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Calculus with Mr. Xiong

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-Notes 1.1 A preview of

Calculus

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AP Calculus Syllabus

Learning Targets

HW: pg 47;4-6,9 1-1

1.1 A preview of calculus What is calculus?

1 Note section

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

Two questions:

1) How do I find instantaneous velocity?

2) How do I find area of different shapes?

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

240( )

3p t t

1) How do I find instantaneous velocity?

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

2) How do I find area of different shapes?

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)

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

What Is Calculus?