slide1
Download
Skip this Video
Download Presentation
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

Loading in 2 Seconds...

play fullscreen
1 / 36

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] - PowerPoint PPT Presentation


  • 64 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]' - sagira


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1
Chabot Mathematics

§2.2 Methods ofDifferentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

review
2.1Review §
  • 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

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 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…

slide32
Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

ad