1. §2.1 Basics of Differentiation
Chabot Mathematics
§2.1 Basics of Differentiation
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 1.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§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 MasterWork Philosophiae Naturalis Principia Mathematica (Principia)
Read it for FREE:
Gottfried Wilhelm von Leibniz Largely Solved the Area-Under-the-Curve Problem

6. Calculus Pioneers
Newton ( )
Leibniz ( )

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 differentiation and integration of 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
Secant line passes thru two KNOWN Points

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. ...as the values of h get closer to Zero
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. 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

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:

23. Delta (∆) Notation
Generally in Math the Greek letter ∆ represents a Difference (subtraction)
Recall the Slope Definition
See Diagram at 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 →

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 is called a DIFFERENCE
∆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 a Geometric Comparison of
∆x, dx, ∆y, dx
∆x & ∆y are associated with the CHORD Slope • dx & dy are associated with the TANGENT Slope

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 is the 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))
Draw conceptual graph showing [f(b) – f(a)]/[b – a]

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
Chord Slope

34. 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([ ]); % 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',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
plot(x2(1),y2(1), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', [ ], '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: =1

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 D w.r.t. p at p = 500:
Negative Derivative???!!!
What does this mean in this 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/Ton
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 of D(p) and the Tangent Line at 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 = − = −3
y’(4) = 2(4) + 7 = = 15
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 Independent 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 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 (think sharp corners). .

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 Slopes are the SAME when approached from EITHER side

51. WhiteBoard Work Problem From §2.1
P2.1 → Find Derivative from Definition
P46 → Declining Marginal Productivity

52. A Different Type of Derivative
All Done for Today
A Different Type of Derivative

53. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

57. P2.1-46