AP CALCULUS

1. 1001 - Limits 1: Local Behavior AP CALCULUS

2. You have 5 minutes to read a paragraph out of the provided magazine and write a thesis statement regarding what you read

3. Activity: Teacher-Directed Instruction

4. Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words

5. REVIEW: ALGEBRA is a _________________ machine that ___________________ a function ___________ a point. function evaluates CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point Limit Describes the behavior of near

6. Limits Review: PART 1: LOCAL BEHAVIOR (1).General Idea: Behavior of a function very near the point where (2). Layman’s Description of Limit(Local Behavior) (3). Notation (4). Mantra L a xayL

7. G N A W 0 Graphically What is the y value? 3 What is the y value? “We Don’t Care” Postulate”: The existence or non-existence of f(a) has no bearing on the

8. G N A W Numerically 2.0001 1.9 2.001 1.999 2 2.01 x 1.99 error 40.561 39.914 40.204 40.239 y 40.268 37.165 40.2

9. Activity: Teacher-Directed Instruction

10. Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words

11. G – Graphically N – Numerically A – Analytically W -- Words

12. The Formal Definition The function has a limit as x approaches a if, given any positive number ε, there is a positive number δ such that for all x, 0< <δ ε Layman’s definition of a limit As x approaches a from both sides (but x≠a) If f(x) approaches a single # L then L is the limit

13. FINDING LIMITS

14. G N A W • Numerically Must write every time .99834 .999 .99999 .9834 .99999 .9999 • Words Mantra: xa, yL Verify these also:

15. (6).FINDING LIMITS • Graphically “We Don’t Care” Postulate….. • The existence or non-existence of f(x) at x = 2 has no bearing on the limit as

16. FINDING LIMITS • Analytically • “a” in the Domain • Use _______________________________ Direct substitution 13 • “a” not in the Domain • This produces ______ called the _____________________ Indeterminate form Rem: Always start with Direct Substitution

17. Method 1: Algebraic - Factorization Rem: Always start with Direct Substitution Creates a hole so you either factor or rationalize Method 2: Algebraic - Rationalization Method 3: Numeric – Chart (last resort!) Method 4: Calculus To be Learned Later !

18. Do All Functions have Limits? Why? Where LIMITS fail to exist. f(x) approaches two different numbers Approaches ∞ Oscillates At an endpoint not coming from both sides

20. Review : • 1) Write the Layman’s description of a Limit. • 2) Write the formal definition. ( equation part) • 3) Find each limit. • 4) Does f(x) reach L at either point in #3?

21. Properties of Limits • Using Direct Substitution • BASIC (kis a constant.xis a variable) • 1) • 2) • 3) • 4) IMPORTANT: Goes BOTH ways!

22. Properties of Limits: cont. OPERATIONS Take the limits of each part and then perform the operations. EX: POLYNOMIAL, RADICAL, and RATIONAL FUNCTIONS all us Direct Substitution as long as a is in the domain

23. Composite Functions REM: Notation THEOREM: and Use Direct Substitution. EX: EX:

24. Limits of TRIG Functions Squeeze Theorem: if f(x) ≤ g(x) ≤ h(x) for x in the interval about a, except possibly at a and the Then exists and also equals L h g f a This theorem allow us to use DIRECT SUBSTIUTION withTrig Functions.