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

AP Calculus BC

Chapter 2: Limits and Continuity2.3: 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 y0 is between ƒ(a) and ƒ(b), then y0 = ƒ(c) for some c in [a, b].

Assignment:

• HW 2.3: #3 – 30 (every 3rd), 36, 39, 42, 46.

• Ch. Two Test: Friday, September 25.