Calculus 1.1-1.3

1. Calculus1.1-1.3 Ms. Hernandez

2. Calculus • Syllabus & Assignments Hard Copy

3. Tangent and Derivative Pblm • See Handout  Calculus Concepts and Contexts, Stewart 3rd ed, 2.1 p92-98 • #1,3,5 QZ 1 due Wed 8-16 • GSP WS

4. Limits • Limiting property is why they are called limits • b/c a function f(x) gets really close to some value – its INDTENDED value • Yet, sometimes may not really get there • Like vertical or horizontal asymptotes y = log(x) y = ex • Limits are not limited to asymptotes

5. Limits • Occur at points of discontinuity • Pg 48, 49

6. Formal Def of a Limit • As function f(x) gets REALLY close to a number, we call this number L (or Bob) doesn’t matter what we call it • The value the function gets really close to (techie term is arbitrarily close) • Is the limit of the function f(x)

7. Formal Def of a Limit • Need a place to go to (anchor – reference) • Let c be that place to go to on your function • So then as your function f(x) gets closer to c (somewhere on your function) from the left and from the right of c • Then f(x) approaches a value L • L stands for Limit

8. Def of limit works for all lim • Not all limits are created equal • Some are nicer than others y=f(x) and in nice f(x) then y=L 1.2 # 9,10, 16 piecewise f(x) 1.2 #11, 12 • So the def has to include the asymptotes • So that is works for everyone

9. Limits can fail • Different • Agree to disagree p50 • Unbounded • Can be confused w/infinite limits p50 • Oscillating • f(x) on crack p51 GSP WS

10. Evaluating Limits • Properties of limits p57 • Basic • Scalar multiple • Sum or Difference • Product • Quotient • Power

11. Theorem 1.1 Some Basic Limits • Let b and c be real numbers and let n be a positive integer

12. Theorem 1.2 Properties of Limits • Let b and c be real numbers and let n be a positive integer, AND let f and g be functions with the following limits.

13. Thm 1.2 cont’d

14. Direct Sub • Direct Substitution property is valid for ALL polynomial & rational functions with nonzero denominators • NO ZERO IN THE DENOMINATOR!!! • Thm 1.3

15. Thm 1.4 f(x) with a radical As long as n is positive integer. If n is odd then it works for all values of c. If n is even, then it only works when c > 0.

16. 1.5 Limit of Composite f(x) • If f and g are functions with the following limits this means that you got a function g(x) as x cwith a limit L and then you take another function f(x) as xL then f(x) limit is really just f(L).

17. Example 4 pg.59 • 4 min discuss in groups • In (a) which is f(x) and which is g(x) • In (b) which is f(x) and which is g(x) • Does it really matter?

18. Thm 1.6 Trig f(x) • Six basic trig limits on pg 59 • Memorize them all • Practice using them all • And oh yeah, get to work on memorizing the trig identities in the back of the book • All but co-function identities

19. Evaluating Limits • Polynomial Limits • Just break it up! • p58 • Polynomial & Rational functions • Polynomial & Rational Functions • Ok as long as denominator is NOT zero!

20. Evaluating Limits • Substitution • Divide out (factor out) • Ex 7 pg 61 • Discuss 4 min. • What is an indeterminate form? • How can technology trip you up?

21. Evaluating Limits • Rationale • Conjugate • Ex 8 pg 62 • Discuss 4 min • How do you get the conjugate? • How do you multiply by the conjugate? • Simplifying, what do you need to watch out for?