AP Calculus AB
2.4 Rates of Change and Tangent Lines
Objectives: Average Rates of Change; Tangent to a Curve; Slope of a Curve; Normal
to a Curve
Procedure:
1. Average Rates of Change:
a. Examples 1 – 3: Find the average rate change of the function over each
interval.
b. Example 4: Growing Drosophila in a laboratory.
Use the points P(23, 150) and Q(45, 340) in the figure below to compute
the average rate of change and the slope of the secant line PQ.
2. Tangent to a Curve:
The solution that mathematician Pierre Fermat found in 1629 proved to be one of
that century’s major contributions to calculus. We still use his method of defining
tangents to produce formulas for slopes of curves and rates of change:
1. We start with what we can calculate, the slope of a secant through P
and a point Q nearby on the curve.
2. We find the limiting value of the secant slope (if it exists) as Q
approaches P along the curve.
3. We define the slope of the curve P to be this number and define the
tangent to the curve at P to be the line through P with this slope.
a. Example 5: Finding slope and tangent line.
Find the slope of the parabola y  x 2 at the point P(2, 4). Write an
equation for the tangent to the parabola at this point.

3. Slope of a Curve:
a. Definition 1: The _______________________ y  f x  at the point
Pa, f a is the number


provided the limit exists.
b. Example 6: Exploring slope and tangent.
(a) Find the slope of the curve at x = a.
(b) Where does the slope equal -1/4?
(c) What happens to the tangent to the curve at the point (a, 1/a) for
different values of a?
4. Normal to a Curve:
a. Definition 2: The _____________________ to a curve at a point is the
line perpendicular to the tangent at the point.
b. Examples 7 & 8: At the indicated point find (a) the slope of the curve, (b)
an equation of the tangent, and (c) an equation of the normal. (d) Then
draw a graph of the curve, tangent line, and normal line in the same
square viewing window.
c. Example 9: Determine whether the curve has a tangent at the indicated
point. If it does, give its slope. If not, explain why not.
d. Example 10: The table below gives the amount of federal funding for the
Immigration and Naturalization Service (INS) for several years.
Federal Funding
INS Funding
Year
($ billions)
1993
1.5
1994
1.6
1995
2.1
1996
2.6
1997
3.1
(a) Find the average rate of change in funding from 1993 to 1995.
(b) Find the average rate of change from 1995 to 1997.
(c) Let x = 0 represent 1990, x = 1 represent 1991, and so forth. Find
the quadratic regression equation for the data and superimpose
its graph on a scatter plot of the data.
(d) Compute the average rates of change in (a) and (b) using the
regression equation.
(e) Use the regression equation to find how fast the funding was
growing in 1997.
5. HW: