SOLVING PROBLEMS USING RATES of CHANGE
(Max & Min)
Introduction
It is often useful to use rates of change to determine the maximum or minimum
values of a function.
y
y
local max
absolute min
x
x
absolute max
local min
The slope of the tangent line at
the max/min point is 0!!

If the slope of the tangent is negative to the left of the point
and positive to the right of the point, then a minimum
exists at that point.
–ve
+ve
m=0 at min

If the slope of the tangent is positive to the left of the point
and negative to the right of the point, then a maximum
exists at that point.
m=0 at max
+ve
–ve
Example 
Use an algebraic strategy to verify that the point (–2, –25) is either
a maximum or minimum of the function, f(x) = x2 + 4x – 21.
Example 
Determine the equation of the tangent line to the curve,
f(x) = x2 – 3x + 1, at the point where x = –2.
Example 
A golf shot is modelled by the function, H(t) = –5t2 + 40t, where H(t)
is the height of the golf ball in metres and t is the time that it is in the
air in seconds. Estimate the time at which the golf ball will be at a
maximum height above the ground.
Homework: p.112–113 #5ac, 6ab, 7, 10, 11