Be seated before the bell rings

DESK

All bags, purse, etc off desk

homework

Be prepare : Have these things out

Warm-up (in your notes)

Glue in learning targets

HW: pg 47;4-6,9

•Enter the classroom – Enter quietly, go to your seat.

Take off hat. – No gum or food, except water– Materials (sharpen pencil) – Have homework out– Work on warm-up quietly.– Go over warmup/homwork – Homework quiz ?

•During Class– Take notes – Remove backpack/purse off disk. – Participate/ Listen / no talking – Stay seated. Ask to move for any

reason

• Group Work – Follow Study Team

Expectations– Stay in your seat

• Leaving class– Only pack up the last

min of class. – Pick up any trash

around you – Straighten up your

seats – Collect group hw- Turn

in your homework in the turn-in basket.

Class Room Rule and Procedures

Notebook

First Page First Page

Table of content

1) 1-1 A preview of Calculus

Page

1

Composition Book (Notebook )

Table of contentPage

1

Skip about 3 Skip about 3 pagepage

1

1) 1-1 A Preview of Calculus

1.1 A preview of calculus What is calculus?

1

http://www.ima.umn.edu/~arnold/calculus/secants/secants1/secants-g.html

Two questions:

1)How do I find instantaneous rate of velocity?

2) How do I find area of different shapes?

What Is Calculus?

Calculus is the mathematics of change. For instance, calculus is the mathematics of velocities, accelerations, tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations.

Although precalculus mathematics also deals with velocities, accelerations, tangent lines, slopes, and so on, there is a fundamental difference between precalculus mathematics and calculus.

Precalculus mathematics is more static, whereas calculus is more dynamic.

What Is Calculus?

What do you think the graph of Mr. X’s travels will look like?

Time

Dis

tanc

e

The tangent line problem

We want the slope of the tangent line because it represents rate

Except for cases involving a vertical tangent line, the problem of finding the tangent line at a point P is equivalent to finding the slope of the tangent line at P.

You can approximate this slope by

using a line through the point of

tangency and a second point on

the curve, as shown in Figure 1.2(a).

Such a line is called a secant line.

The Tangent Line Problem

Figure 1.2 (a)

If P(c, f(c)) is the point of tangency and

is a second point on the graph of f, the slope of the secant line through these two points can be found using precalculus and is given by

The Tangent Line Problem

As point Q approaches point P, the slopes of the secant lines approach the slope of the tangent line, as shown in Figure 1.2(b).

When such a “limiting position” exists,

the slope of the tangent line is said

to be the limit of the slopes of the

secant lines.

The Tangent Line Problem

Figure 1.2 (b)

The area problem

https://www.youtube.com/watch?v=O2d26o8-jcI

As you increase the number of rectangles, the approximation tends to become better and better because the amount of area missed by the rectangles decreases.

Your goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increases without bound.

The Area Problem

You can approximate the area of the region with several rectangular regions as shown in Figure 1.4.

Figure 1.4

The Area Problem

Example problems: Estimating area under curve

https://www.youtube.com/watch?v=O2d26o8-jcI

Answer to the Two questions:

1)How do I find instantaneous rate of velocity?

2) How do I find area of different shapes?

We use secant lines to approximate slopes of tangent line . The slope of the tangent line is said to be the limit of the slopes of the secant line.

We use rectangles to approximate area.

We let the number of rectangles go to infinite