2 4 Rates of Change and Tangent Lines

2. 4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington

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:

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.

The slope of a curve at a point is the same as the slope of the tangent line at that point. 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 normal line is perpendicular. ) )

Example 4: Let a Find the slope at .

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

Example 4: Let b Where is the slope ?

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

On the TI-nspire: tangent. Line(1/x, x, 2) T (Find the tangent line to the function y=1/x when x is 2. ) tangent. Line(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.

Use to toggle to the graph screen. Use Zoom – Standard if you don’t have this window. menu 3 enter 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: These are often mixed up by Calculus students! average slope: slope at a point: average velocity: So are these! instantaneous velocity: velocity = slope If is the position function: p