1. 2.4 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

2. 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).

3. 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?

4. “instantaneous”
slope
slope at
The slope of the curve “at” the point is:

5. 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.

6. 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…

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

8. 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

9. 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

10. 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