bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

[email protected] • MTH15_Lec-07_sec_2-1_Differeniatation-Basics_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§2.1 Basics of

Differentiation



Review §

Any QUESTIONS About• §1.6 → OneSided-Limits & Continuity

Any QUESTIONS About HomeWork• §1.6 → HW-06

1.6

http://www.google.com/url?sa=i&rct=j&q=continuity&source=images&cd=&cad=rja&docid=K9BxdQo2F2dG9M&tbnid=q_zrez_Fs4DNwM:&ved=0CAUQjRw&url=http://www.backupworks.com/business-continuity-overland-storage.aspx&ei=Wg7TUbaKIKOSiALvxIGoDA&bvm=bv.48705608,d.cGE&psig=AFQjCNHeBd0hRepVEOwmHlMxMeLTFK9SoQ&ust=1372872602953988



§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

http://www.google.com/url?sa=i&rct=j&q=derivative+slop&source=images&cd=&cad=rja&docid=sC7jcePwXaHsDM&tbnid=_ZskiGRc5-1boM:&ved=0CAUQjRw&url=http://memegenerator.net/instance/37010489&ei=kxLTUfW7FIKoigLYr4HIAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNFvrwnnunDQOqFu7bSMg7QzxkWR0Q&ust=1372873708853331



Why Calculus? Calculus divides into the Solution of TWO

Main Questions/Problems1. 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



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/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf

Gottfried Wilhelm von Leibniz Largely Solved the Area-Under-the-Curve Problem





Calculus Pioneers Newton (1642-1727) Leibniz (1646-1716)

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=JloOrMsPcFuSoM&tbnid=UdYdUuYB3jF1tM:&ved=0CAgQjRwwAA&url=http://amandakovacs.wordpress.com/2008/11/22/newton-vs-leibniz/&ei=QkXTUby1N-a2igLM5YGIDg&psig=AFQjCNEHtuclmUPF1dlfrs-fucNPZJfg_A&ust=1372886722944900



Origin of Calculus The word

Calculus comes from the Greek word for PEBBLES

Pebbles were used for counting and doing simple algebra…

http://www.google.com/url?sa=i&rct=j&q=counting+pebbles&source=images&cd=&cad=rja&docid=m4TZqdH4mdALBM&tbnid=j9HP8N0IpPE42M:&ved=0CAUQjRw&url=http://gallimafry.blogspot.com/2010/04/story-of-computers.html&ei=tEjTUfLxKKrpiwKzl4HYCg&bvm=bv.48705608,d.cGE&psig=AFQjCNENiA2r_fTSGEmt7oNeaVzuRXhOqg&ust=1372887537660849



“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.”

http://www.google.com/url?sa=i&rct=j&q=tooth+calculus&source=images&cd=&cad=rja&docid=to6eNRmDGWApWM&tbnid=iDMzAiJSLLi02M:&ved=0CAUQjRw&url=http://drmuna.com/dental-calculus/&ei=V0rTUYv5LOOIiAKMmYG4Ag&bvm=bv.48705608,d.cGE&psig=AFQjCNGjODFZ2YIYamPH65Sfj35iclGl6Q&ust=1372887790408530



“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”

http://www.google.com/url?sa=i&rct=j&q=derivatives+and+integrals&source=images&cd=&cad=rja&docid=I0wMJC6iqFWonM&tbnid=XEGiDdkbOq4c-M:&ved=0CAUQjRw&url=http://hyperphysics.phy-astr.gsu.edu/hbase/math/derint.html&ei=KkvTUdGECoqxigLi_4HIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH2ulOOk9A-wstGzDve6VRFLG2ELw&ust=1372888193872186



“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)0

limh

xxlimorlim



Calculus Controversy Who was first; Leibniz or Newton?

We’ll Do DERIVATIVES First

Derivatives Integrals

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=7EhFef8jrDL7IM&tbnid=rDqQxG-AVXoHBM:&ved=&url=http://xkcd.com/626/&ei=gEXTUYq_HIq5iwKsyICQAg&psig=AFQjCNGVX4bZZFUIqGqe_n0f7g3OKm8Ouw&ust=1372886784503539



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

http://www.google.com/url?sa=i&rct=j&q=rate+of+change&source=images&cd=&cad=rja&docid=TJBuc71_hlAYzM&tbnid=Vb6_HPoQZe10XM:&ved=0CAUQjRw&url=http://www.mathcaptain.com/calculus/rate-of-change.html&ei=Y1PTUaawNKqoigK1poCYDQ&bvm=bv.48705608,d.cGE&psig=AFQjCNEDQP6QEwV_sXK5dI2ZhLzpdW-TlQ&ust=1372890230332935



The TANGENT Line

single pointof Interest

x

y



Slope of a Secant (Chord) Line

x

y

h

Slope, m, of Secant Line (− −) = Rise/Run xhx

xfhxfxxyym

12

12

RunRise

x xfy 1

hx

hxfy 2

22 , yx

11, yx



Slope of a Closer Secant Line

x

y

h

xhx

xfhxfxxyym

12

12

RunRise

x xfy 1

hx

xf



Move x Closer & Closer

x

y

xhx

Note that distance h is getting Smaller



Secant Line for Decreasing h

x

y

The slope of the secant line gets closer and closer to the slope of the tangent line...



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

this Translates to…



The Tangent Slope Definition

The Above Equation yields the SLOPE of the CURVE at the Point-of-Interest

With a Tiny bit of Algebra

xhx

xfhxfmh

0tan lim

h

xfhxfmh

0tan lim



Example Parabola Slope

want the slopewhere x=2

2xy



Example Parabola Slope Use the Slope-Calc Definition

hxhx

hxfhxfm

hh

22

00

)(lim)()(lim

hhxh

hxhxhx

hh

)2(lim2lim0

222

0

4222)2(lim0

xhxmh

0

0

42222 xxm



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/



Delta (∆) Notation Generally in Math the Greek letter ∆

represents a Difference (subtraction) Recall the

Slope Definition See

Diagramat Right

xy

xxyy

xym

Δ

in Changein Change

RunRise

12

12

yin Change

1x 2x

x

y

1y

2y

x

y



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 →

yin Change

1x 2x

x

y

1y

2y

x

y

xxfxfyxfyxxx

122

11

12 11

11

12

12Δxxxxfxxf

xxyy

xymAB

0



∆→d Notation Thus as ∆x→0 The

Chord Slope of AB approaches the Tangent slope of AC

Mathematically

Now by Math Notation Convention:

Thus

yin Change

1x 2x

x

y

1y

2y

x

y

xymm

xABxA

00

limlim

x

xfxxfmxA

11

0lim

xfdxd

dxxdf

dxdy

xy

x

0lim

x

xfxxfdxdy

x

11

0lim



∆→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

yin Change

1x 2x

x

y

1y

2y

x

y

dy

dxx



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

dxdym



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

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

x

y =

f(x) =

x2

MTH15 • Bruce Mayer, PE • dy/dx

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m

dxdyx

hxhxm

h

2)(lim

22

0

xxdxdxf

dxdy

dxd

dxdy 22



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.

dxdy

xy

x

0lim

xxdxd 22

http://www.google.com/url?sa=i&rct=j&q=dy/dx&source=images&cd=&cad=rja&docid=kt1-TVu6DM-65M&tbnid=E04jt_bI7GArUM:&ved=0CAUQjRw&url=http://www.math.rutgers.edu/~greenfie/mill_courses/math135/diary2.html&ei=nnbUUajmDOyWigL7tYHgBw&bvm=bv.48705608,d.cGE&psig=AFQjCNHee-t8ZjDZ5NFUMVUEsGhabS3yfg&ust=1372964818395346



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

'lim)()(lim)('00

yxy

hxfhxfxf

xh



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))



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 Slope0 1 2 3 4 5 6 7

0

5

10

15

20

25

30

35

40

45

x

y =

f(x) =

x2

MTH15 • Avg Rate-of-Change

0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

x

y =

f(x) =

x2


XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

82

163535)()( 22

xy

abafbfmavg

y

x



Example Avg Rate-of-Change

3 4 5

10

15

20

25

x

y =

f(x) =

x2



0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

x

y =

f(x) =

x2


0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

x

y =

f(x) =

x2


XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

y

xy

x

ChordSlope



MATLA

B C

ode% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10;% The FUNCTIONx1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2;x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1;% The Total Function by appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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



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

http://www.google.com/url?sa=i&rct=j&q=road+grade&source=images&cd=&cad=rja&docid=q0sOMoNtY6ZEsM&tbnid=K4k12wSVPLMndM:&ved=0CAUQjRw&url=http://www.visualphotos.com/image/1x6344697/road_sign_showing_twelve_percent_grade&ei=9IPUUe6cLMGyiQLcioDQCA&bvm=bv.48705608,d.cGE&psig=AFQjCNGR7mw_d56Zxy6Kd8i5Yuzpn2D7Lw&ust=1372968234875061



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.p

pD 100)(

500'D



Example Rice is Nice SOLUTIONa) Using the definition

of the derivative:

Clear fractions by multiplying by

Simplifying

• Note the Limit is Undefined at h = 0

hpDhpD

dPdD

h

)()(lim0

hpphpp

hphp

dpdD

h

100100

lim0

hpphhpp

dPdD

h

0

lim100



Example Rice is Nice Remove the UNdefinition by multiplying

by the Radical Conjugate of the Numerator: hpp

hpphpp

hpphhpp

dpdDpD

h

0

lim100'

hpphpphhphpphppp

dpdD

h

)(lim100

0

hpphpphhpD

h

0

lim100'



Example Rice is Nice Continue the Limit Evaluation

hpphppdpdD

h

1lim1000

2/350' ppD



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:

2/350' ppD

$Ton10

$Ton

1Ton10

Ton$Ton10

Ton$MTon 2666

dpdD

.00447.050050500' 2/3 D

$

Ton 4470$Ton 1000447.0500'

226

D



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.

Ton$

Ton4470$

Ton 4470500'2

D



Example Rice is Nice SOLUTIONb) 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.



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:

MegaTons 47.4500

100)500( D

00447.050050 2/3500

pdpdD



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:



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

dp

dDdpdD

dpdD 500500

??



“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

7277 22 xxxdxd

dxdf

dxdyxfy

dxdxxxfy



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

CfdxdfCy

dxdy

cxcx

''

CfdxdfCy

dxdy

cc

''



Example: Eval-At bar Consider the Previous

f(x) Example: Using the d notation to find the Slope

(Derivative) for x = −5 & 4

xxxfy 72

dxdyxxx

dxdy

dxd

7272

15742375245

dx

dydxdy

x



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.

.



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

cxcx

cxcx dx

dfKdxdf

limlim

http://www.google.com/url?sa=i&rct=j&q=smooth+curve+graph&source=images&cd=&cad=rja&docid=tfzjQ2ei5f7V4M&tbnid=Eeij-sCAHRRuhM:&ved=0CAUQjRw&url=http://embedded.com/electronics-blogs/programmer-s-toolbox/4228490/Estimation-interruptus&ei=T6PUUe6hGoe_igKDnYHwCg&bvm=bv.48705608,d.cGE&psig=AFQjCNHpQJFMO32V3kC7xrEc6S1fwCnH8g&ust=1372976288178011



WhiteBoard Work Problem From §2.1

• P46 → DecliningMarginalProductivity

0 1 2 3 4 5 60

50

100

150

200

250

L (k-WorkerHours)

Q (k

-Uni

ts)

MTH15 • P2.1-46




All Done for Today

A DifferentType of

Derivative

http://www.google.com/url?sa=i&rct=j&q=derivative&source=images&cd=&cad=rja&docid=P-yGyF4Hn515vM&tbnid=Wz7Ot-JO7f-bOM:&ved=0CAUQjRw&url=http://www.batr.org/negotium/101712.html&ei=WaDUUd6LLYmbiQL9yYCQBg&bvm=bv.48705608,d.cGE&psig=AFQjCNFJTSsVs_zWMdwjIND6ZNLFvr3kug&ust=1372975518458397



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



P2.1-46

bruce mayer, pe licensed electrical & mechanical engineer bmayer@chabotcollege

Documents

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