Bruce Mayer, PE Regsitered Electrical & Mechanical Engineer BMayer@ChabotCollege
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
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Transcript of Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§5.1
Integration
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Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §4.4 → Exp & Log
Math Models Any QUESTIONS
About HomeWork• §4.4 → HW-21
4.4
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Bruce Mayer, PE Chabot College Mathematics
§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
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Bruce Mayer, PE Chabot College Mathematics
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
xfdxxfdxdxF
dxddxxfxF thenif
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Bruce Mayer, PE Chabot College Mathematics
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
dxxfxFxfdxdF or
34xdxxdf
4xxf
34 4xxdxdxf
dxd
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
CxFdxdxfxG
dxd
CxFdxdxG
dxd
?
dxdC
dxdF
dxdG
?
0?
dxdF
dxdG
xfdxdF
dxdGxf
Derivative of a Sum
Derivative of a Const
Transitive Property
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Bruce Mayer, PE Chabot College Mathematics
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
2or
4
4
xxG
xxF
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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 =
7e-5
x/2 +
5x
- 8 +
CMTH15 • Familiy of AntiDerivatives
B. May er • 20Jul13
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Bruce Mayer, PE Chabot College Mathematics
MATLA
B C
ode% 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])
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Bruce Mayer, PE Chabot College Mathematics
MuPA
D C
ode
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)
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Bruce Mayer, PE Chabot College Mathematics
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–
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Bruce Mayer, PE Chabot College Mathematics
Common Fcn Integration Rules1. 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
Cnxdxxn
n
1
1
Cxdxx
ln 1
Cek
dxe kxkx 1
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Bruce Mayer, PE Chabot College Mathematics
Integration Algebra Rules1. Constant Multiple Rule: For any
constant, a
2. The Sum or Difference Rule:
• This often called the Term-by-Term Rule
dxxuadxxua
dxxvdxxudxxvxu
dxxqdxxpdxxqxp
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
Cxxxdxdxxdxx
1012/12
12 1 2
1012/1122/12
Cxxxdxxx 2/33
2
34
3 12
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Example Propensity to Consume 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)
,xPCdxdxMPC
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Bruce Mayer, PE Chabot College Mathematics
Example Propensity to Consume Find PC by
Integrating
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
8.01
Ce x 8.025.1
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Bruce Mayer, PE Chabot College Mathematics
Example Propensity to Consume 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
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Bruce Mayer, PE Chabot College Mathematics
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
)(xfdxdy
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Bruce Mayer, PE Chabot College Mathematics
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
dxxfdxdxdydxxf
dxdyxf
dxdy
1
dxxfydxxfdydxxfdydxxfdy )(1
dxxfy )(
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Bruce Mayer, PE Chabot College Mathematics
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
xexPCdxd 8.0
CexPC x 8.025.1
4.125.1 8.0 xexPC
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Bruce Mayer, PE Chabot College Mathematics
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:
dxxfdyygdxxfdyygygxf
dxdy
dxyg
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
Example Fluid Dynamics SOLUTION: First, TRANSLATE the written
description into an equation:• “rate of change
in volume”• “is proportional to the
square root of volume”• “with constant of
proportionality equal to 0.044”
tVdtd
Vk
044.0k
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Bruce Mayer, PE Chabot College Mathematics
Example Fluid Dynamics 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
VdtdV 044.0
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Bruce Mayer, PE Chabot College Mathematics
Example Fluid Dynamics Integrating
Where
SquaringBoth SidesFind:
1221 CCC
2022.0 CtV
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Bruce Mayer, PE Chabot College Mathematics
Example Fluid Dynamics 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
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work Problems From §5.1
• P58 → Oil Production(not a Gusher…)
• P73 → Car StoppingDistance
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Bruce Mayer, PE Chabot College Mathematics
All Done for Today
LOTS moreon DE’s
in MTH25
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
ConCavity Sign Chart
a b c
−−−−−−++++++ −−−−−− ++++++
x
ConCavityForm
d2f/dx2 Sign
Critical (Break)Points Inflection NO
InflectionInflection
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics