Download POWER SERIES

INCREASING AND
DECREASING FUNCTIONS
AND THE FIRST DERIVATIVE
TEST
Section 3.3
When you are done with your
homework, you should be able
to…
• Determine intervals on which a
function is increasing or decreasing
• Apply the first derivative test to
find relative extrema of a function
Ptolemy lived in 150AD. He devised the
1st “accurate” description of the solar
system. What branch of mathematics
did he create?
A. Calculus
B. Number Theory
C. Trigonometry
D. Statistics
Definitions of Increasing and
Decreasing Functions
• A function f is increasing on an interval if
for any two numbers x1 and x2 in the
interval, x1  x2 implies f  x1   f  x2  .
• A function is decreasing on an interval if
for any two numbers x1 and x2 in the
interval, x1  x2 implies f  x1   f  x2  .
How does this relate to the derivative?!
Theorem: Test for Increasing and
Decreasing Functions
• Let f be a function that is continuous
on the closed interval  a, b and
differentiable on the open interval a, b  .
– If f   x   0 for all x   a, b  , then f is
increasing on  a, b.
– If f  x  0 for all x   a, b  , then f is
decreasing on  a, b.
– If f  x  0 for all x   a, b  , then f is
constant on  a, b.
 
 
Guidelines for Finding Intervals on which
a Function is Increasing or Decreasing
Let f be continuous on the interval  a, b  .To find the
open intervals on which f is increasing or
decreasing, use the following steps.
1. Locate the critical numbers of f  x  in  a, b  and
use these numbers to determine the test
intervals.
2. Determine the sign of f '  x  at one test value in
each of the intervals.
3. Use the previous theorem to determine whether
f is increasing or decreasing on each interval.
*These guidelines are also valid if the interval  a, b 
•
is replaced by an interval of the form
 , b ,  a,  or  , .
The weight (in pounds) of a newborn infant
during its 1st three months of life can be
modeled by the equation below, where t is
measured in months. Determine when the
infant was gaining weight and losing weight.
1 3 5 2 19
W t   t  t  t  8
3
2
6
A.
B.
C.
D.
The infant lost weight for approximately the
first month and then gained for the 2nd and 3rd.
The infant lost weight for approximately .56
month and then gained thereafter.
The infant gained weight for approximately the
first month and then lost weight during the 2nd
and 3rd months.
The infant lost weight for approximately .56
month and then gained weight during the 2nd and
3rd months.
Fibonacci lived in 1200.
Why was he famous?
A. He introduced the Hindu-Arabic system
of numeration (aka “base 10”)
B. He developed the study of numbers in the
form of 1, 1, 2, 3, 5, 8, 13, 21, . . .
2
2
2
x

y

z
C. He found all integer solutions to
D. All of the above.
Theorem: The First Derivative
Test
Let c be a critical number of a function f that is continuous on
an open interval I containing c. If f is differentiable on the
interval, except possibly at c, then f  c  can be classified as
follows:
• If f '  x  changes from negative to positive at c, then f has a
relative minimum at  c, f  c   .
• If f '  x  changes from positive to negative at c, then f has a
relative maximum at  c, f  c   .
• If f '  x  is positive on both sides of c or negative on both
sides of c, then f  c  is neither a relative minimum or
relative maximum.
1.
f   x  0
f   x  0
f   x  0
f   x  0
a
c
b
a
f   x  0
c
b
f   x  0
f   x  0
f   x  0
a
c
b
a
c
b
Find the relative extrema of the function
and the intervals on which it is increasing
or decreasing.
f  x   x  x  5
A.
B.
C.
1
3
Relative min @ (5, 0), relative max @
(5, -75/16), increasing on  ,5 4   5,  ,
decreasing on  5 4,5 .
Relative max @ (5, 0), relative min @
(5, -75/16), increasing on  5 4,5 , decreasing
on  ,5 4   5,   .
None of the above