BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE...

BMayer@ChabotCollege.edu • MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§2.3 Higher Order

Derivatives

Review §

Any QUESTIONS About• §2.2 → Techniques of Differentiation

Any QUESTIONS About HomeWork• §2.2 → HW-8

§2.3 Learning Goals

Use the product and quotient rules to find derivatives

Define and study the second derivative and higher-order derivatives

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

xfhxfxf

)()(lim'

Example Product INequality

Compute Similar-Looking Derivatives

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

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

dgxfxgxf

''' fggfgf

ProductRule Proof

Do OnWhiteBoard

Example Product Rule on

Compute the Derivativeof the Product:

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

Rule so that:

Example Product Rule on

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

22332 32 xxxxxxdx

4432 32 xxxxdx

432 5xxxdx

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:

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

Engaging the Product Rule

21001012' pppdp

xgxfxgxf

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

2.03002.02012 kp

$M 412412

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

0 0.05 0.1 0.15 0.2 0.25 0.30

p ($k/Ph)

)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

p ($k/Ph)

)MTH15 • CellPh Max Revenue

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m $0.1695k

$1.2597M

% 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.

xfdxdf

QuotientRule Proof

Do OnWhiteBoard

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:

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:

Simplifying:

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

332353

1406.064

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.

1406.01 dtdP

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

From the Previous Example

Following the Derivation Sequence

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

dmxxxm 684628 23

Conventionally (dx)2 is written as dx2

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

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

Some Examples

Deriv No. Leibniz Form LaGrange Form

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

dfxxxf 62837 324

684628 232

xfd168684 2

WhiteBoard Work

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

the Product Production Rate

All Done for Today

UNconventionalLiebniz

Notation nn

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

srsrsr 22

Alternative Quotient Rule Restate Quotient as rational Exponent,

then apply Product rule;to whit:

Putting 2nd term over common denom

1 xgxfxg

dgxgxf

dy 121

22 xgdxdf

xgdxdg

BMayer@ChabotCollege.edu MTH15_Lec-08_sec_2-3_Higher_Order_Derivatives_.pptx 1 Bruce Mayer, PE...

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