MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §2.2 Methods of Differentiation

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.1 → Intro to Derivatives  Any QUESTIONS About HomeWork §2.1 → HW

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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 cornell.edu/resource s.php?id=1805

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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) = x 5 ; then plug into Deriv-Def –The F(x+h) & F(x) –Then: F(x+h) − F(x) 4321

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 6 Bruce Mayer, PE Chabot College Mathematics 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 0000

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 7 Bruce Mayer, PE Chabot College Mathematics MuPAD Code

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 9 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Rule Roster  Sum Rule Then by Deriv-Def thus

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Derivative Rules Summarized

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 12 Bruce Mayer, PE Chabot College Mathematics Derivative Rules Summarized  In other words… The derivative of a constant function is zero The derivative of a constant 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 13 Bruce Mayer, PE Chabot College Mathematics Derivative Rules: Quick Examples  Constant Rule   Power Rule   Constant Multiple Rule 

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Sum/Diff & Pwr Rule  Find df/dx for:  SOLUTION  Use the Difference & Power Rules (difference rule)

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Sum/Diff & Pwr Rule  Thus (constant multiple rule) (power rule)

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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 Object ACCELERATION, a(t)

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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)

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 18 Bruce Mayer, PE Chabot College Mathematics RectMotion: Positive/Negative  For the Position Fcn, s(t) Negative s → object is to LEFT of Zero Position Positive s → object is to RIGHT of Zero Position  For the Velocity Fcn, v(t) Negative v → object is moving to the LEFT Positive v → object is moving to the RIGHT  For the Acceleration Fcn, a(t) Negative a → object is SLOWING Down Positive a → object is SPEEDING Up

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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?

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  High Diver  Using Derivative Rules  Thus

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 22 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 23 Bruce Mayer, PE Chabot College Mathematics 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 SomeThing is 1 of SomeThing

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 24 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 25 Bruce Mayer, PE Chabot College Mathematics MultiVariable Sensitivty 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 , Jan/Feb 2001

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 26 Bruce Mayer, PE Chabot College Mathematics 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 , 1996 M. Refai, G. Aral, V. Kudriavtsev, B. Mayer, “Thermal Modeling for APCVD Furnace Calibration Using MATRIXx“, Electrochemical Soc. Proc., vol. 97-9, pp , 1997

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Rice Sensitivity  SOLUTION  By %-RoC Definition  Calculate RoC at p = 500  Using Derivative Rules

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 29 Bruce Mayer, PE Chabot College Mathematics 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.

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 30 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.2 P60 → Rapid Transit P68 → Physical Chemistry

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 31 Bruce Mayer, PE Chabot College Mathematics All Done for Today Power Rule Proof A LOT of Missing Steps…

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 32 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 33 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 34 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 35 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-07_sec_2-2_Differeniatation-Methods_.pptx 36 Bruce Mayer, PE Chabot College Mathematics