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

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Chabot Mathematics

§2.2 Methods ofDifferentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

2.1

- Any QUESTIONS About
- §2.1 → Intro to Derivatives

- Any QUESTIONS About HomeWork
- §2.1 → HW-07

- 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

- 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

- 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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- 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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- 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

- 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

- Then by Deriv-Def
- thus

Q.E.D.

- 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

- Constant Rule
- Power Rule
- ConstantMultiple Rule

- Find df/dx for:
- SOLUTION
- Use the Difference & Power Rules

(difference rule)

- Thus

(constant multiple rule)

(power rule)

- 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)

- 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)

- 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

- Negativea → object is SLOWING Down
- Positivea → object is SPEEDING Up

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- 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?

- 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

- Using Derivative Rules
- Thus

- 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

- 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

- 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

- 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

- The demand for rice in the USA in 2009 approximately followed the function
- Where
- p ≡ Rice Price in $/Ton
- D ≡ Rice Demand in MegaTons

- Where

- 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

- SOLUTION
- By %-RoC Definition
- Calculate RoC at p = 500

- Using Derivative Rules

- 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.

- Problems From §2.2
- P60 → Rapid Transit
- P68 → Physical Chemistry

PowerRuleProof

A LOT of Missing Steps…