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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Presentation transcript:

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

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 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 opposite signed reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)

Example 4: a Find the slope at. Let

The box above the zero is for a + or – sign to indicate direction. Example 4: Note: If it says “Find the limit” on a test, you must show your work! On the TI-nspire: menu 4 Calculus 4 Limit enter

Example 4: b Where is the slope ? Let

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

(Find the tangent line to the function y=1/x when x is 2.) On the TI-nspire: T tangentLine( tangentLine(1/x, x, 2) tangentLine(1/x, x, -2) Hint: Instead of re-entering the formula, use the up arrow to highlight the first formula and then press enter. Then insert the negative sign. Now we will graph the function and the tangent lines.

menu enter3 Use Zoom – Standard if you don’t have this window. Use to toggle to the graph screen. Now go back to the calculate screen, use the up arrow to highlight the first tangent equation, and use. ctrl C Go to the graph screen and input the first tangent equation using. ctrl V Repeat the process to input the second tangent equation.

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 