2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, Washington Photo 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 (1,1) and (4,16). We could get a better approximation if we move the second point closer to (1,1): (3,9) Even closer would be the point (2,4).

The slope of a line is given by: If the second point got really close to (1,1), say at (1.1,1.21), the approximation would get better still! How close can we get to x=1?

“instantaneous” slope 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 on a curve “using the definition of slope,” or “using the difference quotient”, this “limit of the difference quotient” is the technique to use.

The instantaneous slope of a curve “at” a point is the same as the slope of the tangent line “at” that point. For the previous example, once the slope of the tangent was found, the tangent line equation could be written using , point-slope form. To write the equation of the normal line (perpendicular to the tangent line), the normal slope is the opposite reciprocal of the tangent slope…

Example 4: Let a Find the slope at an arbitrary point, .

On the TI-84: Let b Where is the slope ? Y= y = 1 / x WINDOW Example 4: On the TI-84: Let b Where is the slope ? Y= y = 1 / x WINDOW at x = 2: f(2) = 0.5 at x = -2: f(-2) = -0.5 GRAPH

Example 4: Let b Where is the slope ? We can graph f(x), and check graphs of the two point-slope tangent lines: at (2, 0.5): y1= (-1/4)(x – 2) + 0.5 at (-2, -0.5): y2= (-1/4)(x – - 2) – 0.5

Summary: p These are often mixed up by calculus students! average slope over an interval: instantaneous slope at x = a: If is a position function: average velocity over an interval: So are these! instantaneous velocity: velocity = slope along a position graph! p