Section 5.1 The First Derivative
Definitions:
A partition number of f 0 is a number p for which either f 0(p) = 0
or f 0(p) does not exist.
A critical number of f is a number c for which either f 0(c) = 0 or
f 0(c) does not exist AND c is in the domain of f .
Note: Critical numbers are also partition numbers, but partition numbers may not be
critical numbers.
1. Given f(x) =x2 + 3
x� 1
(a) find the domain of f .
(b) find f 0.
(c) find the partition numbers of f 0.
(d) find the critical numbers of f .
-
f ( x ) =X
2t 3
I⇒
x -I to ⇒ x
Domaon.ee/-oeiI1uCt#y
mm ":¥f
'
Ix ) = ¥t3L( x - c)2=2×7%72
⇒ ftxt-YII.ITf
'
= o ⇒ x2
-2x - 3=0 ⇒ ( × - 3)( x t I ) -
-O
⇒ x=3,x=1J
t'
-
- DNE ⇒ I × - , , 2=0 ⇒ × →is
not indown
critical # 's
.
. y=31(in domain )
What is a Sign Chart? We will use a sign chart to track where f 0 > 0 (+) or where f 0 < 0 (-).
Steps for making a Sign Chart
1. Plot all partition and critical numbers on a number line.
2. Choose values to test regions on the number line around the critical/partition numbers.
3. Plug test values into f 0 and record the sign (±).
2. Make a sign chart for the function in Problem 1.⇣Note: f 0(x) =
(x� 3)(x+ 1)
(x� 1)2
⌘
Increasing/Decreasing Test:
(a) If f 0(x) > 0 on an interval, then f is increasing on that interval
(b) If f 0(x) < 0 on an interval, then f is decreasing on that interval
3. Use the sign chart from Problem 2 to find intervals on which f(x) =x2 + 3
x� 1is increasing/decreasing.
2 Spring 2019, Maya Johnson
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Definition: Let c be a number in the domain D of a function f . Then
f(c) is the
Relative Maximum value of f if f(c) � f(x) when x is near c.
Relative Minimum value of f if f(c) f(x) when x is near c.
Note: We say that f(c) is a relative extremum if f(c) is either a relative maximum or
a relative minimum.
The First Derivative Test: Suppose that c is a critical number of a
continuous function f .
(a) If f 0(x) changes from positive to negative around c, then f has a rela-
tive maximum at c.
(b) If f 0(x) changes from negative to positive around c, then f has a rela-
tive minimum at c.
(c) If f 0(x) does not change sign around c (for example, f 0(x) is positive
on both sides of c or negative on both sides), then f has no relative
maximum or minimum at c.
4. Use the sign chart from Problem 2 to find any x-value where f(x) =x2 + 3
x� 1has a relative
extremum.
3 Spring 2019, Maya Johnson
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-
- -
art¥D-
na x -
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- ree.max.atx.IN
RelMm.atx=3J
Steps for finding where f is Increasing/Decreasing or any Relative Extrema
1. Find the domain of f .
2. Find all partition numbers of f 0.
3. Find all critical numbers of f .
4. Make a sign chart to track where f 0 > 0 or f 0 < 0.
5. For the function below, find all intervals on which f is increasing/decreasing and the x-values of
any relative extrema.
f(x) = 5(x2 � 36x+ 180)4/5
4 Spring 2019, Maya Johnson
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,Odd - root facts have domain of too ,D )
.
→negative
power
2)f
'
1×1=5.451×2-36×+180 .(2×2-36)a
= 412×-3671×2-
36×+180745
f'
-
a0 ⇒ 412×-362=0 ⇒ X=t8
§titer
#' s
f'
=D NE ⇒ x'
- 36×+180=0
⇒= o ⇒ X = 30,4=6
3) Critical # 's × = is,
× = go ,×=6 ( in thedomain)
4) fix ) -
- 1411271×-1872 y WC:(6,l87Ul30it( ( x -
3021×-6>515
..±:*.±
..f::÷:::::s
e#
pd.max.at×-
fmin
6. For the function below, find all intervals on which f is increasing/decreasing and the x-values of
any relative extrema.
f(x) = 6x3 + 9x2 � 360x.
7. For the function below, find all intervals on which f is increasing/decreasing and the x-values of
any relative extrema.
f(x) = x5e�0.5x
5 Spring 2019, Maya Johnson
-
- --
1) Domain : L - •,
oo )4) f 't -67 ft to ) f 's )
d d k
2) partition # es :
ftp.t#.tt'
f'
1×7=18×2+18×-360
max mon
ft I x ) -
-O - ⇒ 18×2+18×-360=0
partition # 's
.
⇒ 181×2 t X - 202=0 -
µc:cx,._-)UL4#€
⇒ =0⇒×= -51*-4DEc.es#yf'-rDNE-
→ never ! pd.max.at/x---5J37 Critical # 's X= - 5. x
, pd.mon.atx
1) Domain : L - DID )- O .
5X
2) ft Cx) -
- 5×450.5×+55.1-0.5) .jo.5× x5#=x 0.5×15 - o .s× )
5×4+-0.550-57
f'
so ⇒ x 4=0,
e- " 5×1=0,5-0.5×-0
⇒ & } partition # ' s
f 's DNE → never !
3) Critical # 's : x -01×-10 L in domain )
f 'cx7=x4e -0.5×15-0.5×3
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ft- D f 'Ll) ft fun) DEC:(lO,D€ty÷÷Ty
max
8. Suppose the function f(x) has a domain of all real numbers and its derivative is shown below.
f 0(x) = (x+ 2)2(x� 7)4(x� 8)7.
(a) Find all intervals on which f(x) is increasing.
(b) Find all intervals on which f(x) is decreasing.
9. Suppose the function f(x) has a domain of all real numbers except x = �7. The first derivative
of f(x) is shown below.
f 0(x) =�5(x� 1)
(x+ 7)7
(a) Find all intervals on which f(x) is increasing.
(b) Find all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
6 Spring 2019, Maya Johnson
-
=D-
--
partition I critical # 'S
: x = -2, x -
-7
, x a 8
f 't's ) fIo) f 5) I'M)
wce.es#y> 8
- 2
DEC.L-q-ZSUL-2.DUG.se#f
-
Partition # 's : f '
-0 ⇒ - 5 C x - D ⇒ ⇒ XII,
not in domain
f'
=D NE ⇒ txt77-0 ⇒ ×
critical # ' S : X =L ( in domain )
fats) fly) f'
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wc:cf-
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w
not a min .
since X =- 7
is not in domain .
Partition # not critical #
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10. Use the graph of f 0(x) to answer questions about the function f(x).
This is the graph of f 0(x).
(a) Find all critical values of f(x).
(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
7 Spring 2019, Maya Johnson
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¥0 ,
f '=o ⇒ ×=0,x=7,andX=l÷f
'=D NE → never !
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11. Use the graph of the function f(x) displayed below to answer the following questions.
(a) Find the critical values where f 0(x) does not exist.
(b) Find the critical values where f 0(x) = 0.
(c) Find the x-coordinate(s) of the relative maxima for f(x).
(d) Find the x-coordinate(s) of the relative minima for f(x).
8 Spring 2019, Maya Johnson
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f'
=D NE ⇒ X=-lsx=T
f 1=0 ⇒ x=-3sx=T#
f
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