[email protected] mth15_lec-07_sec_2-1_differeniatation-basics_.pptx 1 bruce mayer, pe chabot...

[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

xfhxf

xx

yym

12

12

Run

Rise

x xfy 1

hx

hxfy 2

22 , yx

11, yx



Slope of a Closer Secant Line

x

y

h

xhx

xfhxf

xx

yym

12

12

Run

Rise

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

xfhxfm

h

0

tan lim

h

xfhxfm

h

0tan lim



Example Parabola Slope

want the slopewhere x=2

2xy



Example Parabola Slope

Use the Slope-Calc Definition

h

xhx

h

xfhxfm

hh

22

00

)(lim

)()(lim

h

hxh

h

xhxhxhh

)2(lim

2lim

0

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

SeeDiagramat Right

x

y

xx

yy

x

ym

Δ

in Change

in Change

Run

Rise

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

xxfxfy

xfy

xxx

122

11

12 11

11

12

12Δ

xxx

xfxxf

xx

yy

x

ymAB

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

x

ymm

xAB

xA

00limlim

x

xfxxfm

xA

11

0lim

xfdx

d

dx

xdf

dx

dy

x

yx

0lim

x

xfxxf

dx

dyx

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

dx

dym



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)

= x

2

MTH15 • Bruce Mayer, PE • dy/dx

XYf cnGraph6x6BlueGreenBkGndTemplate1306.m

dx

dyx

h

xhxm

h

2

)(lim

22

0

xxdx

dxf

dx

dy

dx

d

dx

dy22



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.

dx

dy

x

yx

0lim

xxdx

d22

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

yx

y

h

xfhxfxf

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)

= x

2

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)

= x

2


XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

8

2

16

35

35)()( 22

x

y

ab

afbfmavg

y

x



Example Avg Rate-of-Change

3 4 5

10

15

20

25

x

y =

f(x)

= x

2



0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

x

y =

f(x)

= x

2


0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

x

y =

f(x)

= x

2


XYf cnGraph6x6BlueGreenBkGndTemplate1306.mXYf cnGraph6x6BlueGreenBkGndTemplate1306.m

y

xy

x

ChordSlope



MA

TL

AB

Co

de

% 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

pD100

)(

500'D



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

h

pDhpD

dP

dDh

)()(lim

0

hpp

hpp

h

php

dp

dDh

100100

lim0

hpph

hpp

dP

dDh

0lim100



Example Rice is Nice Remove the UNdefinition by multiplying

by the Radical Conjugate of the Numerator: hpp

hpp

hpp

hpph

hpp

dp

dDpD

h

0

lim100'

hpphpph

hphpphppp

dp

dDh

)(lim100

0

hpphpph

hpD

h

0

lim100'



Example Rice is Nice

Continue the Limit Evaluation

hpphppdp

dDh

1lim100

0

2/350' ppD




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

1

Ton10

Ton$

Ton10

Ton$

MTon 2666

dp

dD

.00447.050050500' 2/3 D

$

Ton 4470

$

Ton 1000447.0500'

226

D




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




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.




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




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

dD

dp

dD

dp

dD 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 xxxdx

d

dx

df

dx

dyxfy

dx

dxxxfy



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

Cfdx

dfCy

dx

dy

cxcx

''

Cfdx

dfCy

dx

dy

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

dx

dyxxx

dx

dy

dx

d 7272

15742375245

dx

dy

dx

dy

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

dfK

dx

df

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 → Declining

MarginalProductivity

0 1 2 3 4 5 60

50

100

150

200

250

L (k-WorkerHours)

Q (

k-U

nits

)

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

[email protected] mth15_lec-07_sec_2-1_differeniatation-basics_.pptx 1 bruce mayer, pe chabot...

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