0 2=2×7%72 ftxt-YII

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

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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.


--

- -

-

- -

art¥D-

na x -

-min

--

- 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


"

,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



f(x) = 6x3 + 9x2 � 360x.



f(x) = x5e�0.5x


-

- --

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).


-

=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

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-

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'

I2)

wc:cf-

IfI

w

not a min .

since X =- 7

is not in domain .

Partition # not critical #

DEC.C-og-DUCI.TW

RI.max.atx-IJNORI.MIL

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).


. . .

,••c•,

•

¥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).


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f'

=D NE ⇒ X=-lsx=T

f 1=0 ⇒ x=-3sx=T#

f

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0 2=2×7%72 ftxt-YII

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