AP Calculus BC Chapter 2: Limits and Continuity 2.3: Continuity

Learning Targets: Students will identify the intervals upon which a given function is continuous. Students will remove removable discontinuities by extending or modifying a function. Students will apply the Intermediate Value Theorem and the properties of algebraic combinations and composites of continuous functions.

Learning Objective: Big Idea 1: Limits Enduring Understanding 1.2: Students will understand that continuity is a key property of functions that is defined using limits. Learning Objective 1.2A: Students will be able to analyze functions for intervals of continuity or points of discontinuity. Essential Knowledge 1.2A1: Students will know that a function ƒ is continuous at x = c provided that ƒ(c) exists,

Learning Objective: Big Idea 1: Limits EU 1.2: Continuity is a key property of functions that is defined using limits. LO 1.2A: Analyze functions for intervals of continuity or points of discontinuity. EK 1.2A2: Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains. EK 1.2A3: Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

Learning Objective: Big Idea 1: Limits Enduring Understanding 1.2: Students will understand that continuity is a key property of functions that is defined using limits. Learning Objective 1.2B: Students will be able to determine the applicability of important calculus theorems using continuity. Essential Knowledge 1.2B1: Students will know that continuity is an essential condition for theorems such as the Intermediate Value Theorem.

Mathematical Practices for AP Calculus MPAC 2: Connecting concepts. Students can relate the concept of a limit to all aspects of calculus.

Quote for today: “One must learn by doing the thing; though you think you know it, you have no certainty until you try it.” Sophocles (496 BCE – 406 BCE )

Continuity at a Point: Interior Point: A function y = ƒ(x) is continuous at an interior point c of its domain if Endpoint: A function y = ƒ(x) is continuous at a left endpoint a or at a right endpoint b of its domain if

Discontinuities: If a function is not continuous at a point c, we say that ƒ is discontinuous at c and c is a point of discontinuity of ƒ. Note that c need not be in the domain of ƒ. Types of discontinuities include removable, jump, infinite, and oscillating. A continuous extension of a function removes a removable discontinuity.

Continuous Functions: A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval. E.g., y = 1/x is not continuous on [-1, 1].

Theorem – Properties of Continuous Functions: If the functions ƒ and g are continuous at x = c, then the following combinations are continuous at x = c: sums ƒ + g, differences ƒ – g, products ƒ·g, constant multiples kƒ, and quotients ƒ/g, provided g(c) ≠ 0.

Theorem – Composite of Continuous Functions: If ƒ is continuous at c and g is continuous at ƒ(c), then the composite function is continuous at x = c.

The Intermediate Value Theorem for Continuous Functions: A function y = ƒ(x) that is continuous on a closed interval [a, b] takes on every value between ƒ(a) and ƒ(b). In other words, if y 0 is between ƒ(a) and ƒ(b), then y 0 = ƒ(c) for some c in [a, b].

Assignment: HW 2.3: #3 – 30 (every 3 rd ), 36, 39, 42, 46. Ch. Two Test: Friday, September 25.