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

Chabot Mathematics. §2.1 Basics of Differentiation. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 1.6. Review §. Any QUESTIONS About §1.6 → OneSided -Limits & Continuity Any QUESTIONS About HomeWork §1.6 → HW-06. §2.1 Learning Goals.

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

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  1. Chabot Mathematics §2.1 Basics ofDifferentiation Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 1.6 Review § • Any QUESTIONS About • §1.6 → OneSided-Limits & Continuity • Any QUESTIONS About HomeWork • §1.6 → HW-06

  3. §2.1 Learning Goals • Examine slopes of tangent lines and rates of change • Define the derivative, and study its basic properties • Compute and interpret a variety of derivatives using the definition • Study the relationship between differentiability and continuity

  4. Why Calculus? • Calculus divides into the Solution of TWO Main Questions/Problems • Calculate the SLOPE of a CURVED-Line Function-Graph at any point • Find the AREA under a CURVED-Line Function-Graph between any two x-values

  5. Calculus Pioneers • Sir Issac Newton Solved the Curved-Line Slope Problem • See Newton’s MasterWorkPhilosophiaeNaturalis Principia Mathematica (Principia) • Read it for FREE: http://archive.org/download/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf • Gottfried Wilhelm von Leibniz Largely Solved the Area-Under-the-Curve Problem

  6. Calculus Pioneers • Newton (1642-1727) • Leibniz (1646-1716)

  7. Origin of Calculus • The word Calculus comes from the Greek word for PEBBLES • Pebbles were used for counting and doing simple algebra…

  8. “Calculus” by Google Answers • “A method of computation or calculation in a special notation (like logic or symbolic logic). (You'll see this at the end of high school or in college.)” • “The hard deposit of mineralized plaque that forms on the crown and/or root of the tooth. Also referred to as tartar.”

  9. “Calculus” by Google Answers • “The branch of mathematics involving derivatives and integrals.” • “The branch of mathematics that is concerned with limits and with the differentiationand integrationof functions”

  10. “Calculus” by B. Mayer • Use “Regular” Mathematics (Algebra, GeoMetry, Trigonometry) and see what happens to the Dependent quantity (usually y) when the Independent quantity (usually x) becomes one of: • Really, Really TINY • Really, Really BIG (in Absolute Value)

  11. Calculus Controversy • Who was first; Leibniz or Newton? • We’ll Do DERIVATIVES First Derivatives Integrals

  12. What is a Derivative? • A function itself • A Mathematical Operator (d/dx) • The rate of change of a function • The slope of the line tangent to the curve

  13. The TANGENT Line single point of Interest

  14. Slope of a Secant (Chord) Line • Slope, m, of Secant Line (− −) = Rise/Run

  15. Slope of a Closer Secant Line

  16. Move x Closer & Closer • Note that distance h is getting Smaller

  17. Secant Line for Decreasing h • The slope of the secant line gets closer and closer to the slope of the tangent line...

  18. Limiting Behavior • The slope of the secant lines get closer to the slope of the tangent line... ...as the values of hget closer to Zero • thisTranslates to…

  19. The Tangent Slope Definition • The Above Equation yields the SLOPE of the CURVE at the Point-of-Interest • With a Tiny bit of Algebra

  20. Example  Parabola Slope want the slope where x=2

  21. Example  Parabola Slope • Use the Slope-Calc Definition 0 0

  22. SlopeCalc ≡ DerivativeCalc • The derivative IS the slope of the line tangent to the curve (evaluated at a given point) • The Derivative (or Slope) is a LIMIT • Once you learn the rules of derivatives, you WILL forget these limit definitions • A cool site for additional explanation: • http://archives.math.utk.edu/visual.calculus/2/

  23. Delta (∆) Notation • Generally in Math the Greek letter ∆ represents a Difference (subtraction) • Recall the Slope Definition • SeeDiagramat Right

  24. Delta (∆) Notation • From The Diagram Notice that at Pt-A the Chord Slope, AB, approaches the Tangent Slope, AC, as ∆x gets smaller • Also: • Then → 0

  25. ∆→d Notation • Thus as ∆x→0 The Chord Slope of AB approaches the Tangent slope of AC • Mathematically • Now by Math Notation Convention: • Thus

  26. ∆→d Notation • The Difference between ∆x & dx: • ∆x ≡ a small but FINITE, or Calcuable, Difference • dx ≡ an Infinitesimally small, Incalcuable, Difference • ∆x is called a DIFFERENCE • dx is called a Differential • See the Diagram above for the a Geometric Comparison of • ∆x, dx, ∆y, dy

  27. Derivative is SAME as Slope • From a y = f(x) graph we see that the infinitesimal change in y resulting from an infinitesimal change in x isthe Slope at the point of interest. Generally: • The Quotient dy/dx is read as: “The DERIVATIVE of y with respect to x” • Thus “Derivative” and “Slope” are Synonymous

  28. d → Quantity AND Operator • Depending on the Context “d” can connote a quantity or an operator • Recall from before the example y = x2 • Recall the Slope Calc • We could also “take the derivative of y = x2 with respect to x using the d/dx OPERATOR

  29. d → Quantity AND Operator • dy & dx (or d?) Almost Always appears as a Quotient or Ratio • d/dx or (d/d?) acts as an OPERATOR that takes the Base-Function and “operates” on it to produce the Slope-Function; e.g.

  30. Prime Notation • Writing dy/dx takes too much work; need a Shorthand notation • By Mathematical Convention define the “Prime” Notation as • The “Prime” Notation is more compact • The “d” Notation is more mathematically Versatile • I almost always recommend the “d” form

  31. Average Rate of Change • The average rate of change of function f on the interval [a,b] is given by • Note that this is simply the Secant, or Chord, slope of a function between two points (x1,y1) = (a,f(a)) & (x2,y2) = (b,f(b))

  32. Example  Avg Rate-of-Change • For f(x) = y = x2 find the average rate of change between x = 3 (Pt-a) and x = 5 (Pt-b) • By the Chord Slope

  33. Example  Avg Rate-of-Change ChordSlope

  34. % Bruce Mayer, PE % MTH-15 • 01Jul13 % XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m % % The Limits xmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10; % The FUNCTION x1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2; x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1; % The Total Function by appending x = [x1, x2]; y = [y1, y2]; % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k', x1(end),y1(end), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', 'b',... 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),... title(['\fontsize{14}MTH15 • Bruce Mayer, PE • 2-Sided Limit',]),... annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7) hold on plot(x2(1),y2(1), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', [0.8 1 1], 'LineWidth', 3) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax]) hold off MATLAB Code

  35. Slope vs. Rate-of-Change • In general the Rate-of-Change (RoC) is simply the Ratio, or Quotient, of Two quantities. Some Examples: • Pay Rate → $/hr • Speed → miles/hr • Fuel Use → miles/gal • Paper Use → words/page • A Slope is a SPECIAL RoC where the UNITS of the Dividend and Divisor are the SAME. Example • Road Grade → Feet-rise/Feet-run • Tax Rate → $-Paid/$-Earned

  36. Example  Rice is Nice • 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 and interpret • Find the equation of the tangent line to D at p = 500.

  37. Example  Rice is Nice • SOLUTION • Using the definition of the derivative: • Clear fractions by multiplying by • Simplifying • Note the Limit is Undefined at h = 0

  38. Example  Rice is Nice • Remove the UNdefinition by multiplying by the Radical Conjugate of the Numerator:

  39. Example  Rice is Nice • Continue the Limit Evaluation

  40. Example  Rice is Nice • Run-Numbers to Find the Change in DEMAND with respect to PRICE • Unit analysis for dD/dp • Finally State: for when p = 500 the Rate of Change of Rice Demand in the USA:

  41. Example  Rice is Nice • Thus The RoC for Dw.r.t. p at p = 500: • Negative Derivative???!!! • What does this mean in the context? • Because the derivative is negative, at a unit price of $500 per ton, demand is decreasing by about 4,470 tons per $1/Ton INCREASE in unit price.

  42. Example  Rice is Nice • SOLUTION • Find the equation of the tangent line to D at p = 500 • The tangent line to a function f is defined to be the line passing through the point and having a slope equal to the derivative at that point.

  43. Example  Rice is Nice • First, find the value of D at p = 500: • So we know that the tangent line passes through the point (500, 4.47) • Next, use the derivative of D for the slope of the tangent line:

  44. Example  Rice is Nice • Finally, we use the point-slope formula for the Eqn of a Line and simplify: • The Graph ofD(p) and theTangent Lineat p = 500 on the Same Plot:

  45. Operation vs Ratio • In the Rice Problem we could easily write D’(500) as indication we were EVALUATING the derivative at p = 500 • The d notation is not so ClearCut. Are these things the SAME? • Generally They are NOT • The d/dx Operator Produces the Slope Function, not a NUMBER • Find dy/dx at x = c DOES make a Number

  46. “Evaluated at” Notation • The d/dx operator produces the Slope Function dy/dx or df/dx; e.g.: • 2x+7 is the Slope Function. It can be used to find the slope at, say, x = −5 & 4 • y’(−5) = 2(−5) + 7 = −10 + 7 = −3 • y’(4) = 2(4) + 7 = 8 + 7 = 5 • Use Eval-At Bar to Clarify a Number-Slope when using the “d” notation

  47. Eval-At BAR • To EVALUATE a derivative a specific value of the Indepent Variable Use the “Evaluated-At” Vertical BAR. • Eval-At BAR Usage → Find the value of the derivative (the slope) at x = c (c is a NUMBER): • Often the “x =” is Omitted

  48. Example: Eval-At bar • Consider the Previous f(x) Example: • Using the d notation to find the Slope (Derivative) for x = −5 & 4

  49. Continuity & Smoothness • We can now define a “smoothly” varying Function • A function f is differentiable at x=a if f’(a) is defined. • e.g.; no div by zero, no sqrt of negNo.s • IF a function is differentiable at a point, then it IS continuous at that point. • Note that being continuous at a point does NOT guarantee that the function is differentiable there. .

  50. Continuity & Smoothness • A function, f(x), is SMOOTHLY Varying at a given point, c, If and Only If df/dx Exists and: • That is, the Slopesare the SAME whenapproached fromEITHER side

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