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Chabot Mathematics. §2.2 Methods of Differentiation. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 2.1. Review §. Any QUESTIONS About §2.1 → Intro to Derivatives Any QUESTIONS About HomeWork §2.1 → HW-07. §2.2 Learning Goals.

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

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Chabot Mathematics

§2.2 Methods ofDifferentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2.1

### Review §

• Any QUESTIONS About

• §2.1 → Intro to Derivatives

• Any QUESTIONS About HomeWork

• §2.1 → HW-07

### §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.cornell.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

### 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)

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

0

0

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### 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

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

### Rule Roster  Sum Rule

• Then by Deriv-Def

• thus

Q.E.D.

### Derivative Rules Summarized

• In other words…

• The derivative of a constant function is zero

• The derivative of aconstant 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

### 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

(difference rule)

### Example  Sum/Diff & Pwr Rule

• Thus

(constant multiple rule)

(power rule)

### 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)

### 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)

### RectMotion: Positive/Negative

• For the Position Fcn, s(t)

• Negatives → object is to LEFT of Zero Position

• Positives → object is to RIGHT of Zero Position

• For the Velocity Fcn, v(t)

• Negativev → object is moving to the LEFT

• Positivev → object is moving to the RIGHT

• For the Acceleration Fcn, a(t)

• Negativea → object is SLOWING Down

• Positivea → object is SPEEDING Up

• -10

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-1

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### 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?

### 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

### 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

### 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 SomeThingis 1 of SomeThing

### 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

### MultiVariableSensitivty 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

• 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

### Example  Rice Sensitivity

• SOLUTION

• By %-RoC Definition

• Calculate RoC at p = 500

• Using Derivative Rules

### 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.

### WhiteBoard Work

• Problems From §2.2

• P60 → Rapid Transit

• P68 → Physical Chemistry

### All Done for Today

PowerRuleProof

A LOT of Missing Steps…

Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu