[email protected] MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical

[email protected] • MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§2.2 Methods of

Differentiation



Review §

Any QUESTIONS About• §2.1 → Intro to Derivatives

Any QUESTIONS About HomeWork• §2.1 → HW-07

2.1

http://epubs.siam.org/doi/book/10.1137/1.9780898717761

http://www.google.com/url?sa=i&rct=j&q=applied+derivative+problems&source=images&cd=&cad=rja&docid=T8hVnVMy5xqvfM&tbnid=IvsajrDGwwhKuM:&ved=0CAUQjRw&url=http://www.wikihow.com/Take-Derivatives-in-Calculus&ei=qajVUf-HOsSLiAKtpYCoCg&bvm=bv.48705608,d.cGE&psig=AFQjCNGeAMl7sr0ofqqzZBrXPrbOE2y2vQ&ust=1373043206091958



§2.2 Learning Goals

Use the constant multiple rule, sum rule, and power rule to find derivatives

Find relative and percentage rates of change

Study rectilinear motion and the motion of a projectile

http://kmoddl.library.cornel

l.edu/resources.php?

id=1805



Rule Roster Constant Rule

• For Any Constant c

• The Derivative of any Constant is ZERO

• Prove Using Derivative Definition

For f(x) = c

Example f(x) =73• By Constant Rule

h

xfhxf

dx

dfh

0lim

0cdx

d

00

lim

lim

0

0

h

h

cc

dx

dc

h

h

073 dx

d



Rule Roster

Power Rule• For any constant

real number, n• Proof by Definition is VERY tedious, So Do

a TEST Case instead • Let F(x) = x5; then plug into Deriv-Def

– The F(x+h) & F(x)

– Then: F(x+h) − F(x)

1 nn nxxdx

d

&

4 3 2 1



Rule Roster Power Rule• Then the Limit for h→0

• Finally for n = 5

• The Power Rule WILL WORK for every other possible Test Case

432234

0510105lim xhxxhxhh

dx

dFh

0 0 0 0

145 5 nn nxxxdx

dx

dx

d



MuPAD Code



Rule Roster Constant Multiple

Rule• For Any Constant c,

and Differentiable Function f(x)

• Proof: Recall from Limit Discussion the Constant Multiplier Property:

• Thus for the Constant Multiplier

xfdx

dcxfc

dx

d

xfkxfkcxcx

limlim

xfdx

dcxfc

dx

dh

xfhxfc

h

xfhxfcxfc

dx

d

h

h

0

0

lim

lim

Q.E.D



Rule Roster

Sum Rule• If f(x) and g(x) are Differentiable, then the

Derivative of the sum of these functions:

• Proof: Recall from Limit Discussion the “Sum of Limits” Property

xgdx

dxf

dx

dxgxf

dx

d

xgxfxgxfcxcxcx

limlimlim



Rule Roster Sum Rule• Then by Deriv-Def

• thus

xgdx

dxf

dx

d

h

xghxg

h

xfhxf

h

xghxg

h

xfhxfxgxf

dx

d

hh

h

00

0

limlim

lim

Q.E.D. xg

dx

dxf

dx

dxgxf

dx

d



Derivative Rules Summarized

0 constant, a is If cdx

dc

1 constant, a is If nn xnxdx

dn

)()( constant, a is If xfdx

dcxfc

dx

dc

)()()()( xgdx

dxf

dx

dxgxf

dx

d



Derivative Rules Summarized

In other words… • The derivative of a constant function is zero• The derivative of a

constant times a function is that constant times the derivative of the function

• The derivative of the sum or difference of two functions is the sum or difference of the derivative of each function

http://www.google.com/url?sa=i&rct=j&q=derivative+rules&source=images&cd=&cad=rja&docid=LuedzTPGY1biuM&tbnid=Y26y05h7XlIWlM:&ved=0CAUQjRw&url=http://ghcimcv4u.weebly.com/section-21.html&ei=N8_VUezAN4PTiwLli4DQAg&bvm=bv.48705608,d.cGE&psig=AFQjCNEaoRiMWFNl65fF9C-aPsB_ksz3pg&ust=1373053041336538



Derivative Rules: Quick Examples

Constant Rule

Power Rule

ConstantMultiple Rule



Example Sum/Diff & Pwr Rule

Find df/dx for:

SOLUTION Use the Difference & Power Rules

xxxf 2

(difference rule)



Example Sum/Diff & Pwr Rule

Thus

15.0 2' xdx

dx

dx

dxf

25.0 5.0 x

(constant multiple rule)

(power rule)

24

225.02 5.0 x

xxxdx

d



RectiLinear (StraightLine) Motion

If the position of an Object moving in a Straight Line is described by the function s(t) then:

• The Object VELOCITY, v(t)

• The ObjectACCELERATION,a(t)

dt

dststv '

dt

dvtvta '



RectiLinear (StraightLine) Motion

Note that:• The Velocity (or Speed) of the Object is the

Rate-of-Change of the Object Position• The Acceleration of the Object is the Rate-

of-Change of the Object Velocity

To Learn MUCH MORE about Rectilinear Motion take Chabot’s PHYS4A Course (it’s very cool)

http://www.google.com/url?sa=i&rct=j&q=scott+hildreth&source=images&cd=&cad=rja&docid=Edx6UbOaMgrrHM&tbnid=ltQpknZ7074bgM:&ved=0CAUQjRw&url=http://www.chabotcollege.edu/faculty/shildreth/&ei=0tjVUbylMcS0iQKD7YCQDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH_uR6XG5Ca3qH5j8gORG8mxcVsUg&ust=1373055547062789



RectMotion: Positive/Negative For the Position

Fcn, s(t)• Negative s → object

is to LEFT of Zero Position

• Positive s → object is to RIGHT of Zero Position

For the Velocity Fcn, v(t)

• Negative v → object is moving to the LEFT

• Positive v → object is moving to the RIGHT

For the Acceleration Fcn, a(t)• Negative a → object is

SLOWING Down• Positive a → object is

SPEEDING Up

10-9 -7 -5 -3 -1 1 3 5 7 9-10 -8 -4 0 4 8-10 -2 6-6 102

ts ts



Example High Diver

A High-Diver’s height, in meters, above the surface of a pool t seconds after jumping is given by by Math Model

For this situation Determine how quickly diver is rising (or falling) after 0.2 seconds? After 1 second?

105.49.4)( 2 ttth

http://www.google.com/url?sa=i&rct=j&q=Stanford+high+dive&source=images&cd=&cad=rja&docid=b-7Wu75sCwENOM&tbnid=zR0aKX88VE39RM:&ved=0CAUQjRw&url=http://americascup.tumblr.com/post/40856599651&ei=Gt_VUfykC8OniALYyoCoDA&bvm=bv.48705608,d.cGE&psig=AFQjCNGFKa4kdQZftXX10RWmIUT9SmXcOg&ust=1373057148620890



Example High Diver

SOLUTION Assuming that the Diver Falls Straight

Down, this is then a Rect-Mtn Problem• In other Words this a

Free-Fall Problem

Use all of the Derivative rules Discussed previously to Calculate the derivative of the height function



Example High Diver

Using Derivative Rules

Thus

tvttdt

dth 105.49.4' 2

015.429.4 t

m/s of unitsin 5.48.9' tthtv



Example High Diver

Use the Derivative fcn for v(t) to find v(0.2s) & v(1s)

• The POSITIVE velocity indicates that the diver jumps UP at the Dive Start

• The NEGATIVE velocity indicates that the diver is now FALLING toward the Water

m/s 2.54 5.4)2.0(8.92.0'2.0 hv

m/s 3.5 5.418.91'1 hv



Relative & %-age Rate of Change

The Relative Rate of Change of a Quantity Q(z) with Respect to z:

The Percentage RoC is simply the Relative Rate of Change Converted to the PerCent Form• Recall that 100% of SomeThing is 1 of

SomeThing

zQdzdQ

relRoC

1

%100

1

%100RoCRoC rel%

zQ

dzdQ



Relative RoC, a.k.a. Sensitivity

Another Name for the Relative Rate of Change is “Sensitivity”

Sensitivity is a metric that measures how much a dependent Quantity changes with some change in an InDependent Quantity relative to the BaseLine-Value of the dependent Quantity



MultiVariable Sensitivty Analysis

B. Mayer, C. C. Collins, M. Walton, “Transient Analysis of Carrier Gas Saturation in Liquid Source Vapor Generators”, Journal of Vacuum Science Technolgy A, vol. 19, no.1, pp. 329-344, Jan/Feb 2001



Sensitivity: Additional Reading

For More Info on Sensitivity see• B. Mayer, “Small Signal Analysis of Source

Vapor Control Requirements for APCVD”, IEEE Transactions on Semiconductor Manufacturing, vol. 9, no. 3, pp. 344-365, 1996

• M. Refai, G. Aral, V. Kudriavtsev, B. Mayer, “Thermal Modeling for APCVD Furnace Calibration Using MATRIXx“, Electrochemical Soc. Proc., vol. 97-9, pp. 308-316, 1997



Example Rice Sensitivity The demand for rice

in the USA in 2009 approximately followed the function

• Where– p ≡ Rice Price in

$/Ton– D ≡ Rice Demand in

MegaTons

Use this Function to find the percentage rate of change in demand for rice in the United States at a price of 500 dollars per ton

http://www.google.com/url?sa=i&rct=j&q=rice+bags&source=images&cd=&cad=rja&docid=GPa8IDeXm-gJqM&tbnid=z3-U5kavjT2RCM:&ved=0CAUQjRw&url=http://tcktcktck.org/2013/01/bags-of-rice-photo/40429&ei=Ee_VUZXMMaiyiQL6yoCQCg&bvm=bv.48705608,d.cGE&psig=AFQjCNGt_se9DVU0kGCk133UaTjwn7jQ9g&ust=1373061198008578



Example Rice Sensitivity SOLUTION By %-RoC Definition

Calculate RoC at p = 500

Using Derivative Rules

pD

dpdD100RoC%

500

100500RoC 500% D

dpdD

pdp

d

dp

dD 100

2/350 p



Example Rice Sensitivity

Finally evaluate the percentage rate of change in the expression at p=500:

In other words, at a price of 500 dollars per ton demand DROPS by 0.1% per unit increase (+$1/ton) in price.

)500(

100 500

D

dpdD% 1.0



WhiteBoard Work

Problems From §2.2• P60 → Rapid

Transit

• P68 → Physical Chemistry

http://www.google.com/url?sa=i&rct=j&q=bart+transit&source=images&cd=&cad=rja&docid=dFktGAsjaVM6MM&tbnid=E0QAP25lVvQq6M:&ved=0CAUQjRw&url=http://subwaynut.com/bart/&ei=cPTVUdaRMqepiALfnIGIBA&bvm=bv.48705608,d.cGE&psig=AFQjCNFdBTyEnVJ5GDbpRGKw8ejqH9TdsQ&ust=1373062622616888

http://www.google.com/url?sa=i&rct=j&q=DeBye+physical+chemistry&source=images&cd=&cad=rja&docid=R-C0NzUbA47EQM&tbnid=iHVVzi3YN9vGuM:&ved=0CAUQjRw&url=http://izquotes.com/quote/223421&ei=0fTVUcn5Kqj7igKE2YDQAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNEIOKG2eMr8Zlw6BFNdokEsLKPtWA&ust=1373062702831190



All Done for Today

PowerRuleProof

A LOT of Missing Steps…

http://www.google.com/url?sa=i&rct=j&q=Power+Rule&source=images&cd=&cad=rja&docid=fIqwbtmDdZxh-M&tbnid=K-LPnGJumYAekM:&ved=0CAUQjRw&url=http://smallstones09p5.wikispaces.com/Fall+Week+05&ei=DUzWUYLsA6rjiAKHt4HwDw&bvm=bv.48705608,d.cGE&psig=AFQjCNFq3ukKEGmOmaop8qPLNeEJ6u7zYQ&ust=1373085009801390



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22









Documents

[email protected] MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical