[email protected] MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§3.3 Curve

Sketching



Review §

Any QUESTIONS About• §3.2 → ConCavity & InflectionPoints

Any QUESTIONS About HomeWork• §3.2 →

HW-14

3.2

http://www.google.com/url?sa=i&rct=j&q=concavity%20second%20derivative&source=images&cd=&cad=rja&docid=ktGGLinfesRG5M&tbnid=3BqGMXcYqriGiM:&ved=0CAUQjRw&url=http://mathsfirst.massey.ac.nz/Calculus/Sign2ndDer/Sign2DerPOI.htm&ei=BnnhUY32OY_oiAL0qIHIAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH_7i_-z0nsoBcOiPC-X2f2JdV9Vg&ust=1373816673539841



§3.3 Learning Goals

Determine horizontal and vertical asymptotes of a graph

Use Algebra to find Axes InterCepts on a Funciton Graph

Use Derivatives to find Significant Points on the graph

Discuss and apply a general procedure for sketching graphs

http://www.google.com/url?sa=i&rct=j&q=sketching+math+graphs&source=images&cd=&cad=rja&docid=zPxnl0h14_UB6M&tbnid=o8-BUsS4w5MktM:&ved=0CAUQjRw&url=http://www.softmath.com/tutorials-3/relations/math-1051-precalculus-i.html&ei=V3zhUeSdMobKiwLshICYBQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH6SN55FvJ4whSPTPuAQGuLr0u-Nw&ust=1373818125582140



T-Table Can Miss Features

Consider the Function

Make T-Table,Connect-Dots

210

810

x

xxyxf

x Y-5 -6.00-4 -4.44-3 -3.06-2 -1.88-1 -0.860 0.001 0.742 1.393 1.954 2.455 2.89 -5 -4 -3 -2 -1 0 1 2 3 4 5

-6

-5

-4

-3

-2

-1

0

1

2

3

x

y =

f(x)

= 1

0x(

x+8

)/(x

+1

0)2

MTH15 • GraphSketching

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 13Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m% ref:%% The Limitsxmin = -35; xmax = 25; ymin = -15; ymax = 40;% The FUNCTIONx = linspace(xmin,xmax,500); y = 10*x.*(x+8)./(x+10).^2;% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = = 10x(x+8)/(x+10)^2'),... title(['\fontsize{16}MTH15 • GraphSketching',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)hold onplot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)plot([-10 -10], [ymin, ymax], '-- m', [xmin xmax],[10 10], '-- m', 'LineWidth', 2) set(gca,'XTick',[xmin:5:xmax]); set(gca,'YTick',[ymin:5:ymax])



T-Table Can Miss Features

But Using Methods to be Discussed, Find

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25-15

-10

-5

0

5

10

15

20

25

30

35

40

x

y =

f(x)

= =

10

x(x+

8)/

(x+

10

)2MTH15 • GraphSketching




MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 23Jun13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m% ref:%% The Limitsxmin = -5; xmax = 5; ymin = -6; ymax = 3;% The FUNCTIONx = [-5 -4 -3 -2 -1 0 1 2 3 4 5];y = [-6 -4.444444444 -3.06122449 -1.875 -0.864197531 0 0.743801653 1.388888889 1.952662722 2.448979592 2.888888889]% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 10x(x+8)/(x+10)^2'),... title(['\fontsize{16}MTH15 • GraphSketching',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)hold onplot(x,y, 'x m', 'MarkerSize', 15, 'LineWidth', 3)plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax]hold off



T-Table Can Miss Features In Order for

T-Tables & ConnectDots to properly Characterize the Fcn Graph, the Domain (x) Column must• Cover sufficiently

Wide values• Have sufficiently

small increments

Unfortunately the Grapher does NOT know a-priori the• x Span • ∆x Increment Size

-5 -4 -3 -2 -1 0 1 2 3 4 5-6

-5

-4

-3

-2

-1

0

1

2

3

x

y =

f(x)

= 1

0x(

x+8

)/(x

+1

0)2



-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25-15

-10

-5

0

5

10

15

20

25

30

35

40

x

y =

f(x)

= =

10

x(x+

8)/

(x+

10

)2



x-SpanInSufficent



Better Graphing GamePlan

1. Find THE y-Intercept, if Anya. Set x = 0, find y

b. Only TWO Functions do NOT have a y-intercepts

– Of the form 1/x– x = const; x ≠ 0

2. Find x-Intercept(s), if Anya. Set y = 0, find x

b. Many functions do NOT have x-intercepts

http://www.google.com/url?sa=i&rct=j&q=y-intercept&source=images&cd=&cad=rja&docid=U5fPM8lv1BGA2M&tbnid=ot8F1iJNRRbBUM:&ved=0CAUQjRw&url=http://www.mathsisfun.com/algebra/finding-intercepts-equation.html&ei=IYvhUbv2EaWTiAKQ84HYDg&bvm=bv.48705608,d.cGE&psig=AFQjCNEd49FWCJrBaqFfbC5vU4in8_033w&ust=1373822032204241




3. Find VERTICAL (↨) Asymptotes, If Anya. Exist ONLY when fcn has a denom

b. Set Denom = 0, solve for x– These Values of x are the Vertical Asymptote

(VA) Locations

4. Find HORIZONTAL (↔) Asymptotes (HA), If Any

a. HA’s Exist ONLY if the fcn has a finite limit-value when x→+∞, or when x→−∞




b. Find y-value for:– These Values of y are the HA Locations

5. Find the Extrema (Max/Min) Locationsa. Set dy/dx = 0, solve for xE

b. Find the corresponding yE = f(xE)

c. Determine by 2nd Derivative, or ConCavity, to test whether (xE, yE) is a Max or a Min

– See Table on Next Slide

xfyx

limHA



Better Graphing GamePlan– Determine Max/Min By Concavity

6. Find the Inflection Pt Locationsa. Set d2y/dx2 = 0, solve for xi

b. Find the corresponding yi = f(xi)

c. Determine by 3rd Derivative test The Inflection-concavity form: ↑-↓ or ↓- ↑

𝒅𝟐𝒚𝒅𝒙𝟐ቚ𝒙𝑬 Sign Concavity Max or Min

POSitive Up ↑ Min NEGative Down ↓ Max

Neither (Zero) No Information Flat Spot




7. Find the Inflection Pt Locationsa. Set d2y/dx2 = 0, solve for xi

b. Find the corresponding yi = f(xi)

c. Determine by 3rd Derivative test The Inflection form: ↑-↓ or ↓- ↑

– Determine Inflection form by 3rd Derivative𝒅𝟑𝒚𝒅𝒙𝟑ቚ𝒙𝒊 Sign ConCavity Change Inflection Form

POSitive Down-to-Up ↓-↑ NEGative Up-to-Down ↓ ↑-↓

Neither (Zero) No Information ↑-↑ OR ↓-↓




8. Sign Charts for Max/Min and ↑-↓/↓-↑a. To Find the “Flat Spot” behavior for dy/dx

= 0, when d2y/dx2 exists, but [d2y/dx2]xE = 0 use the Direction-Diagram

a b c

−−−−−−++++++ −−−−−− ++++++

x

Slope

df/dx Sign

Critical (Break)Points Max NO

Max/MinMin




9. Sign Charts for Max/Min and ↑-↓/↓-↑a. To Find the ↑-↑ or ↓-↓ behavior for d2y/dx2

= 0, when d3y/dx3 exists, but [d3y/dx3]xi = 0 use the Dome-Diagram

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection



Example Sketch Rational Fcn

Sketch

Set x = 0 to Find y-intercept

• Thus y-intercept → (0, 4/3)

Set y = 0 to Find x-intercept(s), if any

31

2122

2

xx

xxxfy

3

4

3

4

31

21

3010

201020 2

2

2

2

y




y=0:

Solving for x: Finding y(x):

1

31

31

2120

31

2120

2

2

2

2

2

xx

xx

xx

xx

xx

22 201202120 xorxxx

2or21 xx

051

05

3212

221222

02523

250

321121

2211212

2

1

2

2

2

2

2

2

2

2

y

y




The x-Intercepts• (½,0); Multiplicity = 1 (Linear Form)• (−2,0); Multiplicity = 2 (Parabolic Form)

The Horizontal Intercept(s)

3

3

2

2

2

2

1

1

31

212lim

31

212limlim

x

x

xx

xx

xx

xxy

xxx




Continuing with the Limit

• Thus have a HORIZONTAL asymptote at y = 2

xx

xx

xx

xx

xx

xx

yxxx 3

11

1

21

12

lim31

212

limlim 2

2

2

2

2

2

211

12

0101

0102limlim 2

2

xx

y




To Find VERTICAL asymptote(s) set the DeNom/Divisor = 0

• Using Zero Products

• Thus have VERTICAL Asymptotes at – x = −1– x = 3

31031

212 2

2

2

xxxx

xxxy

3or1310 2 xxxx




Use Computer Algebra System, MuPAD to find and Solve Derivatives

From the Derivatives Find• Min at (−2,0) → ConCave UP• Inflection Points

– ↓-to-↑ at (−2.63299, 0.16714)– ↑-to-↓ at (0.63299, −0.29213)



Th

e Grap

h

-4 -3 -2 -1 0 1 2 3 4 5 6-12

-8

-4

0

4

8

12

16

20

x

y =

f(x)

= =

(2

x+1

)(x+

2)2 /(

x+1

)2 (x-3

)MTH15 • GraphSketching




WhiteBoard Work

Problems From §3.3• P46 → Inventory Cost• P60 → Immunization

http://www.google.com/url?sa=i&rct=j&q=inventory&source=images&cd=&cad=rja&docid=WzP9zayMRxVMZM&tbnid=fkhW8vuhHTXPLM:&ved=0CAUQjRw&url=http://promis-mohfw.gov.in/&ei=_RHjUeiNGoOviAKi14HQCw&bvm=bv.48705608,d.cGE&psig=AFQjCNGW4N5r-b0IJ-nMNjt8ForESf-Mbw&ust=1373922137247345

http://www.google.com/url?sa=i&rct=j&q=inoculation&source=images&cd=&cad=rja&docid=82jOpzujpnuhFM&tbnid=V3j5UpNm_MfAiM:&ved=0CAUQjRw&url=http://www.scirra.com/store/music/music-singles/dr-oglesbys-innoculation&ei=lhLjUbf9B4KpiAKWtIGQCw&bvm=bv.48705608,d.cGE&psig=AFQjCNEZT8I9Jpj527IDHPs7D2oRAcEeag&ust=1373922228718400



All Done for Today

A GraphicScalingMachine

http://www.google.com/url?sa=i&rct=j&q=sketch+graph&source=images&cd=&cad=rja&docid=66IObyy_IDLoQM&tbnid=nguYYPGv3oQoQM:&ved=0CAUQjRw&url=http://hummeltattoo.wordpress.com/author/hummeltattoo/&ei=8djhUYn2L8f7iwKS0oHwCQ&bvm=bv.48705608,d.cGE&psig=AFQjCNE8bsy0MSasZKBxGICf_OAiex7VxA&ust=1373841964173129



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign





ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

























P33-46

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

x

y =

f(x)

= 2

x +

80

k/x

MTH15 • P3.3-46 • Bruce Mayer, PE












P3.3-56



http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=NjbfKj_JZBSzwM&tbnid=jJkxz7X095T-NM:&ved=0CAUQjRw&url=http://mathworld.wolfram.com/Derivative.html&ei=TfPWUe_nA6akiQKZ64DIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

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http://www.google.com/url?sa=i&rct=j&q=relative+extrema+of+a+function&source=images&cd=&cad=rja&docid=r50JwQWumQU79M&tbnid=cBBX7LSiOy-pcM:&ved=0CAUQjRw&url=http://www.ltcconline.net/greenl/courses/115/applications/frsttst.htm&ei=C87eUYSWEoWpiQLjwoCYAg&bvm=bv.48705608,d.cGE&psig=AFQjCNETiKTT033nRi_5KJkrhn3WjmAx4w&ust=1373642329390150

Documents

[email protected] MTH15_Lec-15_sec_3-3_Curve_Sketching.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical