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2.4 Rates of Change and Tangent Lines - PowerPoint PPT Presentation


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Photo by Vickie Kelly, 1993. Greg Kelly, Hanford High School, Richland, Washington. 2.4 Rates of Change and Tangent Lines. Devil’s Tower, Wyoming. The slope of a line is given by:. The slope at (1,1) can be approximated by the slope of the secant through (4,16).

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Presentation Transcript
slide1

Photo by Vickie Kelly, 1993

Greg Kelly, Hanford High School, Richland, Washington

2.4 Rates of Change and Tangent Lines

Devil’s Tower, Wyoming

slide2

The slope of a line is given by:

The slope at (1,1) can be approximated by the slope of the secant through (4,16).

We could get a better approximation if we move the point closer to (1,1). ie: (3,9)

Even better would be the point (2,4).

slide3

The slope of a line is given by:

If we got really close to (1,1), say (1.1,1.21), the approximation would get better still

How far can we go?

slide4

slope at

The slope of the curve at the point is:

slope

slide5

is called the difference quotient of f at a.

The slope of the curve at the point is:

If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.

slide6

In the previous example, the tangent line could be found using .

If you want the normal line, use the negative reciprocal of the slope. (in this case, )

The slope of a curve at a point is the same as the slope of the tangent line at that point.

(The normal line is perpendicular.)

slide7

Let

a

Find the slope at .

Note:

If it says “Find the limit” on a test, you must show your work!

Example 4:

slide8

These are often mixed up by Calculus students!

If is the position function:

velocity = slope

Review:

average slope:

slope at a point:

average velocity:

So are these!

instantaneous velocity:

p