Chapter 1Limits and Their Properties

Unit Outcomes – At the end of this unit you will be able to:

•Understand what calculus is and how it differs from precalculus

•Understand that the tangent line and area problems are basic to calculus

•Estimate a limit numerically and graphically

•Determine when a limit does not exist

• Learn and use a formal definition of a limit

• Use properties of limits to evaluate limits

• Develop and use a strategy for finding limits

• Evaluate limits by “dividing out” and “rationalizing”

• Evaluate a limit using the “Squeeze Theorem”

• Determine continuity at a point and continuity on an open interval

• Determine one-sided limits and continuity on a closed interval

Use properties of continuityKnow and use the Intermediate Value

TheoremDetermine infinite limits from the left and

the rightFind and sketch the vertical asymptotes of

the graph of a function

1.1 A Preview of Calculus

What is Calculus????????????????????

•Calculus is a branch of mathematics that deals with rates of change like velocity and acceleration. What we know of Calculus today, began in the 17th century with Newton and Leibnitz

•Calculus deals primarily with limits, derivatives and integrals

•How does Calculus differ from Precalculus?

Precalculus is static while Calculus is dynamic

See the chart on page 43

What is the purpose of Calculus? Finding the slope of curves Calculating the area of bizarre shapes Justifying old formulas Calculating complicated x-intercepts Visualizing graphs Finding the average value of a function Calculating optimal values (optimization)

On the straight incline the slope remains the same, therefore the force that must be used to push it up the hill remains static. On the curved incline, however, the slope does not remain the same, so the force changes and therefore is dynamic

Other examples of regular math vs calculus

Examples taken from What is Calculus http://media.wiley.com/product_data/

Two problems are basic to the study of calculus: The tangent line problem and the area problem

The Tangent Line Problem

The graph of a linear equation has a constant slope, but the graph of a quadratic equation does not. So, to find the slope of the curve at a certain point, we find the slope of the tangent line at that point

We begin by drawing a secant line, then bring the point of intersection closer and closer to the point of tangency. This helps us to get a good approximation of the slope of the tangent line

How will we determine the slope of the tangent line?

So, as Δx gets smaller the slope gets smaller and best approximates the slope of the tangent line

There is a limit to how small the slope can be.

0

limx

f c x f c

x

The Area Problem

As you increase the number of rectangles, the area is a better approximation of the area under the curve

In AP Calculus, we will be approaching problems in three different ways:

Analytically (using the equation)

Numerically

Graphically

Finding Limits Graphically and Numerically

Finding Limits Graphically

The informal definition of a limit is “what is happening to y as x gets close to a certain number”

If we are concerned with the limit of f(x) as we approach some value c from the left hand side, we write lim

x cf x

If we are concerned with the limit of f(x) as we approach some value c from the right hand side, we write lim

x cf x

In order for a limit to exist at c

=

and we write:

limx c

f x

limx c

f x

limx c

f x L

When limits fail to exist

1. When the right hand and left hand limits do not agree

2. When there is unbounded behavior

(as we have just seen)

3. When there is oscillating behavior

Homework p.54 – 57: 3 – 5 odd, 9 – 19, 43, 44, 49, 52.