[email protected] MTH15_Lec-22_sec_5-1_Integration.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§5.1Integrati

on



Review §

Any QUESTIONS About• §4.4 → Exp & Log

Math Models

Any QUESTIONS About HomeWork• §4.4 → HW-21

4.4

http://www.google.com/url?sa=i&rct=j&q=exponential%20model&source=images&cd=&cad=rja&docid=ZVtiL18BSoUT8M&tbnid=UKM87aYJnz5UeM:&ved=0CAUQjRw&url=http://www.bio.georgiasouthern.edu/bio-home/harvey/lect/lectures.html?flnm=grop&ttl=Population%20growth&ccode=el&mda=prnt&ei=68btUcaFE-GqyAHe_4DYAg&bvm=bv.49641647,d.aWc&psig=AFQjCNGw9Lfk_1MdKf6WDQksUX_bHiq0rQ&ust=1374623809263731



§5.1 Learning Goals

Define AntiDerivative Study and compute

indefinite integrals Explore differential

equations and Initial/Boundary value problems

Set up and solve Variable-Separable differential equations

http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=3ce8dous2MQz2M&tbnid=NNZxXfYXcbfDKM:&ved=0CAgQjRwwAA&url=http://www.eisd.net/Page/8825&ei=qs7qUbzGEOi1igKx34HIBQ&psig=AFQjCNF4QG8JdgAi0_xqKtYCZUsXmKoodQ&ust=1374429226304869



Fundamental Theorem of Calculus

The fundamental theorem* of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral.• Part-1: Definite Integral

(Area Under Curve)

• Part-2: AntiDerivative

* The Proof is Beyond the Scope of MTH15

b

aaFbFdxxf

xfdxxfdx

dxF

dx

ddxxfxF thenif

http://www.google.com/url?sa=i&rct=j&q=fundamental+theorem+of+calculus&source=images&cd=&cad=rja&docid=DYvanKV4wzKwXM&tbnid=bwRxp2p-EotnVM:&ved=0CAUQjRw&url=http://www.cafepress.com/+fundamental_theorem_of_calculus_mousepad,103246114&ei=LtDqUYVUwayIAoKVgaAI&bvm=bv.49478099,d.cGE&psig=AFQjCNFV1xR2p95VoqLwNq7HaHePr6k-ew&ust=1374429202323479



AntiDifferentiation

Using the 2nd Part of the Theorem

F(x) is called the AntiDerivative of f(x)• Example:

Find f(x) when

• ONE Answer is

• As Verified by

dxxfxFxfdx

dFor

34xdx

xdf

4xxf

34 4xxdx

dxf

dx

d



Fundamental Property of Antiderivs

The Process of Finding an AntiDerivavite is Called: InDefinite Integration

The Fundamental Property of AntiDerivatives:• If F(x) is an AntiDerivative of the

continuous fcn f(x), then any other AntiDerivative of f(x) has the formG(x) = F(x) + C, for some constant C



Fundamental Property of Antiderivs

Proof of G(x) = F(x) + C Assertion: both G(x) & F(x)+C are

AntiDerivatives of f(x); that is:

Using DerivativeRules

CxFdx

dxfxG

dx

d

CxFdx

dxG

dx

d

?

dx

dC

dx

dF

dx

dG

?

0?

dx

dF

dx

dG

xfdx

dF

dx

dGxf

Derivative of a Sum

Derivative of a Const

Transitive Property



The Indefinite Integral

The family of ALL AntiDerivatives of f(x) is written

The result of ∫f(x)dx is called the indefinite integral of f(x)

Quick Example for:

• u(x) has in INFINITE NUMBER of Results, Two Possibilities:

CxFdxxf )( )(

Cxdxxdxxu 43 4

2

or4

4

xxG

xxF



The Meaning of “C”

The Constant, C, is the y-axis “Anchor Point” for the “natural Response” fcn F(x) for which C = 0.• C is then the y-intercept

of F(x)+C; i.e.,

Adding C to F(x) creates a “family” of functions, or curves on the graph, with the SAME SHAPE, but Shifted VERTICALLY on the y-axis

CFG 00



The Meaning of “C” Graphically

-4 -3 -2 -1 0 1 2 3 4-10

-5

0

5

10

15

20

x

y =

G(x

) =

F(x

)+C

= 7

e-5x/

2 + 5

x -

8 +

CMTH15 • Familiy of AntiDerivatives

B. May er • 20Jul13



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 20Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = -4; xmax = 4; ymin = -10; ymax = 20;% The FUNCTIONx = linspace(xmin,xmax,1000); y = 7*exp(-x/2.5) + 5*x -8;% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg(['white']) % whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y, x,y+9,x,y-pi,x,y+sqrt(13),x,y-7, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = G(x) = F(x)+C = 7e^-^5^x^/^2 + 5x - 8 + C'),... title(['\fontsize{16}MTH15 • Familiy of AntiDerivatives',]),... annotation('textbox',[.71 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'B. Mayer • 20Jul13','FontSize',7)hold onplot(zxv,zyv, 'k', zxh,zyh, 'k', [-1.4995, -1.4995], [ymin,ymax], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])



Mu

PA

D C

od

e

Bruce Mayer, PEMTH15 20Jul13F(x) = 7*exp(-2*x/5) + 5*x -8 f(x) = int(G, x)G := 7*exp(-2*x/5) + 5*x -8dgdx := diff(G, x)assume(x > -6):xmin := solve(dgdx, x)xminNo := float(xmin)Gmin := subs(G, x = xmin)GminNo := float(Gmin)plot(G, x=-4..4, GridVisible = TRUE,LineWidth = 0.04*unit::inch)



Evaluating C by Initial/Boundary

A number can be found for C if the situation provides a value for a SINGLE known point for G(x) → (x, G(x)); e.g., (xn, G(xn)) = (73.2, 4.58)• For Temporal (Time-Based) problems the

known point is called the INITIAL Value– Called Initial Value Problems

• For Spatial (Distance-Based) problems the known point is called the BOUDARY Value– Called Boundary Value Problems–



Common Fcn Integration Rules

1. Constant Rule: for any constant, k

2. Power Rule:for any n ≠ −1

3. Logarithmic Rule:for any x ≠ 0

4. Exponential Rule:for any constant, k

Cxkdxk

Cn

xdxx

nn

1

1

Cxdxx

ln 1

Cek

dxe kxkx 1



Integration Algebra Rules

1. Constant Multiple Rule: For any constant, a

2. The Sum or Difference Rule:

• This often called the Term-by-Term Rule

dxxuadxxua

dxxvdxxudxxvxu

dxxqdxxpdxxqxp



Example Use the Rules

Find the family of AntiDerivatives corresponding to

SOLUTION: First Term-by-Term → break up each

term over addition and subtraction:

dxxx 122

dxdxxdxxdxxx 1 2 12 22



Example Use the Rules

Move out the constant in the 2nd integral (2), and state sqrt as fractional power

Using the Power Rule

CleaningUp →

dxdxxdxxdxxx 1 2 12 2122

Cxxx

dxdxxdxx

1012/12

12 1 2

1012/1122/12

Cxxx

dxxx 2/33

2

3

4

3 12



Example Propensity to Consume

The propensity to consume (PC) is the fraction of income dedicated to spending (as opposed to saving).

A Math Model for the marginal propensity to consume (MPC) for a certain population:

• Where – MPC is the rate of change in PC – x is the fraction of income that is disposable.

xexMPC 8.0




If the propensity to consume is 0.8 when disposable income is 0.92 of total income, find a formula for PC(x)

SOLUTION: From the Problem Statement that the

MPC is a marginal function discern that

Thus the PC fcn is the AntiDerivative of MPC(x)

,xPCdx

dxMPC




Find PC byIntegrating

This is satisfactory for a general solution, but need the particular solution so that PC(0.92) = 0.8

dxxMPCxPC

dxe x 8.0

Ce x

8.0

8.0

1

Ce x 8.025.1




Use the (x,PC) = (0.92,0.8) Boundary Value to Find a NUMBER for the Constant of Integration, C

With C ≈ 1.4, state the particular solution to this Boundary Value Problem

4.1

4.125.1 8.0 xexPC



Differential Equations (DE’s)

A Differential Equation is an equation that involves differentials or derivatives, and a function that satisfies such an equation is called a solution

A Simple Differential Equation is an equation which includestwo differentials in the formof a derivative

)(xfdx

dy



Differential Equations (DE’s)

For some function f. Such a Simple Differential Equation can be solved by integrating:

In summary the Solution, y, to a Simple DE can be found by the integration

dxxfdx

dx

dydxxf

dx

dyxf

dx

dy

1

dxxfydxxfdydxxfdydxxfdy )(1

dxxfy )(



Example Simple DE

From the Previous Example

As previously solved for the general solution by Integration:

Then used the Boundary Value, (0.92, 0.8), to find the Particular Solution

xexPCdx

d 8.0

CexPC x 8.025.1

4.125.1 8.0 xexPC



Variable-Separable DE’s

A Variable Separable Differential equation is a differential equation of the form• For some integrable functions f and g

Such a differential equation can be solved by separating the single-variable functions and integrating:

dxxfdyygdxxfdyygyg

xf

dx

dy

dx

yg



Example Fluid Dynamics

The rate of change in volume (in cubic centimeters) of water in a draining container is proportional to the square root of the depth (in cm) of the water after t seconds, with constant of proportionality 0.044.

Find a model for the volume of water after t seconds, given that initially the container holds 400 cubic centimeters.




SOLUTION: First, TRANSLATE the written

description into an equation:

• “rate of change in volume”

• “is proportional to thesquare root of volume”

• “with constant of proportionality equal to 0.044”

tVdt

d

Vk

044.0k




So the (Differential)Equation

Note that the right side does not explicitly depend on t, so we can’t simply integrate with respect to t. • Instead move the expression

containing V to the left side:

The Variables are now Separated, allowing simple integration

Vdt

dV044.0




Integrating

Where

SquaringBoth SidesFind:

122

1CCC

2022.0 CtV




For The particular solution find the a number for C using the Initial Value: when t = 0, V = 400 cc:• Sub (0,400) into

DE Solution

Thus the volume of water in the Draining Container as a fcn of time:

20400 C

220022.0 ttV



WhiteBoard Work

Problems From §5.1• P58 → Oil Production

(not a Gusher…)• P73 → Car Stopping

Distance

http://www.google.com/url?sa=i&rct=j&q=oil+well+pump&source=images&cd=&cad=rja&docid=aL6ySRiqR7OE7M&tbnid=HmZYXDTzDZN-WM:&ved=0CAUQjRw&url=http://science.lotsoflessons.com/oil-gas.html&ei=op3sUcKlDYXJiQL3sYH4CA&bvm=bv.49478099,d.cGE&psig=AFQjCNGW2itrPfr6sdADr1B87HOFELaBdg&ust=1374547241742893

http://www.google.com/url?sa=i&rct=j&q=car%20stopping%20distances&source=images&cd=&cad=rja&docid=f2MwvGeVBZss0M&tbnid=fqBNXdcsFkzznM:&ved=0CAUQjRw&url=http://www.bstreetsmart.org/Road-Safety/Stopping-Distances/&ei=O57sUaXnDOWjiAKKn4Fo&bvm=bv.49478099,d.cGE&psig=AFQjCNHjtHlFzNeSh2NWh2wG2SrgpwRzTg&ust=1374547809093600



All Done for Today

LOTS moreon DE’s

in MTH25



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection



http://www.google.com/url?sa=i&rct=j&q=oil+well+pump&source=images&cd=&cad=rja&docid=7K2ZbrJjvMR6ZM&tbnid=Adfw5YbrOm5MtM:&ved=0CAUQjRw&url=http://www.igsb.uiowa.edu/browse/oilex/oilex.htm&ei=JZzsUcG5MoigigKzooCgDQ&bvm=bv.49478099,d.cGE&psig=AFQjCNGW2itrPfr6sdADr1B87HOFELaBdg&ust=1374547241742893









http://www.google.com/url?sa=i&rct=j&q=stopping+distances&source=images&cd=&cad=rja&docid=lvH0ToVt4isE-M&tbnid=0-MQ4DHkaKybSM:&ved=0CAUQjRw&url=http://www.drivingtesttips.biz/stopping-distances.html&ei=-b_tUfCqKJPPqQHGqoGABw&bvm=bv.49478099,d.aWc&psig=AFQjCNEdrODs1hPJ31TcXn6lODNE-inClA&ust=1374622027484895



















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Documents

[email protected] MTH15_Lec-22_sec_5-1_Integration.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical