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AP CALCULUS

1002 - Limits 1: Local Behavior. AP CALCULUS. REVIEW:. ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point. Limits Review:

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AP CALCULUS

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  1. 1002 - Limits 1: Local Behavior AP CALCULUS

  2. REVIEW: ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point

  3. 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) L a (3). Notation (4). Mantra

  4. G N A W Graphically “We Don’t Care” Postulate”:

  5. G N A W Numerically

  6. The Formal Definition

  7.  (5). Formal Definition ( Equation Part) Graphically: Find a If 3 2 1 1 2 3 4

  8. Analytically Find a if given and for ------------------------------------------

  9. Find a for any

  10. Day 2

  11. FINDING LIMITS

  12. G N A W • Numerically • Words Mantra: Verify these also:

  13. (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

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

  15. Method 1: Algebraic - Factorization Rem: Always start with Direct Substitution Method 2: Algebraic - Rationalization Method 3: Numeric – Chart (last resort!) Method 4: Calculus To be Learned Later !

  16. Do All Functions have Limits? Why? Where LIMITS fail to exist.

  17. 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?

  18. Homework Problems • From the figure, • determine a • such that

  19. Review: (5). The graph of the function displays the graph of a function with Estimate how close x must be to 2 in order to insure that f(x) is within 0.5 of 4. (6). Find a such that

  20. Last Update: • 08/12/10

  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.

  25. Limits of TRIG Functions:cont. In a UNIT CIRCLE measured in RADIANS: Defn. of radians! THEREFORE:

  26. Exponential and Logarithmic Limits Use DIRECT SUBSTITUTION. REM: the Domain of the functions For a > 0 REM: Special Exponential Limit

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