Blog #8

This week in Calculus, we learned how to find extreme values, critical numbers, absolute max and min, and relative max and min.

First, we need to remember the definitions of all of these terms:
Extreme values – max and min
Absolute max and min – the highest and lowest point on an interval
Relative max and min – any max or min not on an interval
Critical number – any max or min (often referred to as x = c)

Now, to find a critical number, etc. is the same as finding a horizontal tangent (something we did in the last chapter). You take the derivative, then set equal to zero. If you have an interval and it asks for the max and min, plug your x-values you get from solving the derivative and your endpoints from the interval into the original equation. The biggest number is your max and the smallest number is your min.

Ex. Find the max and min on the graph f(x) = cos π x on the interval of [0, 1/6].
f ’(x) = (-sin π x) (π) = -π sin π x
-π sin π x = 0
sin π x = 0
π x = sin^-1(0)
π x = 0, π, 2π
x = 0, 1, 2
f(0) = cos π (0) = cos 0 = 1
f(1/6) = cos π (1/6) = cos π/6 = ½
f(1) = cos π (1) = cos π = -1
f(2) = cos π (2) = cos 2π = 1
max = 1
min = -1