1. Chapter 1 Limits and Their Properties Unit Outcomes – At the end of this unit you will be able to: Understand what calculus is and how it differs from precalculus Understand that the tangent line and area problems are basic to calculus Estimate a limit numerically and graphically Determine when a limit does not exist

2. Learn and use a formal definition of a limit Use properties of limits to evaluate limits Develop and use a strategy for finding limits Evaluate limits by “dividing out” and “rationalizing” Evaluate a limit using the “Squeeze Theorem” Determine continuity at a point and continuity on an open interval Determine one-sided limits and continuity on a closed interval

3.  Use properties of continuity  Know and use the Intermediate Value Theorem  Determine infinite limits from the left and the right  Find and sketch the vertical asymptotes of the graph of a function

4. 1.1 A Preview of Calculus What is Calculus????????????????????

5. Calculus is a branch of mathematics that deals with rates of change like velocity and acceleration. What we know of Calculus today, began in the 17 th century with Newton and Leibnitz Calculus deals primarily with limits, derivatives and integrals How does Calculus differ from Precalculus? Precalculus is static while Calculus is dynamic See the chart on page 43

6. What is the purpose of Calculus?  Finding the slope of curves  Calculating the area of bizarre shapes  Justifying old formulas  Calculating complicated x-intercepts  Visualizing graphs  Finding the average value of a function  Calculating optimal values (optimization)

7. On the straight incline the slope remains the same, therefore the force that must be used to push it up the hill remains static. On the curved incline, however, the slope does not remain the same, so the force changes and therefore is dynamic

8. Other examples of regular math vs calculus

9. Examples taken from What is Calculus http://media.wiley.com/product_data /

10. Two problems are basic to the study of calculus: The tangent line problem and the area problem The Tangent Line Problem The graph of a linear equation has a constant slope, but the graph of a quadratic equation does not. So, to find the slope of the curve at a certain point, we find the slope of the tangent line at that point

12. We begin by drawing a secant line, then bring the point of intersection closer and closer to the point of tangency. This helps us to get a good approximation of the slope of the tangent line

13. How will we determine the slope of the tangent line?

14. So, as Δx gets smaller the slope gets smaller and best approximates the slope of the tangent line There is a limit to how small the slope can be.

15. The Area Problem

17. As you increase the number of rectangles, the area is a better approximation of the area under the curve

18. In AP Calculus, we will be approaching problems in three different ways:  Analytically (using the equation)  Numerically  Graphically

19. Finding Limits Graphically and Numerically

20. Finding Limits Graphically The informal definition of a limit is “what is happening to y as x gets close to a certain number”

21. If we are concerned with the limit of f(x) as we approach some value c from the left hand side, we write

22. If we are concerned with the limit of f(x) as we approach some value c from the right hand side, we write

23. In order for a limit to exist at c = and we write:

24. When limits fail to exist 1.When the right hand and left hand limits do not agree 2.When there is unbounded behavior (as we have just seen) 3. When there is oscillating behavior

25. Homework p.54 – 57: 3 – 5 odd, 9 – 19, 43, 44, 49, 52.