1. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.1 Basics of Differentiation

2. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.6 → OneSided-Limits & Continuity  Any QUESTIONS About HomeWork §1.6 → HW-06 1.6

3. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 4 Bruce Mayer, PE Chabot College Mathematics Why Calculus?  Calculus divides into the Solution of TWO Main Questions/Problems 1.Calculate the SLOPE of a CURVED-Line Function-Graph at any point 2.Find the AREA under a CURVED-Line Function-Graph between any two x-values

5. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 5 Bruce Mayer, PE Chabot College Mathematics Calculus Pioneers  Sir Issac Newton Solved the Curved- Line Slope Problem See Newton’s MasterWork Philosophiae Naturalis Principia Mathematica (Principia) –Read it for FREE: http://archive.org/download/newtonspmathema0 0newtrich/newtonspmathema00newtrich.pdf http://archive.org/download/newtonspmathema0 0newtrich/newtonspmathema00newtrich.pdf  Gottfried Wilhelm von Leibniz Largely Solved the Area-Under-the-Curve Problem

6. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 6 Bruce Mayer, PE Chabot College Mathematics Calculus Pioneers  Newton (1642-1727)  Leibniz (1646-1716)

7. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Origin of Calculus  The word Calculus comes from the Greek word for PEBBLES  Pebbles were used for counting and doing simple algebra…

8. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 8 Bruce Mayer, PE Chabot College Mathematics “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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 9 Bruce Mayer, PE Chabot College Mathematics “Calculus” by Google Answers  “The branch of mathematics involving derivatives and integrals.”  “The branch of mathematics that is concerned with limits and with the differentiation and integration of functions”

10. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 10 Bruce Mayer, PE Chabot College Mathematics “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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 11 Bruce Mayer, PE Chabot College Mathematics Calculus Controversy  Who was first; Leibniz or Newton?  We’ll Do DERIVATIVES First DerivativesIntegrals

12. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 13 Bruce Mayer, PE Chabot College Mathematics The TANGENT Line single point of Interest

14. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 14 Bruce Mayer, PE Chabot College Mathematics Slope of a Secant (Chord) Line  Slope, m, of Secant Line (− −) = Rise/Run

15. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 15 Bruce Mayer, PE Chabot College Mathematics Slope of a Closer Secant Line

16. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 16 Bruce Mayer, PE Chabot College Mathematics Move x Closer & Closer  Note that distance h is getting Smaller

17. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 17 Bruce Mayer, PE Chabot College Mathematics Secant Line for Decreasing h  The slope of the secant line gets closer and closer to the slope of the tangent line...

18. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 18 Bruce Mayer, PE Chabot College Mathematics Limiting Behavior  The slope of the secant lines get closer to the slope of the tangent line......as the values of h get closer to Zero  this Translates to…

19. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Parabola Slope want the slope where x=2

21. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Parabola Slope  Use the Slope-Calc Definition 0 0

22. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 22 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 23 Bruce Mayer, PE Chabot College Mathematics Delta (∆) Notation  Generally in Math the Greek letter ∆ represents a Difference (subtraction)  Recall the Slope Definition  See Diagram at Right

24. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 24 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 25 Bruce Mayer, PE Chabot College Mathematics ∆→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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 26 Bruce Mayer, PE Chabot College Mathematics ∆→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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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 is the Slope at the point of interest. Generally:  The Quotient dy/dx is read as: DERIVATIVE “The DERIVATIVE of y with respect to x”  Thus “Derivative” and “Slope” are Synonymous

28. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 28 Bruce Mayer, PE Chabot College Mathematics d → Quantity AND Operator  Depending on the Context “d” can connote a quantity or an operator  Recall from before the example y = x 2  Recall the Slope Calc  We could also “take the derivative of y = x 2 with respect to x using the d/dx OPERATOR

29. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 29 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 30 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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 (x 1,y 1 ) = (a,f(a)) & (x 2,y 2 ) = (b,f(b))

32. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Avg Rate-of-Change  For f(x) = y = x 2 find the average rate of change between x = 3 (Pt-a) and x = 5 (Pt-b)  By the Chord Slope

33. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Avg Rate-of-Change Chord Slope

34. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 34 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % 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([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrow PieceWise'),... 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

35. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 35 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 36 Bruce Mayer, PE Chabot College Mathematics 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: a)Find and interpret b)Find the equation of the tangent line to D at p = 500.

37. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 37 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  SOLUTION a)Using the definition of the derivative:  Clear fractions by multiplying by  Simplifying Note the Limit is Undefined at h = 0

38. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  Remove the UNdefinition by multiplying by the Radical Conjugate of the Numerator:

39. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 39 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  Continue the Limit Evaluation

40. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 40 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 41 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  Thus The RoC for D w.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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 42 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  SOLUTION b)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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 43 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 44 Bruce Mayer, PE Chabot College Mathematics Example  Rice is Nice  Finally, we use the point-slope formula for the Eqn of a Line and simplify:  The Graph of D(p) and the Tangent Line at p = 500 on the Same Plot:

45. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 45 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 46 Bruce Mayer, PE Chabot College Mathematics “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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 47 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 48 Bruce Mayer, PE Chabot College Mathematics Example: Eval-At bar  Consider the Previous f(x) Example:  Using the d notation to find the Slope (Derivative) for x = −5 & 4

49. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 49 Bruce Mayer, PE Chabot College Mathematics 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 neg No.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. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 50 Bruce Mayer, PE Chabot College Mathematics 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 Slopes are the SAME when approached from EITHER side

51. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 51 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problem From §2.1 P46 → Declining Marginal Productivity

52. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 52 Bruce Mayer, PE Chabot College Mathematics All Done for Today A Different Type of Derivative

53. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 53 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

54. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 54 Bruce Mayer, PE Chabot College Mathematics

55. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 55 Bruce Mayer, PE Chabot College Mathematics

56. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 56 Bruce Mayer, PE Chabot College Mathematics

57. BMayer@ChabotCollege.edu MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx 57 Bruce Mayer, PE Chabot College Mathematics P2.1-46