03_05

7/21/2019 03_05

http://slidepdf.com/reader/full/030556d6bf641a28ab3016960fcd 1/28

Copyright © Cengage Learning. All rights reserved.

3Applications of

Differentiation

7/21/2019 03_05


Copyright © Cengage Learning. All rights reserved.

3.5 Summary of Curve Sketching

7/21/2019 03_05


3

Guidelines for Sketching a Curve

7/21/2019 03_05


4


The folloing checklist is intended as a guide to sketchinga curve y ! f " x # $y hand. %ot every item is relevant to every

function. "&or instance' a given curve might not have an

asymptote or possess symmetry.#

(ut the guidelines provide all the information you need to

make a sketch that displays the most important aspects of

the function.

A. Domain )t*s often useful to start $y determining the

domain D of f ' that is' the set of values of for hich f " x # is

defined.

7/21/2019 03_05


+


B. Intercepts The y ,intercept is f "-# and this tells us herethe curve intersects the y ,ais. To find the x ,intercepts' e

set y ! - and solve for x . "/ou can omit this step if the

e0uation is difficult to solve.#

C. Symmetry

"i# )f f "1 x # ! f " x # for all x in D' that is' the e0uation of the

curve is unchanged hen x is replaced $y 1 x ' then f is an

even function and the curve is symmetric a$out the y ,ais.

7/21/2019 03_05


2


This means that our ork is cut in half. )f e kno hat thecurve looks like for x -' then e need only reflect a$out

the y ,ais to o$tain the complete curve see &igure 3"a#.

5ere are some eamples6 y ! x 7' y ! x 4' y ! 8 x|, and

y ! cos x .

"a# 9ven function6 reflectional symmetry

Figure 3

7/21/2019 03_05


:


"ii# )f f "1 x # ! 1f " x # )f for all in x in D' then f is an oddfunction and the curve is symmetric a$out the origin.

Again e can o$tain the complete curve if e kno hat it

looks like for x -.

;otate <=-> a$out the origin?

see &igure 3"$#.

Some simple eamples of odd functions are y ! x ' y ! x 3'

y ! x +' and y ! sin x .

"$# @dd function6 rotational symmetry

Figure 3

7/21/2019 03_05


=


"iii# )f f " x + p# ! f " x # for all x in D' here p is a positiveconstant' then f is called a periodic function and the

smallest such num$er p is called the period.

&or instance' y ! sin x has period 7π

and y ! tan x hasperiod π . )f e kno hat the graph looks like in an interval

of length p' then e can use translation to sketch the entire

graph "see &igure 4#.

eriodic function6 translational symmetryFigure 4

7/21/2019 03_05


B


D. Asymptotes

"i# Horizontal Asymptotes. )f either lim x →

f " x # ! L or

lim x → f " x # ! L' then the line y ! L is a horiontal asymptote

of the curve y ! f " x #.

)f it turns out that lim x →

f " x # ! "or #' then e do not

have an asymptote to the right' $ut that is still useful

information for sketching the curve.

7/21/2019 03_05


<-


"ii# Vertical Asymptotes. The line x ! a is a verticalasymptote if at least one of the folloing statements is true6

"&or rational functions you can locate the verticalasymptotes $y e0uating the denominator to - after

canceling any common factors. (ut for other functions this

method does not apply.#

7/21/2019 03_05


<<


&urthermore' in sketching the curve it is very useful to knoeactly hich of the statements in is true.

)f f "a# is not defined $ut a is an endpoint of the domain of f '

then you should compute lim x →a 1 f " x # or lim x →aD f " x #' hether

or not this limit is infinite.

"iii# Slant Asymptotes.

7/21/2019 03_05


<7


E. Intervals of Increase or Decrease Ese the )F Test.Compute f ′ " x # and find the intervals on hich f ′ " x # is

positive "f is increasing# and the intervals on hich f ′ " x # is

negative "f is decreasing#.

F. ocal !a"imum and !inimum #alues &ind the critical

num$ers of f the num$ers c here f ′ "c # ! - or f ′ "c # does

not eist.

Then use the &irst erivative Test. )f f ′ changes from

positive to negative at a critical num$er c ' then f "c # is a

local maimum.

7/21/2019 03_05


<3


)f f ′ changes from negative to positive at c ' then f "c # is alocal minimum. Although it is usually prefera$le to use the

&irst erivative Test' you can use the Second erivative

Test if f ′ "c # ! - and f ″ "c # ≠ -.

Then f ″ "c # H - implies that f "c # is a local minimum' hereas

f ″ "c # I - implies that f "c # is a local maimum.

$. Concavity and %oints of Inflection Compute f ″ " x # and

use the Concavity Test. The curve is concave upard

here f ″ " x # & - and concave donard here f ″ " x # ' -.

)nflection points occur here the direction of concavity

changes.

7/21/2019 03_05


<4


(. S)etc* t*e Curve Esing the information in items A1G'dra the graph. Sketch the asymptotes as dashed lines.

lot the intercepts' maimum and minimum points' and

inflection points.

Then make the curve pass through these points' rising and

falling according to 9' ith concavity according to G' and

approaching the asymptotes.

)f additional accuracy is desired near any point' you can

compute the value of the derivative there. The tangent

indicates the direction in hich the curve proceeds.

7/21/2019 03_05


<+

9ample <

Ese the guidelines to sketch the curve

A. The domain is

J x | x 7 1 < K - ! J x | x K <

! " ' 1<# ∪ "1<' <#' ∪ "<' #

B. The x , and y ,intercepts are $oth -.

C. Since f "1 x # ! f " x #' the function f is even. The curve is

symmetric a$out the y ,ais.

7/21/2019 03_05


<2

9ample <

D.

Therefore the line y ! 7 is a horiontal asymptote.

Since the denominator is - hen x ! <' e compute the

folloing limits6

cont*d

7/21/2019 03_05


<:

9ample <

Therefore the lines x ! < and x ! 1< are verticalasymptotes.

This information a$out limits and asymptotes ena$les us to

dra the preliminary sketch in &igure +' shoing the partsof the curve near the asymptotes.

cont*d

reliminary sketch

Figure 5

7/21/2019 03_05


<=

9ample <

E.

Since f ′ " x # & - hen x ' - " x ≠ 1<# and f ′ " x # ' - hen

x & - " x ≠ <#' f is increasing on " ' 1<# and "1<' -# and

decreasing on "-' <# and "<' #.

F. The only critical num$er is x ! -.

Since f ′ changes from positive to negative at -' f "-# ! -

is a local maimum $y the &irst erivative Test.

cont*d

7/21/2019 03_05


<B

9ample <

$.

Since <7 x 7 D 4 H - for all x ' e have

f ″ " x # H - x 7 1 < & - 8 x 8 & <

and f

″ " x # ' - 8 x 8 ' <. Thus the curve is concave upardon the intervals " ' 1<# and "<' # and concave

donard on "1<' <#. )t has no point of inflection since <

and 1< are not in the domain of f .

cont*d

7/21/2019 03_05


7-

9ample <

(. Esing the information in 91G' e finish the sketch in&igure 2.

&inished sketch of y !

Figure +

cont*d

7/21/2019 03_05


7<

Slant Asymptotes

7/21/2019 03_05


77

Slant Asymptotes

Some curves have asymptotes that are oblique, that is'neither horiontal nor vertical. )f

then the line y ! mx D b is called

a slant asymptote $ecause the

vertical distance $eteen the

curve y ! f " x # and the line

y ! mx + b approaches -' as in&igure <-.

"A similar situation eists if e let x → .#

Figure ,-

7/21/2019 03_05


73

Slant Asymptotes

&or rational functions' slant asymptotes occur hen thedegree of the numerator is one more than the degree of the

denominator.

)n such a case the e0uation of the slant asymptote can $efound $y long division as in the net eample.

7/21/2019 03_05


74

9ample 4

Sketch the graph of

A. The domain is ! " ' #.

B. The x , and y ,intercepts are $oth -.

C. Since f "1 x # ! 1f " x #' f is odd and its graph is symmetric

a$out the origin.

D. Since x 7 D < is never -' there is no vertical asymptote.

Since f " x # → as x → and f " x # → as x → '

there is no horiontal asymptote.

7/21/2019 03_05


7+

9ample 4

(ut long division gives

So the line y ! x is a slant asymptote.

cont*d

7/21/2019 03_05


72

9ample 4

E.

Since f ′ " x # & - for all x "ecept -#' f is increasing

on " ' #.

F. Although f ′ "-# ! -' f ′ does not change sign at -' so there

is no local maimum or minimum.

cont*d

7/21/2019 03_05


7:

9ample 4

$.

Since f ″ " x # ! - hen x ! - or x ! e set up the

folloing chart6

The points of inflection are "-' -#' and

cont*d

7/21/2019 03_05


7=

9ample 4

(. The graph of f is sketched in &igure <<.

cont*d

Figure ,,

Documents

03_05