1.1-1.2 Introduction to Calculus and Limits

Objectives: Understand what calculus is, tangent line problem and area problem; Find limits graphically and numerically

Miss BattagliaAB/BC Calculus

Very advanced algebra and geometry Look at the two pictures, the problem in both

cases is to determine the amount of energy required to push the crate to the top.

calculus problem regular math problem things are constantly changing unchanging force/unchanging speed

What is calculus?

Precalculus Mathematics

Limit Process Calculus

The Tangent Line Problem Find the slope of the tangent line at P

msec=

Approximate the area of the region

As you increase the number of rectangles, the approximation becomes better and better.

The Area Problem

An Introduction to Limits

x -.2 -.0015 -.00027 0 .00002 .008 .01 .5

f(x) 3.052 2.720 2.718 2.718 2.707 2.705 2.25

x approaches 0 from the left

f(x) approaches e

x approaches 0 from the right

f(x) approaches e

Suppose you are asked to find

Estimating a limit numerically

x -.05 -.03 -.01 0 .01 .03 .05

f(x)

Use the table on the calculator

Evaluate the function atseveral points near x=0 and use the results to estimate the limit

Finding a Limit

Find the limit of f(x) as x approaches 2, where f is defined as

Behavior that Differs from the Right and from the Left Show that the limit does not exist

Unbounded behavior

Discuss the existence of the limit

Oscillating Behavior Discuss the existence of the limit.

1. f(x) approaches a different number from the right side of c than it approaches from the left side.

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

Common Types of Behavior Associated with Nonexistence of a Limit

Use the graph of f to identify the values of c for which the limit as x approaches c exists.

Read 1.1 and 1.2

Page 54 #1, 3, 11, 13 (use table on calculator), 15-33 all, 35, 46

Classwork/Homework