[email protected] MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical

[email protected] • MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§2.3 Higher Order

Derivatives



Review §

Any QUESTIONS About• §2.2 → Techniques of Differentiation

Any QUESTIONS About HomeWork• §2.2 → HW-8

2.2

http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=LuedzTPGY1biuM&tbnid=Y26y05h7XlIWlM:&ved=0CAUQjRw&url=http://ghcimcv4u.weebly.com/section-21.html&ei=bfPWUe3hB-reigLNzoHwDQ&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906



§2.3 Learning Goals

Use the product and quotient rules to find derivatives

Define and study the second derivative and higher-order derivatives

http://www.google.com/url?sa=i&rct=j&q=QUOTIENT+RULE&source=images&cd=&cad=rja&docid=AIGV-0zZPInkKM&tbnid=31JwoknTMh3VPM:&ved=0CAUQjRw&url=https://calculusclass1.wikispaces.com/Quotient+Rule&ei=1PTWUbvGKMjziQKhoIGoDQ&bvm=bv.48705608,d.cGE&psig=AFQjCNHnTsLUwhXakokMsg9dXsdoqKn6Yg&ust=1373128244380300



Product INequality

The Derivative Defintion (at right) is NONLinear Such That:

In other words, the derivative of a product of functions does NOT EQUAL the Product of the individual Derivatives

h

xfhxfxf

h

)()(lim'

0

xg

dx

dxf

dx

dxgxf

dx

d



Example Product INequality

Compute Similar-Looking Derivatives

&

Notice that the two expressions, 5x4 & 6x3, are NOT EQUAL

45x

36x



Rule Roster – Product Rule

If f(x) and g(x) are differentiable at x, then so is their product, f(x)·g(x), and

Or in LaGrange Notation

The Summary Statement:• The 1st times the Derivative of the 2nd Plus

the 2nd times the Derivative of the 1st

dx

dfxg

dx

dgxfxgxf

dx

d

''' fggfgf



ProductRule Proof

Do OnWhiteBoard

http://www.google.com/url?sa=i&rct=j&q=whiteBoard&source=images&cd=&cad=rja&docid=fIN_eRoWHj0G0M&tbnid=b1OWxS6TRyGRJM:&ved=0CAUQjRw&url=http://www.beyond.com/articles/timeline-of-training-technology-part-2-1970s-11106-article.html&ei=uyHXUdnbBsLNiwK53oGwBA&bvm=bv.48705608,d.cGE&psig=AFQjCNHffTCfKrbIKlaSGatZAQ7VO-BdVQ&ust=1373139714877810



Example Product Rule on

Compute the Derivativeof the Product:

SOLUTION Let: f(x) = x2 & g(x) = x3 in the Product

Rule so that:

32 xx

32 xx



Example Product Rule on

Or:

This is the SAME as the correct answer in the Previous Example

32 xx

22332 32 xxxxxxdx

d

4432 32 xxxxdx

d

432 5xxxdx

d



Example CellPhone Revenue

A Smart Industrial Engineer at Apple© Develops a Model Math Function for the Demand for SmartPhones:

• Where–D ≡ Phone-Demand in k-Phones– p ≡ Phone-Price in $k

21001012 pppD




Use the IE’s Demand Model to Find At what rate is revenue changing With Respect To (W.R.T.) price when Selling phones at 0.2 $k ($200 per phone)?

SOLUTION First construct a revenue function as the

product of the price per phone and number of phones sold:

pDpPR




Subbing for D(p) find for R(p):

• Note that R has units of ($k/Ph)·(kPh) = $M– i.e.; R has units of MegaBucks

Recall the RoC is simply the Derivative• Find dR/dp using the product Rule

21001012 ppppDppR

21001012 pppdp

d

dp

dR




Engaging the Product Rule

21001012' pppdp

dpR

dp

dpR

xgxfxgxf

pppdp

dpp

dp

dp

' '

10010121001012 22




Next determine the rate of change in revenue at a unit price of $200.

In other words need to find dR/dp at a price of $0.2k

pppppR 2001010010121' 2 22 200101001012 pppp

.3002012 2pp




23002012 ppdp

dR

2

2.0$

2.03002.02012 kp

dp

dR

Ph$k

$M 412412

2.0$

kp

dp

dR




The Calculation Shows

Thus we can say that at a Selling Price of $0.2k per phone Revenue will DEcrease $4,000 for every $1 INcrease in the Phone Price

Ph$

$k4

ph$k

$M 4

2.0$

kp

dp

dR




0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

p ($k/Ph)

R (

$M

)MTH15 • CellPh Revenue Sensitivity

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m

RoC (Sensitivity) is Tangent Line Slope




0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

p ($k/Ph)

R (

$M

)MTH15 • CellPh Max Revenue

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m $0.1695k

$1.2597M



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 05Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 0.3; ymin =0; ymax = 1.4;% The FUNCTIONx = linspace(xmin,xmax,500); y1 = x.*(12-10*x-100*x.^2); y2 = -4*(x-.2) +1.2% % 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,y1, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 • CellPh Revenue Sensitivity',]),... annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)hold onplot(x,y2, '-- m', 0.2,1.2, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2)set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax])hold off%disp('showning first plot - HIT ANY KEY to continue')pauseaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y1, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}p ($k/Ph)'), ylabel('\fontsize{14}R ($M)'),... title(['\fontsize{16}MTH15 • CellPh Max Revenue',]),... annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraph6x6BlueGreenBkGndTemplate1306.m','FontSize',7)hold onplot([0.1695,0.1695], [0,1.2597], '-- m', [0,0.1695], [1.2597,1.2597], '-- m', 'LineWidth', 2)set(gca,'XTick',[xmin:.05:xmax]); set(gca,'YTick',[ymin:.2:ymax])%[C,I] = max(y1)x(I)



Rule Roster – Quotient Rule

If f(x) and g(x) are differentiable functions with g(x) ≠ 0, then

In particular, the derivative of the quotient of f(x) and g(x) is NOT df/dx divided by dg/dx.

xg

dxdg

xfdxdf

xg

xg

xf

dx

d2



QuotientRule Proof

Do OnWhiteBoard

http://www.google.com/url?sa=i&rct=j&q=whiteBoard&source=images&cd=&cad=rja&docid=fIN_eRoWHj0G0M&tbnid=b1OWxS6TRyGRJM:&ved=0CAUQjRw&url=http://www.beyond.com/articles/timeline-of-training-technology-part-2-1970s-11106-article.html&ei=uyHXUdnbBsLNiwK53oGwBA&bvm=bv.48705608,d.cGE&psig=AFQjCNHffTCfKrbIKlaSGatZAQ7VO-BdVQ&ust=1373139714877810



Example RoC in a Population

One population model for deer on an island suggests that t years after initial observation, the population • Where P is the fraction of the carrying

capacity on the island. – e.g.; P(0) = 2/5 = 0.4, meaning 40% of the

Island’s total carrying capacity

Find, and Interpret the Meaning of:

53

23

t

ttP

1tdt

dP



Example RoC in a Population SOLUTION The function’s formula is a ratio of

expressions containing variables (and there’s no nice way to simplify the fraction), so use the quotient rule:

t

t

dt

d

dt

dP

35

32




Simplifying:

Now need to compute P’(1) and interpret the result

235

332353

t

tt

dt

dP

1406.064

9

)1(35

91' 2

1

dt

dPP




Units Analysis for dP/dt

Thus the Interpretation of

After 1 year the Deer population is growing at a rate of about 14.06% of the carrying capacity per year.

year

CC-%

t

P

dt

dP

1406.01 dtdP

http://www.google.com/url?sa=i&rct=j&q=deer&source=images&cd=&cad=rja&docid=HaCFhgyV8VJk4M&tbnid=3WCR977R2pIp1M:&ved=0CAUQjRw&url=http://animals.nationalgeographic.com/animals/mammals/white-tailed-deer/&ei=qzPXUbnbLIbmigLAzoHwCw&bvm=bv.48705608,d.cGE&psig=AFQjCNGXAexXFLy5WzwDEHbirzzdGVRgLg&ust=1373144343360475



Higher Order Derivatives

Q) What is the Derivative of a Derivative? A) Just another Function Quick Example recalling that the 1st

Derivative is just the Slope, m

The Derivative of the Slope is Called the “Curvature” or “Concavity”

xmxxdx

dfxxxf 62837 324

xCdx

xdm




From the Previous Example

Following the Derivation Sequence

If we “fudge” and treat the differentials “d” and “dx” as algebraic quantities…

xCxdx

dmxxxm 684628 23

xf

dx

d

dx

d

dx

df

dx

dm

dx

d

dx

dmxC




Then

Thus

Conventionally (dx)2 is written as dx2

Thus if y = f(x) the 2nd Derivative of y W.R.T. x:

xfdx

dxf

dxdx

ddxf

dx

d

dx

dxf

dx

d

dx

d2

2

2

2

2

2

2

2

2

2

1 dx

fd

dx

xfdxf

dx

dxf

dx

d

2

2

2

2

dx

yd

dx

ydy

dx

d

dx

d




In general the conventional notation for the nth derivative of y W.R.T. x

Some Examples

n

n

dx

yd

Deriv No. Leibniz Form LaGrange Form

First

𝑑𝑦𝑑𝑥 𝑦′ Second 𝑑2𝑦𝑑𝑥2 𝑦′′ Third 𝑑3𝑦𝑑𝑥3 𝑦′′ Fourth 𝑑3𝑦𝑑𝑥3 𝑦′′ nth 𝑑𝑛𝑦𝑑𝑥𝑛 𝑦ሺ𝑛ሻ




Back to the Previous Example

Then the 2nd derivative

Then the 3rd derivative

xxdx

dfxxxf 62837 324

684628 232

2

xxx

dx

d

dx

df

dx

d

dx

xfd

xxdx

d

dx

fd

dx

d

dx

xfd168684 2

2

2

3

3



WhiteBoard Work

Problem From §2.3• P54 → Profit Sensitivity With Respect to

the Product Production Rate

http://www.google.com/url?sa=i&rct=j&q=production+rate&source=images&cd=&cad=rja&docid=BVtE6lP1Yd5ymM&tbnid=rmULsy5VnEsWyM:&ved=0CAUQjRw&url=http://www.aviationnews.eu/2013/01/29/boeing-737-program-starts-building-at-higher-production-rate/&ei=sFXXUdjEI8j3iwKvgYGQCg&bvm=bv.48705608,d.cGE&psig=AFQjCNHuawMC7X7YQoE-MIhE9OTkcBdtsg&ust=1373153002305685



All Done for Today

UNconventionalLiebniz

Notation nn

n

n

dy

yd

dy

yd



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22

































Alternative Quotient Rule Restate Quotient as rational Exponent,

then apply Product rule;to whit:

Then

Putting 2nd term over common denom

1 xgxfxg

xfxy

dx

dfxg

dx

dgxgxf

dx

dy 121

22 xgdxdf

xg

xgdxdg

xf

xg

xf

dx

d

dx

dy



http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=nseUoaK-vZAUSM&tbnid=esGz0hOxLgT1pM:&ved=0CAUQjRw&url=http://betterexplained.com/articles/derivatives-product-power-chain/&ei=tvLWUcHABsmdiAKFpYDgBQ&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

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=higher+order+derivatives+worksheet&source=images&cd=&cad=rja&docid=UzYAWOYbX2jrHM&tbnid=eEfIS0ov0gspKM:&ved=0CAUQjRw&url=http://www.hollywoodhighschool.net/apps/pages/index.jsp?uREC_ID=116947&type=u&pREC_ID=210421&ei=m_TWUYr3O8qoigLJ3YHIAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNHYr8PZTh_Skq5LPhpN-8lENIexrA&ust=1373128179068489

Documents

[email protected] MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical