Welcome to Calculus BC!

1. Welcome to Calculus BC! • Books • Syllabus and website

2. Advise from last year’s crew: • Do the homework (even if it’s not graded!) • Check the blog • Buy an extra prep book • Keep a 3 ring binder to stay organized • Stick with it • Stay for extra help • Bring a calculator every day • Calculator sections are not a joke – prepare for them! • Let’s get started!

3. Limits (1.2 and 1.3) What happens to as x approaches 1?

4. Limit • A limit is a y-value that the graph of a function approaches as x gets closer and closer to a particular value from either direction • Limits can be used more generally to describe the behavior of a function near a particular x-value • Notation:

5. Does a limit necessarily exist? No. • The function might approach a different y-value from one side verses the other. • If there is a vertical asymptote at x=c, then the function is going to ±∞

6. Assuming the limit does exist, how can you find it? • Try direct substitution (review your unit circle!) • If that results in the indeterminate form , then the limit still exists and you can find it by… * Using the table from your calculator * Factoring out a common term. Example:

7. The Big Ideas • When we talk about limits, we don’t necessarily care what’s happened to f and x=c, but right around x=c. • Limits are the foundation for everything we talk about in calculus.

8. One-Sided Limits refers to the limit from both sides of the function around c refers to the limit as x approaches c from the left. refers to the limit as x approaches c from the right.

11. How can we use one-sided limits to determine whether the limit of a piecewise function as x approaches a “change –over” value occurs?

12. Special Limits Just know them….

13. Continuity (1.4) There is a strong connection between limits and continuity A function f(x)is continuous at x=c if… 1. must exist 2. must exist 3. A function is continuous if you can draw its graph without picking up your pencil!

14. Which functions are always continuous? • Polynomial (constant, linear, quadratic, cubic..) • Sine, cosine • Absolute value • Radical functions of odd degrees • Exponential

15. Which functions are continuous, but have a restricted domain? • Radical functions of even degrees • Logarithmic functions • Rational functions

16. Which functions are not always continuous? • Tangent, secant, cosecant, cotangent • Piecewise *

17. Investigating Continuity

18. Example Find the value of a and bso that the following function is continuous at x= 1

19. Vertical Asymptotes (1.5) If you plug c into f(x) and get… f has a removable discontinuity at x=c (hole) and exists. f has vertical asymptote at x=c and DNE. (nonremovable discontinuity)

20. Big Idea Even though limits around vertical asymptotes don’t exist, we can use them to describe the behavior of the graph on either side.

21. Examples on notes

22. Special Function . • This function has a jump discontinuity at x =0. • What are the various limits? • Look at another example:

23. Horizontal Asymptotes (3.5)

24. Definition of H.A. The graph of a function f(x) has a horizontal asymptote at y = a if, as x gets infinitely large in either or both directions, the graph gets closer and closer to the line y = a

25. Big Idea • We can use limits to describe end behavior • Not all functions have a finite end behavior

26. What would be an example of a function whose graph had no horizontal asymptote?

27. What would be an example of a function whose graph has one horizontal asymptote?

28. What would be an example of a function whose graph has two horizontal asymptotes?

29. Note The graph of a function can still cross over the line y = k and still Example:

30. Summary If the higher degree polynomial is…. • In the denominator, then and the H.A. is along the x – axis • In the numerator, then D.N.E. (no H.A.) • A tie between the numerator and denominator, then the H.A. can be found in the leading coefficients.

31. Examples 1

32. Examples 2

33. Slowest to fastest: • Logarithmic • Polynomials (grow fastest by degree) • Exponentials • Factorials