Download 3.3 Increasing and Decreasing and the First Derivative Test

Miss Battaglia
AP Calculus AB/BC
A function f is increasing on an interval for any two
numbers x1 and x2 in the interval, x1<x2 implies f(x1)<f(x2)
A function f is decreasing on an interval for any two
numbers x1 and x2 in the interval, x1<x2 implies f(x1)>f(x2)
Increasing! Pierre the Mountain
Climbing Ant is climbing the
hill from left to right.
Decreasing! Pierre is walking
downhill.
Let f be a function that is continuous on the
closed interval [a,b] and differentiable on the
open interval (a,b).
1.
2.
3.
If f’(x)>0 for all x in (a,b), then f is
increasing on [a,b]
If f’(x)<0 for all x in (a,b), then f is
decreasing on [a,b]
If f’(x)=0 for all x in (a,b), then f is contant
on [a,b]
Find the open intervals on which f (x) = x 3 - 3 x 2 is
2
increasing or decreasing.
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Find the first derivative.
Set the derivative equal to zero and solve for x.
Put the critical numbers you found on a
number line (dividing it into regions).
Pick a value from each region, plug it into the
first derivative and note whether your result is
positive or negative.
Indicate where the function is increasing or
decreasing.
1
Find the relative extrema of the function f (x) = x - sin x
2
in the interval (0,2π)
Find the relative extrema of
f (x) = (x 2 - 4)2/3
x 4 +1
Find the relative extrema of f (x) =
2
x
 Read
3.3 Page 179 #1, 8, 12,
21, 27, 29, 35, 43, 45, 63, 67,
79, 99-103