**Unit 2.**

Limits and Continuity.

Limits and Continuity.

Storytelling with graphs.

Examples of Limits.

How fast is the

car traveling?

Strategy for finding

instantaneous speed.

Use a Limit!!!

**Essential Questions:**

How might things look

differently if I stood

impossibly close?

What would happen

if this pattern goes

to infinity?

How might things look

differently if I stood

impossibly close?

What would happen

if this pattern goes

to infinity?

Day 1:

Exploring Limits.

Day 2: Being Precise about Limits.

What happens to f(x) as x approaches a?

Ways to answer this question...

What exactly are we asking?

When is this a 'real' question?

Is it always answerable?

What information will I have to work with?

Lecture 2.1a.Limit Properties

Objectives:

1)

Use algebra to find and prove the limit of polynomial and rational functions.

2)

Use a graph to find and prove a limit.

Formal Definition Of a Limit

Two Basic Limits

Theorem 1. Limit Properties

Premise.... ? (check)

Conclusion....? (use)

Theorem 2: Limits of Polynomial

and Rational Functions.

Premise...?

Conclusion...?

When Algebraic methods fail...

Theorem 3. One Sided Limits.

Premise..?

Conclusion...?

Lecture 2.1b Trickier Limits

Objective:

Use algebraic tricks to solve elusive limits:

- Product Rule

- Conjugates

- U-Substitution

- Squeeze Theorem

Lecture 2.2a Finite Limits at Infinity.

Objective:

I can estimate, explain, and prove limits at infinity.

Squeeze Theorem Revisited.

Lecture 2.2b. Vertical Asymptotes.

Objective:

I can evaluate infinite limits and prove vertical asymptotes.

Lecture 2.2c End Behavior.

Objectives:

1)

Prove that a function is an end behavior model.

2)

Use end behavior to solve limits at infinity and vice versa.

3)

Discover and apply the hierarchy of end behaviors.

Lecture 2.3a. Introduction to Continuity

Objectives:

1)

I can determine and prove continuity at a point, on an interval and as a function.

2)

I can categorize types of discontinuity.

3)

I can extend functions to remove discontinuities.

Lecture 2.3b Continuous Functions.

Objectives:

1)

Identify continuous functions from their graphs and equations.

2)

Apply the Intermediate Value Theorem.

Lecture 2.4a. Average and Instantaneous Change.

Objective:

Use secant lines to calculate average change and estimate instantaneous change.

Lecture 2.4b Slope of a Tangent

Objective:

Calculate the precise slope of a tangent line or normal line (and write their equations)

.