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

1. Chabot Mathematics §2.2 Methods ofDifferentiation Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 2.1 Review § • Any QUESTIONS About • §2.1 → Intro to Derivatives • Any QUESTIONS About HomeWork • §2.1 → HW-07

3. §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

4. 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

5. 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

6. 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

7. MuPAD Code

8. 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

9. 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

10. Rule Roster  Sum Rule • Then by Deriv-Def • thus Q.E.D.

11. Derivative Rules Summarized

12. 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

13. Derivative Rules: Quick Examples • Constant Rule  • Power Rule  • ConstantMultiple Rule 

14. Example  Sum/Diff & Pwr Rule • Find df/dx for: • SOLUTION • Use the Difference & Power Rules (difference rule)

15. Example  Sum/Diff & Pwr Rule • Thus (constant multiple rule) (power rule)

16. 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)

17. 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)

18. 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

19. 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?

20. 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

21. Example  High Diver • Using Derivative Rules • Thus

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. Example  Rice Sensitivity • SOLUTION • By %-RoC Definition • Calculate RoC at p = 500 • Using Derivative Rules

29. 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.

30. WhiteBoard Work • Problems From §2.2 • P60 → Rapid Transit • P68 → Physical Chemistry

31. All Done for Today PowerRuleProof A LOT of Missing Steps…

32. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –