01 Limits

The first "big idea" in calculus is the limit of a function. Incremental changes in a natural system are measured over small intervals. As the interval becomes smaller, our accuracy in modeling the system improves. The mathematical concept of a limit allows us to make the interval as small as we choose - essentially infinitesimal. The limit provides the mathematical foundation for the other two big ideas of calculus (the derivative and the integral).

You will be assessed on the following standards.

1.5 Can estimate limits from graphs and tables

1.6 Can compute limits using the limit laws and algebra

Learning Goals for Chapter 1 - Limits

Videos and Khan Academy Homework (KHW)

1.4 The Tangent and Velocity Problem

We will analyze and discuss what information the tangent to a curve gives us about a function.

The tangent line viewed as the limit of secant lines.
The concepts of average versus instantaneous velocity, described numerically, graphically, and in physical terms.
The tangent line as the line obtained by “zooming in” on a smooth function; local linearity.
Approximating the slope of the tangent line using slopes of secant lines.

Introduction to Limits

KHW: Finding Limits Numerically

1.5 The Limit of a Function

We will use calculators and Excel to help us understand the concept of a limit and to estimate limiting values for functions.

Estimating limits from graphs and tables.
Understanding asymptotes and describing asymptotic behavior in terms of limits involving infinity.
Using computing devices and limits.

Estimating Limits from Graphs

KHW: One-Sided Limits from Graphs

KHW: Two-sided Limits from Graphs

1.6 The Limit Laws

We will learn how to use the limit laws to determine exact values of limits analytically using algebra.

The algebraic computation of limits: manipulating algebraically, examining left- and right-hand limits, using the limit laws to break monstrous functions into pieces, and analyzing the pieces.
The evaluation of limits from graphical representations.
Examples where limits don’t exist (using algebraic and graphical approaches).
The computation of limits when the limit laws do not apply, and the use of direct substitution property when they do.

Finding Limits Algebraically

KHW: Two-sided Limits using Algebra

KHW: Two-sided Limits Using Advanced Algebra

1.8 Continuity

You will understand the concept of continuity, and its formal mathematical definition.

The graphical and mathematical definitions of continuity, and the basic principles.
Examples of discontinuity.
The Intermediate Value Theorem: mathematical statement, graphical examples, and applied examples.

Continuity Using Limits

KHW: Continuity

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Suggested Textbook Problems	Comments
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Suggested Textbook Problems	Comments
[u1.5 core: 1, 3, 5, 11, 13, 19, 29, 31, 39] and [u1.5 rec: 1, 3, 5, 7, 9, 11, 13, 17, 19, 29, 31, 32, 35, 39]	Problems with tabular data are common on the AP test. Carefully check notation on limit problems.
[u1.4 core: 1, 5] and [u1.4 rec: 1, 3, 5, 7]
[u1.8 core: 1, 3, 13, 15, 23, 27, 43, 51] and [u1.8 rec: 1, 3, 7, 9, 13, 15, 19, 23, 27, 33, 35, 43, 51, 53]	Continuity and IVT are always somewhere on the AP test!
[u1.6: 1, 3, 5, 11-21 odd, 63] and [u1.6 rec: 1, 7, 9, 11, 15, 17, 21, 27, 41, 51, 53]	Practice! Problem 1.6#63 is very much like an AP question.
[u2.1: 1-21 odd]	practice on concepts and calculations on Instantaneous rate of change.