is the ______of a ___________.

  12. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Graphically, an average speed is the ______of a ___________.

  13. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Graphically, an average speed is the ______of a ___________. slope

  14. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Graphically, an average speed is the ______of a ___________. slope secant line

  15. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Instantaneous speed is the ______of a ___________.

  16. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Instantaneous speed is the ______of a ___________. slope

  17. Ex 1 (continued). We can graph the distance the book has traveled with respect to time. Instantaneous speed is the ______of a ___________. slope tangent line

  18. In general, the ____________________of with respect to over the interval is:

  19. In general, the ____________________of with respect to over the interval is: average rate of change

  20. Ex 2. Find the average rate of change of over the interval .

  21. Ex 3. Find the slope of the curve at the point .

  22. Ex 3. Find the slope of the curve at the point .

  23. Ex 3. Find the slope of the curve at the point .

  24. Ex 3. Find the slope of the curve at the point .

  25. Ex 3. Find the slope of the curve at the point .

  26. Ex 3 (continued). Now find an equation of the tangent line at .

  27. Ex 3 (continued). Now find an equation of the tangent line at .

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