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2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

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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).

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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?

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

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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.

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In the previous example, the tangent line could be found using. The slope of a curve at a point is the same as the slope of the tangent line at that point. If you want the normal line, use the negative reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)

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Example 4: a Find the slope at. Let On the TI-89: limit ((1/(a + h) – 1/ a) / h, h, 0) F3Calc Note: If it says “Find the limit” on a test, you must show your work!

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Example 4: b Where is the slope ? Let On the TI-89: Y= y = 1 / x WINDOW GRAPH

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Example 4: b Where is the slope ? Let On the TI-89: Y= y = 1 / x WINDOW GRAPH We can let the calculator plot the tangent: F5Math A: Tangent ENTER 2 Repeat for x = -2 tangent equation

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Review: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope

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2.7 Tangents, Velocities, & Rates of Change

2.7 Tangents, Velocities, & Rates of Change

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