2. Calculus Section 1.1 A Preview of Calculus

3. What is Calculus? Calculus is the mathematics of change Two classic types of problems: The Tangent Line Problem Area Under A Curve Problem

4. The Tangent Line Problem Secant line - a line that intersects the graph at two points (easy to find the slope given two points) Tangent line - a line that intersects the graph at one point Using the understanding of limits, one can limit the distance between two points to achieve a tangent line, thus finding the slope of a curve These are Derivatives See fig. 1.1 & fig. 1.2 on page 45 of textbook

5. The Area Problem With the understanding of limits, one can limit the width of rectangles to increase the number of rectangles to more accurately find the area between a curve and the x-axis These are Integrals See fig. 1.3 & fig. 1.4 on page 46 of textbook

6. Calculus Section 1.2 Finding Limits Graphically and Numerically

7. An Introduction to Limits Notation: Read as “the limit as x approaches c of f(x) is L” If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit is L.

8. Evaluating Limits Numerically Example: What is the domain of the function? Domain: (-∞, 2) (2, ∞) So, what happens at 2? Set up a table x 1.51.91.99922.0012.012.5 f(x) x 1.51.91.99922.0012.012.5 f(x)

9. Graphing Calculator: Table Type function into y= Go to 2nd TBLSET, Ind - Ask Go to TABLE Type in values for the independent variable

10. Evaluating Limits Numerically x 1.51.91.99922.0012.012.5 f(x) 3.53.93.999?4.0014.014.5 So, as x approaches 2 from either side, the function value approaches 4 (CONVERGE) We say “the limit as x approaches 2 is 4”

11. Limits That Fail to Exist Common types of behavior associated with nonexistence of a limit f(x) approaches a different value from the left and from the right (they do not CONVERGE ) f(x) increases or decreases without bound as x approaches c (asymptotes) f(x) oscillates between two fixed values as x approaches c

12. Limits and Their Properties Section 1.3 Evaluating Limits Analytically

13. Properties of Limits Theorem 1.1 Some Basic Limits

14. Properties of Limits Theorem 1.2 Properties of Limits

15. Properties of Limits Theorem 1.3 If p(x) is a polynomial function and c is a real number, then In r(x) is a rational function, given by r(x)=p(x)/q(x) and c is a real number such that q(c)≠0, then

16. In Other Words… DIRECT SUBSTITUTE if the function is algebraic (polynomial, rational, radical, etc.) and the function is continuous at c (meaning the function is defined to the left AND to the right of the value c)

17. Cannot DIRECT SUBSTITUTE when… the function is not defined at c, or c is not part of the domain. Therefore, ask: “Is c part of my domain?” AND “Is the function defined to the left AND to the right of c?”

18. Strategies for Finding Limits If f(x) is not defined at c, then find a new function, g(x), that agrees with the original function for all values except at c Two techniques: Dividing Out Technique Rationalization Technique

19. Dividing Out Technique Factor both numerator and denominator Cancel like factors Ask, “Is c now part of my domain?” This is your new function, g(x) DIRECT SUBSTITUTE to find the limit

20. Example 1 Find: Factor: Cancel:

21. Example 1 cont. So, the “new” function is Now, take the limit of the “new” function

22. Example 1 cont. Therefore, we conclude that:

23. Rationalization Technique For functions with radicals Multiply both numerator and denominator by the conjugate Cancel like factors Keep denominator factored (DO NOT MULTIPLY OUT)

24. Example 2 Find: Rationalize:

25. Example 2 cont. Now, take the limit of

26. Example 2 cont. Therefore, we conclude that: