Rates of Change and Tangent Lines Devil’s Tower, Wyoming.

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Presentation transcript:

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 slope at The slope of the curve at the point is:

is called the difference quotient of f at a. 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. The normal line can be found the same way, except by using the opposite reciprocal of the slope. The normal line is perpendicular to the tangent line. The slope of a curve at a point is the same as the slope of the tangent line at that point.

Example 4: a Find the slope at. Let

Example 4: Note: The general derivative can only be found on a CAS calculator. Also, if it says “Find the limit” on a test, you must show your work! On the calculator:

Example 4: b Where is the slope ? Let

Example 4: c What are the tangent line equations when and ?

Review: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up. Be Careful! So are these! velocity = slope 