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[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
§5.1Integrati
on
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 2
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
xfdxxfdx
dxF
dx
ddxxfxF 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
dxxfxFxfdx
dFor
34xdx
xdf
4xxf
34 4xxdx
dxf
dx
d
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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
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 7
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
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
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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
2
or4
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
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 10
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
= 7
e-5x/
2 + 5
x -
8 +
CMTH15 • Familiy of AntiDerivatives
B. May er • 20Jul13
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Bruce Mayer, PE Chabot College Mathematics
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])
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Bruce Mayer, PE Chabot College Mathematics
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)
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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–
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 14
Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 16
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
Cxxx
dxdxxdxx
1012/12
12 1 2
1012/1122/12
Cxxx
dxxx 2/33
2
3
4
3 12
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 18
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
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 19
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)
,xPCdx
dxMPC
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 20
Bruce Mayer, PE Chabot College Mathematics
Example Propensity to Consume
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
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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
)(xfdx
dy
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 23
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
dxxfdx
dx
dydxxf
dx
dyxf
dx
dy
1
dxxfydxxfdydxxfdydxxfdy )(1
dxxfy )(
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 24
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
xexPCdx
d 8.0
CexPC x 8.025.1
4.125.1 8.0 xexPC
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 25
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:
dxxfdyygdxxfdyygyg
xf
dx
dy
dx
yg
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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.
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 27
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 thesquare root of volume”
• “with constant of proportionality equal to 0.044”
tVdt
d
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
Vdt
dV044.0
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Bruce Mayer, PE Chabot College Mathematics
Example Fluid Dynamics
Integrating
Where
SquaringBoth SidesFind:
122
1CCC
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 Stopping
Distance
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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
[email protected] • MTH15_Lec-22_sec_5-1_Integration.pptx 48
Bruce Mayer, PE Chabot College Mathematics