Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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Chabot Mathematics. §2.2 Methods of Differentiation. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] 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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Chabot Mathematics

§2.2 Methods ofDifferentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

2.1Review §
• §2.1 → Intro to Derivatives
• §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)

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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

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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

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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
• 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 [email protected]