Download AP Calculus BC - 4J Blog Server

AP Calculus BC
Chapter 2: Limits and Continuity 2.2: Limits Involving Infinity Learning Targets: • Write and verify end behavior models for various functions. • Calculate limits as x —> ±∞. • Identify vertical and horizontal asymptotes. Learning Objectives: •  Big Idea 1: Limits •  Enduring Understanding 1.1: Students will understand that the concept of a limit can be used to understand the behavior of functions. •  Learning Objective 1.1A (a): Students will be able to express limits symbolically using correct notation. •  LO 1.1A (b): Students will be able to interpret limits expressed symbolically. Essential Knowledge: •  EK 1.1A1: Students will know that, given a function ƒ, the limit of ƒ(x) as x approaches c is a real number R if ƒ(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is lim f ( x ) = R
x→c
•  EK 1.1A2: The concept of a limit can be extended to include one-­‐sided limits, limits at infinity, and infinite limits. Essential Knowledge: •  EK 1.1A3: A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right. Learning Objectives: •  Big Idea 1: Limits •  Enduring Understanding 1.1: Students will understand that the concept of a limit can be used to understand the behavior of functions. •  Learning Objective 1.1B: Students will be able to estimate limits of functions. •  EK 1.1B1: Students will know that numerical and graphical information can be used to estimate limits. Learning Objectives: •  Big Idea 1: Limits •  Enduring Understanding 1.1: Students will understand that the concept of a limit can be used to understand the behavior of functions. •  LO 1.1C: Students will be able to determine limits of functions. Essential Knowledge: •  EK 1.1c1: Limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules. •  EK 1.1C2: The limit of a function may be found by using algebraic manipulation … or the squeeze theorem. Learning Objectives: •  Big Idea 1: Limits •  Enduring Understanding 1.1: Students will understand that the concept of a limit can be used to understand the behavior of functions. •  LO 1.1D: Students will be able to deduce and interpret behavior of functions using limits. •  EK 1.1D1: Asymptotic and unbounded behavior of functions can be explained and described using limits. Mathematical Practices for AP Calculus •  MPAC 6: Communicating. •  Students can use accurate and precise language and notation. Quote for today: • “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” • Paul Erdös Horizontal Asymptote: • The line y = b is a horizontal asymptote of the graph of a function y = ƒ(x) if either lim f ( x ) = b
x→∞
or lim f ( x ) = b
x→−∞
Properties of Limits: •  Sum Rule •  Difference Rule •  Product Rule •  Constant Multiple Rule •  Quotient Rule •  Power Rule Vertical Asymptote: •  The line x = a is a vertical asymptote of the graph of a function y = ƒ(x) if either lim+ f ( x ) = ±∞
x→a
or lim− f ( x ) = ±∞
x→a
End Behavior Model: •  The function g is •  a right-­‐end behavior model for a function ƒ iff f ( x)
lim
=1
x→∞ g x
( )
•  a left-­‐end behavior model for a function ƒ iff f ( x)
lim
=1
x→−∞ g x
( )
•  If one function provides both a left-­‐ and a right-­‐end behavior model, it is called an end behavior model. Assignments: • CW/HW 2.2A: Exploration 1 and 2.2 Quick Review. • CW/HW 2.2B: #3, 12, 18, 24, 30, 39, 48, 54, 57, 59.