1.1 A Preview of Calculus and 1.2 Finding Limits Graphically and Numerically

1. 1.1 A Preview of Calculus and 1.2 Finding Limits Graphically and Numerically

2. Objectives • Understand what calculus is and how it compares to precalculus. • Estimate a limit using a numerical or graphical approach. • Learn different ways that a limit can fail to exist.

3. Swimming Speed • Swimming Speed: Taking it to the Limit • Questions 1-5

4. Preview of Calculus • Diagrams on pages 43 and 44

5. Two Areas of Calculus: Differentiation • Animation of Differentiation

6. Two Areas of Calculus: Integration • Animation of Integration

7. Limits • Both branches of calculus were originally explored using limits. • Limits help define calculus.

8. 1.2 Finding Limits Graphically and Numerically

9. Find the Limit x approaches 1 from the left x approaches 1 from the right Limits are independent of single points.

10. Exploration (p. 48) From the graph, it looks like f(2) is defined. Look at the table. On the calculator: tblstart 1.8 and ∆Tbl=0.1. Look at the table again. What does f approach as x gets closer to 2 from both sides?

11. Example Look at the graph and the table.

12. Example Limits are NOT affected by single points!

13. Three Examples of Limits that Fail to Exist If the left-hand limit doesn't equal right-hand limit, the two-sided limit does not exist.

14. Three Examples of Limits that Fail to Exist If the graph approaches ∞ or -∞ from one or both sides, the limit does not exist.

15. Three Examples of Limits that Fail to Exist Look at the graph and table. As x gets close to 0, f(x) doesn't approach a number, but oscillates back and forth. If the graph has an oscillating behavior, the limit does not exist.

16. Limits that Fail to Exist • f(x) approaches a different number from the right side of c than it approaches from the left side. • f(x) increases or decreases without bound as x approaches c. • f(x) oscillates as x approaches c.

17. Homework 1.2 (page 54) #5, 7, 15-23 odd