1. 1. What is Calculus? The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and curvatures 2. Pre-Calculus vs Calculus The mathematics of change (Velocities and Acceleration) Look at page 206 of your text for the 4 examples given Pre-CalculusLimit ProcessCalculus What we will be studying this unit: Section 3.2: Finding Limits Numerically and Graphically Section 3.3: Finding Limits Algebraically Section 3.4: Continuity and One-Sided Limits

2. Objectives: 1.Be able to state the limit notation. 2.Be able to describe where limits are used. 3.Be able to find the limit of a function numerically. Critical Vocabulary: Limit

3. I. Limit Notation “The limit of f(x) as x approaches c is L” II. Where are Limits Used 1. Define the tangent line to a curve 2.Define the velocity of an object that moves along a straight line

4. III. Finding limits Numerically Example 1: Evaluate As x approaches 1 from the left x f(x) -2 -4 -2 0 0.9 1.80.99 1.98.999 1.998 What is y approaching from the left As x approaches 1 from the right What is y approaching from the right 4 8 3 6 2 4 1.1 2.2 1.01 2.02 1.001 2.002 1 ?

5. III. Finding limits Numerically Example 2: Evaluate As x approaches 1 from the left x f(x) -2 0 0 1.9 1.9.99 1.99.999 1.999 What is y approaching from the left As x approaches 1 from the right What is y approaching from the right 4 5 3 4 2 3 1.1 2.1 1.01 2.01 1.001 2.001 1 ?

6. III. Finding limits Numerically Example 3: Evaluate As x approaches 1 from the left x f(x) -2 0.9.99.999 What is y approaching from the left As x approaches 1 from the right What is y approaching from the right 4 1 3 1 2 1 1.1 1 1.01 1 1.001 1 1 ?

7. III. Finding limits Numerically Example 4: Evaluate As x approaches 1 from the left x f(x) -2 0 0.9.99.999 What is y approaching from the left As x approaches 1 from the right What is y approaching from the right 4 4 3 3 2 2 1.1 1.01 1.001 1 ?

8. Page 217 – 218 #1-6 all, 21-31 odd Direction Change: #21-31 Find the limit numerically (you only need to find three values on each side)

9. Objectives: 1.Be able to find the limit of a function graphically. 2.Be able to summarize the big ideas of Limits. Critical Vocabulary: Limit Warm Up: Find the limit Numerically

10. WARM UP: Evaluate x f(x) -2 0 1 0 1.9.9 1.99.99 1.999.999 5 4 4 3 3 2 2.1 1.1 2.01 1.01 2.001 1.001 2 ?

11. I. Finding limits Graphically Example 1: Evaluate x f(x) -2 0 0 1.9 1.9.99 1.99.999 1.999 4 5 3 4 2 3 1.1 2.1 1.01 2.01 1.001 2.001 1 ?

12. I. Finding limits Graphically Example 2: Evaluate x f(x) -2 0.9.99.999 4 1 3 1 2 1 1.1 1 1.01 1 1.001 1 1 ?

13. I. Finding limits Graphically Example 3: Evaluate x f(x) -2 0 0.9 -9.99.999 4 4 3 3 2 2 1.1 1.01 1.001 1 ?

14. II. Summarize the Big Ideas!!!! 1. A limit is a y-value if it exists 2.When you say lim f(x) it means we choose “x’s” very close to “c” and look at the behavior of the function. xcxc 3.For a limit to exist, you must allow “x’s” to approach “c” from both sides of “c”. If f(x) approaches a different number from the left and right, the limit does not exist. 4.We don’t care how the function is defined at “c” We do care about the behavior surrounding where x = c. (Journey) *even if x = c is undefined *even if x = c doesn’t equal the limit

15. Page 217 – 218 #7-12 all, 33, 35 Direction Change: #33, 35 Find the limit numerically and graphically (using your calculator)