Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[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



MATLA

B C

ode% 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



Example Graphical Concavity 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).



Example Graphical Concavity To determine where the function is

INcreasing, we look for the graph to “Rise to the Right (RR)”

Rising



Example Graphical Concavity Similarly, the function is DEcreasing

where the graph “Falls to the Right (FR)”:

Falling



Example Graphical Concavity 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



Example Graphical Concavity 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



Example Graphical Concavity 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



Example Graphical Concavity The graph is “curved DOWNward” for

values of x on the outer edges of the domain.

f is ConCave DOWN f is ConCave DOWN



Example Graphical Concavity 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



MATLA

B C

ode% 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

xxxf

0' xfdxdf

2

2

)1(12)1(

x

xxxdxdf



Example Apply 2nd Deriv Test 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

xxdxdf



Example Apply 2nd Deriv Test 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

12

xxx

dxd

dxyd

4

22

11122221

x

xxxxx



Example Apply 2nd Deriv Test 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

11122221

x

xxxxxdxfd

3

2

2

2

11222211

xx

xxxxxdxfd

3

2

2

2

122221

x

xxxxdxfd



Example Apply 2nd Deriv Test Now expand BiNomials:

Now Check Value of f’’’(0) & f’’’(−2) 3

22

2

2

1422222

x

xxxxxdxfd

3)1(2

x

210

20''

212

22''

30

2

2

32

2

2

x

x

dxfdf

dxfdf 0

0



Example Apply 2nd Deriv Test 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

02

2

22

2

x

x

dxfd

dxfd



Example Apply 2nd Deriv Test 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



Example Dome-Diagram 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




Example Dome-Diagram The

FcnPlotShowingInflectionPoint 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) =

3x5 -

5x4 -

1

MTH15 • Dome-Diagram

(1,−3)



MATLA

B C

ode% 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



Example Population Growth 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



Example Population Growth 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

tPdtd

dtd

3t



Example Population Growth 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 tttP

dtdP

10)3(18)3(33' 2

3

PdtdP

t

171054273'3

PdtdP

t



Example Population Growth 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

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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