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Bell Ringer

Learn about the slope of a line, approximations, difference quotients, tangent lines, and how to find the slope using definitions and limits.

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Bell Ringer

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  1. Bell Ringer Solve even #’s

  2. 2.4 Rates of Change and Tangent Lines

  3. 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).

  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?

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

  6. 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.

  7. In the previous example, the tangent line could be found using . If you want the normal line, use the opposite signed 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.)

  8. Let F3 a Find the slope at . On the TI-89: limit ((1/(a + h) – 1/ a) / h, h, 0) Note: If it says “Find the limit” on a test, you must show your work! Example 4: Calc

  9. Let b Where is the slope ? Example 4:

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

  11. WINDOW Y= GRAPH Example 4: Graph the curve and the tangents on theTI-89: y = 1 / x

  12. WINDOW F5 ENTER ENTER Y= GRAPH Example 4: Graph the curve and the tangents on theTI-89: We can let the calculator plot the tangents: Math y = 1 / x A: Tangent 2 Repeat for x= -2 tangent equation

  13. 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

  14. Homework: 2.4a 2.4 p92 1,7,13 2.3 p84 45 2.4b 2.4 p92 9,15,19,25,37,31

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