2.4 Rates of Change and Tangent Lines

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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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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).

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).

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?

slope at

The slope of the curve at the point is:

slope

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.

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.)

Let

a

Find the slope at .

Note:

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

Example 4:

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