[email protected] MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

[email protected] • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§3.2 Concavity

& Inflection



Review §

Any QUESTIONS About• §3.1 → Relative Extrema

Any QUESTIONS About HomeWork• §3.1 →

HW-13

3.1

http://www.google.com/url?sa=i&rct=j&q=relative+extrema+of+a+function&source=images&cd=&cad=rja&docid=Jk9C15lRbTeR_M&tbnid=46FkbtmfG_JEoM:&ved=0CAUQjRw&url=http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap03/section03.htm&ei=VM3eUfCEE6qfiQKFrIGoBg&bvm=bv.48705608,d.cGE&psig=AFQjCNETiKTT033nRi_5KJkrhn3WjmAx4w&ust=1373642329390150



§3.2 Learning Goals

Introduce Concavity (a.k.a. Curvature) Use the sign of the second derivative to

find intervals of concavity Locate and examine

inflection points Apply the second

derivatives test for relative extrema

http://www.google.com/url?sa=i&rct=j&q=andy+grove+inflection+point&source=images&cd=&cad=rja&docid=kZ9FVInxCqBWmM&tbnid=wwrv_hb3vTiGOM:&ved=0CAUQjRw&url=http://www.lizeversoll.com/2011/01/24/inflection-point/&ei=zs_eUcuSG86ajAKM0oDIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNFbAUk3XrHXZIzjGtpc3pVSQj9bww&ust=1373642867342925



ConCavity Described

Concavity quantifies the Slope-Value Trend (Sign & Magnitude) of a fcn when moving Left→Right on the fcn Graph

1 2 3 4-5

-4

-3

-2

-1

0

1

2

3

Position, x

m =

df/d

x

MTH15 • BLUE

1 2 3 4-5

-4

-3

-2

-1

0

1

2

3

Position, x

MTH15 • RED

m≈+

2.2

m≈0

m≈−1.4

m≈−4.4

m≈−4.4

m≈−1.4

m≈+

2.2

http://www.google.com/url?sa=i&rct=j&q=CONCAVITY&source=images&cd=&cad=rja&docid=WRY96fw190tugM&tbnid=5rsyf4SakP_BoM:&ved=0CAUQjRw&url=http://www.nabla.hr/Z_IntermediateAlgebraIntroductionToFunctCont_3.htm&ei=bdHeUdipEIigigLtpYGIDQ&bvm=bv.48705608,d.cGE&psig=AFQjCNEF3Zf8TgQr8J9loNmNs7jS79LAbQ&ust=1373643269301465



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 •11Jul133% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m% % The datablue =[2.2 0 -1.4 -4.4]red = [-4.4 -1.4 0 2.2]%% the 6x6 Plotaxes; set(gca,'FontSize',12);subplot(1,2,1)bar(blue, 'b'), grid, xlabel('\fontsize{14}Position, x'), ylabel('\fontsize{14}m = df/dx'),... title(['\fontsize{16}MTH15 • BLUE',]), axis([0 5 -5,3])subplot(1,2,2)bar(red, 'r'), grid, xlabel('\fontsize{14}Position, x'), axis([0 5 -5,3]),... title(['\fontsize{16}MTH15 • RED',])set(get(gco,'BaseLine'),'LineWidth',4,'LineStyle',':')



ConCavity Defined

A differentiable function f on a < x < b is said to be:

… concave DOWN (↓) if df/dx is DEcreasing on the interval

…concave up if df/dx is INcreasing on the interval.

http://www.google.com/url?sa=i&rct=j&q=CONCAVITY&source=images&cd=&cad=rja&docid=F2D7Bc5_kcey3M&tbnid=RvTtNZNSiSewtM:&ved=0CAUQjRw&url=http://www.analyzemath.com/calculus/concavity/concavity.html&ei=iOTeUcDrKOThiALCioHYBw&psig=AFQjCNG-r-LLFyd4RvBuevArxEBRHlf_gg&ust=1373648386645706

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=F2D7Bc5_kcey3M&tbnid=DNlbRWzuhHO-AM:&ved=&url=http://www.analyzemath.com/calculus/concavity/concavity.html&ei=v-TeUZXoO-S6iwK5vYG4Dg&psig=AFQjCNGVg4t9PBtG6hsniniU9TMosBsoxQ&ust=1373648448018536



Example Graphical Concavity

Consider the function f given in the graph and defined on the interval (−4,4).

Approximate all intervals on which the function is INcreasing, DEcreasing, concave up, or concave down




SOLUTION Because we have NO equation for the

function, we need to use our best judgment: • around where the

graph changes directions (increasing/decreasing)

• where the derivative of the graph changes directions (concave up or down).




To determine where the function is INcreasing, we look for the graph to “Rise to the Right (RR)”

Rising




Similarly, the function is DEcreasing where the graph “Falls to the Right (FR)”:

Falling




Conclude that f is increasing on the interval (0,4) and decreasing on the interval (−4,0)

Now ExamineConcavity.

Falling to Rt Rising to Rt




A function is concave UP wherever its derivative is INcreasing. Visually, we look for where the graph is“curved upward”, or “Bowl-Shaped”Similarly, A function is concave DOWN wherever its derivative is DEcreasing. Visually, we look for where the graph is “curved downward”, or “Dome-Shaped”

http://www.google.com/url?sa=i&rct=j&q=bowl&source=images&cd=&cad=rja&docid=repVicNyYatbuM&tbnid=7tI9LtS0UxVVXM:&ved=0CAUQjRw&url=http://www.flyingcurls.com/Cothren2.html&ei=m-zeUfnrN4rWiwLG-oHoBQ&psig=AFQjCNHvCE-7lhvdvAYcxxvCh5fTc3AQ7g&ust=1373650361649189

http://www.google.com/url?sa=i&rct=j&q=dome&source=images&cd=&cad=rja&docid=M4At32qZT446SM&tbnid=NckOlur7yIAYJM:&ved=0CAUQjRw&url=http://sensingarchitecture.com/2675/10-ways-to-design-architecture-that-defies-gravity-slideshow/dome-night-structure-image-2/&ei=Xe3eUbeXBKGciQLu84HwDQ&psig=AFQjCNFi_dyAjugLXhZl65u05Y7j7MX5Vw&ust=1373650568273100




The graph is “curved UPward” for values of x near zero, and might guess the curvature to be positive between −1 & 1

f is ConCave UP




The graph is “curved DOWNward” for values of x on the outer edges of the domain.

f is ConCave DOWN f is ConCave DOWN




Thus the function is concave UP approximately on the interval (−1,1) and concave DOWN on the intervals (−4, −1) & (1,4)

f is ConCave UPf is ConCave DOWN f is ConCave DOWN



Inflection Point Defined

A function has an inflection point at x=a if f is continuous and the CONCAVITY of f CHANGES at Pt-a

-2 -1 0 1 2 3 4 5 6 7 8 9-50

-40

-30

-20

-10

0

10

20

30

40

50

x

y =

f(x)

MTH15 • Inflection Point

ConCave DOWN

ConCave UP

InflectionPoint



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 10Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = -2; xmax = 9; ymin =-50; ymax = 50;% The FUNCTIONx = linspace(xmin,xmax,1000); y =(x-4).^3/4 + (x+5).^2/7;yOf4 = (4-4).^3/4 + (4+5).^2/7 % % 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', 5),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Inflection Point',])hold onplot(4, yOf4, 'd r', 'MarkerSize', 9,'MarkerFaceColor', 'r', 'LineWidth', 2)plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:10:ymax])hold off



Example Inflection Graphically

The function shown above has TWO inflection points.

change from concave

down to up

change from concave

up to down



2nd Derivative Test

Consider a function for Which is Defined on some interval containing a critical Point (Recall that ) Then:• If , then is Concave UP at so is a

Relative MIN• If , then is Concave DOWN at so is

a Relative MAX



Example Apply 2nd Deriv Test

Use the 2nd Derivative Test to Find and classify all critical points for the Function

SOLUTION Find the

critical points by solving:

1

2

x

xxf

0' xfdx

df

2

2

)1(

12)1(

x

xxx

dx

df




By Zero-Products:

Also need to check for values of x that make the derivative undefined.• ReCall the

1st Derivative: • Thus df/dx is UNdefined for x = −1, But the

ORIGINAL function is ALSO Undefined at the this value– Thus there is NO Critical Point at x = −1

2OR020 xxxx

2

2

)1(

2

x

xx

dx

df




Thus the only critical points are at −2 & 0 Now use the second derivative test to

determine whether each is a MAXimum or MINimum (or if the test is InConclusive):

2

2

2

2

1

2

x

xx

dx

d

dx

yd

4

22

1

1122221

x

xxxxx




Before expanding the BiNomials, note that the numerator and denominator can be simplified by removing a common factor of (x+1) from all terms:

4

22

2

2

1

1122221

x

xxxxx

dx

fd

3

2

2

2

11

222211

xx

xxxxx

dx

fd

3

2

2

2

1

22221

x

xxxx

dx

fd




Now expand BiNomials:

Now Check Value of f’’’(0) & f’’’(−2)

3

22

2

2

1

422222

x

xxxxx

dx

fd3)1(

2

x

210

20''

212

22''

3

0

2

2

3

2

2

2

x

x

dx

fdf

dx

fdf 0

0




The 2nd Derivative is NEGATIVE at x = −2• Thus the orginal fcn is ConCave

DOWN at x = −2, and aRelative MAX exists at this Pt

Conversely, 2nd Derivative is POSITIVE at x = 0• Thus the orginal fcn is ConCave UP at x = 0

and a Relative MIN exists at this Pt

2

2

0

2

2

2

2

2

x

x

dx

fd

dx

fd




Confirm by Plot → Note the relative

MINimum at 0, relative MAXimumat −2, and a vertical asymptote where the function is undefined at x=−1 (although the vertical line is not part of the graph of the function)



ConCavity Sign Chart A form of the df/dx (Slope) Sign Chart

(Direction-Diagram) Analysis Can be Applied to d2f/dx2 (ConCavity)

Call the ConCavity Sign-Charts “Dome-Diagrams” for INFLECTION Analysis

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection



Example Dome-Diagram

Find All Inflection Points for • Notes on this (and all other) PolyNomial

Function exists for ALL x

Use the ENGR25 Computer Algebra System, MuPAD, to find • Derivatives• Critical Points

153 45 xxxfy



Example Dome-Diagram The Derivatives

The Critical Points

The ConCavity Values Between Break Pts• At x = −1

• At x = ½

• At x = ½



MyPAD Code




Draw Dome-Diagram

The ConCavity Does NOT change at 0, but it DOES at 1• Since Inflection requires Change, the

only Inflection-Pt occurs at x = 1

0 1

−−−−−−−−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points NO





TheFcnPlotShowingInflectionPoint at(1,y(1))= (1,−3)

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-15

-10

-5

0

5

10

15

x

y =

f(x)

= 3

x5 - 5

x4 - 1

MTH15 • Dome-Diagram

(1,−3)



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 11Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = -1.5; xmax = 2.5; ymin =-15; ymax = 15;% The FUNCTIONx = linspace(xmin,xmax,1000); y =3*x.^5 - 5*x.^4 - 1;% % 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', 5),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 3x^5 - 5x^4 - 1'),... title(['\fontsize{16}MTH15 • Dome-Diagram',])hold onplot(1,-3, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2)plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:5:ymax])hold off



Example Population Growth

A population model finds that the number of people, P, living in a city, in kPeople, t years after the beginning of 2010 will be:

Questions • In what year will the population be

decreasing most rapidly? • What will be the population at that time?

105109 23 ttttP




SOLUTION: “Decreasing most rapidly” is a phrase

that requires some examination. “Decreasing” suggests a negative derivative.

“Decreasing most rapidly” means a value for which the negative derivative is as negative as possible. In other words, where the derivative is a MIN




Need to find relative minima of functions (derivative functions are no exception) where the rate of change is equal to 0.

“Rate of change in the population derivative, set equal to zero” TRANSLATES mathematically to

0

tPdt

d

dt

d

3t




The only time at which the second derivative of P is equal to zero is the beginning of 2013.• Need to verify that the derivative is, in fact,

negative at that point:

10183' 2 tttPdt

dP

10)3(18)3(33' 2

3

Pdt

dP

t

171054273'3

Pdt

dP

t




Thus the function is decreasing most rapidly at the inflection point at the beginning of 2013:

The Model Predicts 2013 Population:

x

Peoplek 81105)3(10)3(9)3(3 23 P



WhiteBoard Work

Problems From §3.2• P45 → Sketch Graph using General

Description• P66 → Spreading a Rumor

http://www.google.com/url?sa=i&rct=j&q=rumor+mill&source=images&cd=&cad=rja&docid=EJJMwlEQuPC0sM&tbnid=EqRv2ScPIXx6DM:&ved=0CAUQjRw&url=http://kentuckysportsradio.com/?p=109228&ei=BYvgUcKFPKTKiwL9xYGIBQ&bvm=bv.48705608,d.cGE&psig=AFQjCNEw7VDvrOw_bCefnVzft9_vo2HHqw&ust=1373756475137976



All Done for Today

RememgeringConCavity:

cUP & frOWN

http://www.google.com/url?sa=i&rct=j&q=concavity&source=images&cd=&cad=rja&docid=bKMDwJOUShu37M&tbnid=u0Lsq9v4Jt2rzM:&ved=0CAUQjRw&url=http://mathwithbaddrawings.com/2013/05/30/how-to-remember-concavity/&ei=kz_fUaThAYTiiAKuiIG4DA&bvm=bv.49147516,d.cGE&psig=AFQjCNEPKcN1t-oI4Z8jN_qllv8IGpVOZg&ust=1373669229784687

http://www.google.com/url?sa=i&rct=j&q=concavity&source=images&cd=&cad=rja&docid=bKMDwJOUShu37M&tbnid=u0Lsq9v4Jt2rzM:&ved=0CAUQjRw&url=http://mathwithbaddrawings.com/2013/05/30/how-to-remember-concavity/&ei=kz_fUaThAYTiiAKuiIG4DA&bvm=bv.49147516,d.cGE&psig=AFQjCNEPKcN1t-oI4Z8jN_qllv8IGpVOZg&ust=1373669229784687



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO




Max/Min Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

Slope

df/dx Sign

Critical (Break)Points Max NO

Max/MinMin















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

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=wSGtOAIrRxIoIM&tbnid=l4Q1f5xFVNRraM:&ved=0CAUQjRw&url=http://www.cantonschools.org/~lforastiere/ap%20calculus?Templates=Printable&ei=2TLcUY-gOa2UjALp4IHgDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=nuuUeFKHM5pqjM&tbnid=x7raEhpCGJ8edM:&ved=0CAUQjRw&url=http://designatedderiver.wikispaces.com/Games+and+fun+stuff&ei=IzPcUYPxBaOLiwLjs4GYAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

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

[email protected] MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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