Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

[email protected] • ENGR-25_Lec-28_Excel-1.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/MATH/Physics 25

Sketch FcnGraphs by

MuPAD



ReCall from MTH1 Graph Sketching Determine horizontal and vertical

asymptotes of a graph Use Algebra to find Axes InterCepts on a

Function Graph Use Derivatives to find

Significant Points on the graph Discuss and apply a

general procedure forsketching 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

xxxyxf

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) =

10x

(x+8

)/(x+

10)2

MTH15 • GraphSketching

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m



MATLA

B C

ode% 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) =

= 1

0x(x

+8)/(

x+10

)2MTH15 • GraphSketching




MATLA

B C

ode% 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) =

10x

(x+8

)/(x+

10)2


XYfcnGraph6x6BlueGreenBkGndTemplate1306.m

-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) =

= 1

0x(x

+8)/(

x+10

)2



x-SpanInSufficent



Better Graphing GamePlan1. Find THE y-Intercept, if Any

a. Set x = 0, find yb. 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 xb. 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



Better Graphing GamePlan3. Find VERTICAL (↨) Asymptotes, If Any

a. Exist ONLY when fcn has a denomb. 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→−∞



Better Graphing GamePlanb. 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, 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 form: ↑-↓ or ↓-↑

𝒅𝟐𝒚𝒅𝒙𝟐ቚ𝒙𝑬 Sign Concavity Max or Min

POSitive Up ↑ Min NEGative Down ↓ Max

Neither (Zero) No Information Flat Spot



Better Graphing GamePlan7. Find the Inflection Pt Locations

a. 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 ↓-↓



Better Graphing GamePlan8. 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



Better Graphing GamePlan9. 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

xxxxxfy

3

434

3121

3010201020 2

2

2

2

y



Example Sketch Rational Fcn y=0:

Solving for x: Finding y(x):

1

31312120

312120

2

2

2

2

2

xx

xxxx

xxxx

22 201202120 xorxxx

2or21 xx

05105

3212221222

02523

2503211212211212

21

2

2

2

2

2

2

2

2

y

y



Example Sketch Rational Fcn The x-Intercepts

• (½,0); Multiplicity = 1 (LINE-Like)• (−2,0); Multiplicity = 2 (PARABOLA Like)

The Horizontal Intercept(s)

3

3

2

2

2

2

11

31212lim

31212limlim

xx

xxxx

xxxxy

xxx



Example Sketch Rational Fcn Continuing with the Limit

• Thus have a HORIZONTAL asymptote at y = 0

xx

xx

xx

xx

xx

xx

yxxx 3111

2112lim

31

212

limlim 2

2

2

2

2

2

21112

01010102limlim 2

2

xx

y



Example Sketch Rational Fcn To Find VERTICAL asymptote(s) set the

DeNom/Divisor = 0

• Using Zero Products

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

31031212 2

2

2

xxxxxxxy

3or1310 2 xxxx



Example Sketch Rational Fcn 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)



The Graph

-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)





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, PE Licensed Electrical & Mechanical Engineer [email protected]

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Transcript of Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]