Chabot Mathematics
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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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Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics

§2.2 Methods ofDifferentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


Review

2.1

Review §

  • Any QUESTIONS About

    • §2.1 → Intro to Derivatives

  • Any QUESTIONS About HomeWork

    • §2.1 → HW-07


2 2 learning goals

§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

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 roster1

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

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

0

0


Mupad code

MuPAD Code


Rule roster2

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 roster3

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

Rule Roster  Sum Rule

  • Then by Deriv-Def

  • thus

Q.E.D.


Derivative rules summarized

Derivative Rules Summarized


Derivative rules summarized1

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

Derivative Rules: Quick Examples

  • Constant Rule 

  • Power Rule 

  • ConstantMultiple Rule 


Example sum diff pwr rule

Example  Sum/Diff & Pwr Rule

  • Find df/dx for:

  • SOLUTION

  • Use the Difference & Power Rules

(difference rule)


Example sum diff pwr rule1

Example  Sum/Diff & Pwr Rule

  • Thus

(constant multiple rule)

(power rule)


Rectilinear straightline motion

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 motion1

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

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

    -10

    -9

    -8

    -7

    -6

    -5

    -4

    -3

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    10


    Example high diver

    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 diver1

    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 diver2

    Example  High Diver

    • Using Derivative Rules

    • Thus


    Example high diver3

    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

    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

    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

    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


    Sensitivity additional reading

    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

    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 sensitivity1

    Example  Rice Sensitivity

    • SOLUTION

    • By %-RoC Definition

    • Calculate RoC at p = 500

    • Using Derivative Rules


    Example rice sensitivity2

    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

    WhiteBoard Work

    • Problems From §2.2

      • P60 → Rapid Transit

      • P68 → Physical Chemistry


    All done for today

    All Done for Today

    PowerRuleProof

    A LOT of Missing Steps…


    Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

    Chabot Mathematics

    Appendix

    Bruce Mayer, PE

    Licensed Electrical & Mechanical [email protected]


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