[email protected] mth15_lec-24_sec_5-3_fundamental_theorem.pptx 1 bruce mayer, pe chabot...
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[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics§5.3
FundamentalTheorem of
Calc
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §5.2 → AntiDerivatives by Substitution
Any QUESTIONS About HomeWork• §5.22 → HW-23
5.2
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 3
Bruce Mayer, PE Chabot College Mathematics
§5.3 Learning Goals Show how area under a curve can be
expressed as the limit of a sum Define the definite integral and explore its
properties State the fundamental theorem of calculus,
and use it to compute definite integrals Use the fundamental theorem to solve applied
problems involving net change Provide a geometric justification of the
fundamental theorem
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 4
Bruce Mayer, PE Chabot College Mathematics
Area Under the Curve (AUC)
The AUC has many Applications in Business, Science, and Engineering
Calculation of Geographic Areas
River ChannelCross Section
Wind-ForceLoading
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 5
Bruce Mayer, PE Chabot College Mathematics
Area Under Function Graph
For a Continuous Function, approximate the area between the Curve and the x-Axis by Summing Vertical Strips• Use Rectangles of Equal Width– Three Possible Forms
( )y f x
Left end points Right end points Midpoints
( )y f x( )y f x
x
a b
b ax
n
Strip Width
(n strips)
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 6
Bruce Mayer, PE Chabot College Mathematics
Example: Strip Sum
Approximate the area under the graph of
Use • n = 4
(4 strips)• Strip
MidPoints
2( ) 2 on 0,2f x x
x
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
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8Bruce May er, PE • 24JUul13
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 7
Bruce Mayer, PE Chabot College Mathematics
Example: Strip Sum GamePlan
x
y =
f(x)
= 2
x2MTH15 • Area by Strip Addition
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x
y =
f(x)
= 2
x2MTH15 • Area by Strip Addition
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8Bruce May er, PE • 24JUul13Bruce May er • 24Jul13
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 8
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 24Jul13% XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m%% The FUNCTIONxmin = 0; xmax = 2; ymin = 0; ymax = 8;% The FUNCTIONx = linspace(xmin,xmax,20); y = 2*x.^2;% % 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-Green% Now make AREA Plotarea(x,y, 'FaceColor', [1 .8 1] , 'LineWidth', 3), axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24JUul13','FontSize',7)hold onset(gca,'Layer','top')plot(x,y, 'LineWidth', 3),
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 9
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
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% Bruce Mayer, PE% MTH-15 • 24Jul13%% The Limitsxmin = 0; xmax = 2; ymin = 0; ymax = 8;% The FUNCTIONx = linspace(xmin,xmax,500); y = 2*x.^2;x1 = [0.25:.5:1.75]; y1 = 2*x1.^2% % 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(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7)hold onarea([(x1(1)-.25)*ones(1,100),(x1(1)+.25)*ones(1,100)],[y1(1)*ones(1,100),y1(1)*ones(1,100)],'FaceColor',[1 .8 1])area([(x1(2)-.25)*ones(1,100),(x1(2)+.25)*ones(1,100)],[y1(2)*ones(1,100),y1(2)*ones(1,100)],'FaceColor',[1 .8 1])area([(x1(3)-.25)*ones(1,100),(x1(3)+.25)*ones(1,100)],[y1(3)*ones(1,100),y1(3)*ones(1,100)],'FaceColor',[1 .8 1])area([(x1(4)-.25)*ones(1,100),(x1(4)+.25)*ones(1,100)],[y1(4)*ones(1,100),y1(4)*ones(1,100)],'FaceColor',[1 .8 1])plot(x,y, 'LineWidth', 4)set(gca,'Layer','top')plot(x1,y1,'g d', 'LineWidth', 4)plot([x1(1)-.25,x1(1)+.25],[y1(1),y1(1)], 'm', [x1(2)-.25,x1(2)+.25],[y1(2),y1(2)], 'm',... [x1(3)-.25,x1(3)+.25],[y1(3),y1(3)], 'm', [x1(4)-.25,x1(4)+.25],[y1(4),y1(4)], 'm','LineWidth',2)plot([x1(1)-.25,x1(1)-.25],[0,y1(1)], 'm',[x1(2)-.25,x1(2)-.25],[0,y1(2)], 'm',... [x1(3)-.25,x1(3)-.25],[0,y1(3)], 'm', [x1(4)-.25,x1(4)-.25],[0,y1(4)], 'm', 'LineWidth', 2)set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax])
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 10
Bruce Mayer, PE Chabot College Mathematics
Example: Strip Sum
The Algebra
1 2 3 4( ) ( ) ( ) ( )A x f m f m f m f m
midpoints
1 1 3 5 7
2 4 4 4 4A f f f f
1 1 9 25 49 21
2 8 8 8 8 4A
x
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
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2
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x
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
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8Bruce May er, PE • 24JUul13Bruce May er • 24Jul13
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 11
Bruce Mayer, PE Chabot College Mathematics
Area under a Curve
GOAL: find the exact area under the graph of a function; i.e., the curve
PLAN: Use an infinite number of strips of equal width and compute their area with a limit.
a b
( )y f xWidth:
b ax
n
(n strips)
x
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 12
Bruce Mayer, PE Chabot College Mathematics
Area Under a Curve
Function, f(x), oninterval [a,b] is:• Continuous• NonNegative
Then the Area Under the Curve, A
The x1, x2, …, xn-1,xn are arbitrary, n SubIntervals each with width (b − a)/n
a b
( )y f x
1 2lim ( ) ( ) ... ( )nn
A f x f x f x x
kx
kxf
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 13
Bruce Mayer, PE Chabot College Mathematics
Riemann Sum ∑f(xk)·∆x
For a Continuous, NonNeg fcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be approximated by the sum the area of Vertical Strips
nnk AAAAAAUC 121
n
kkAUC
1
dthConstantWiHEIGHT
xxfxxfxxfxxfxxfAUC nnk 121
Constfor11
xxxfxxfAUCn
kk
n
kk
Riemann ∑
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 14
Bruce Mayer, PE Chabot College Mathematics
Riemann ∑ → Definite Integral
For a Continuous, NonNeg fcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be calculated EXACTLY by the Riemann sum as the number of strips becomes infinite.
This Process of finding an Infinite Sum is called “Integration”; • "to render (something) whole," from Latin
integratus, past participle of integrare "make whole,"
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 15
Bruce Mayer, PE Chabot College Mathematics
Riemann ∑ → Definite Integral As the No. of Strips increase the AUC
Calculation becomes more accurate
The Riemann-Sum to Definite-Integralx
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
0 0.5 1 1.5 20
1
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x
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
0 0.5 1 1.5 20
1
2
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x
y =
f(x)
= 2
x2
MTH15 • Area by Strip Addition
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8Bruce May er • 24Jul13Bruce May er • 24Jul13Bruce May er • 24Jul13
xy
= f(
x) =
2x2
MTH15 • Area by Strip Addition
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2
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xy
= f(
x) =
2x2
MTH15 • Area by Strip Addition
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8Bruce May er • 24Jul13Bruce May er • 24Jul13
Twenty Strips Fifty Strips
dxxfxxfn
abxf
b
a
n
kk
n
n
kk
n
11
limlim
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 16
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
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% Bruce Mayer, PE% MTH-15 • 24Jul13%% The Limitsxmin = 0; xmax = 2; ymin = 0; ymax = 8;% The FUNCTIONx = linspace(xmin,xmax,500); y = 2*x.^2;x1 = [1/20:1/10:39/20]; y1 = 2*x1.^2; % 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(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),... title(['\fontsize{16}MTH15 • Area by Strip Addition',]),... annotation('textbox',[.13 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7)hold onbar(x1,y1, 'BarWidth',1, 'FaceColor', [1 .8 1], 'EdgeColor','b', 'LineWidth', 2)set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax])set(gca,'Layer','top')plot(x,y, 'LineWidth', 3)
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 17
Bruce Mayer, PE Chabot College Mathematics
b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrand
variable of integration(dummy variable)
It is called a dummy variable because the answer does not depend on the
symbol chosen; it depends only on a&b
Definite Integral Symbology
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 18
Bruce Mayer, PE Chabot College Mathematics
Recall Fundamental Theorem
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
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 19
Bruce Mayer, PE Chabot College Mathematics
Fundamental Theorem – Part2
Previously we stated that the AntiDerivative of f(x) is F(x), so then
Now consider the definite Integral (AUC) Relationship to the AntiDerivative
xfxfxfdxfddxxfdx
dxF
dx
d 1
b
a
b
axFaFbFdxxf
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 20
Bruce Mayer, PE Chabot College Mathematics
DefiniteIntegral↔AntiDerivative
That is, The AUC for a continuous Function, f(x), spanning domain [a,b] is the AntiDerivative evaluated at b minus the AntiDerivative evaluated at a.
b
a
b
axFaFbFdxxf
–D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp. 179-181, pg. 770
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 21
Bruce Mayer, PE Chabot College Mathematics
Example Find AUC
Find the area under the graph of y = 2x3
Then
2 3
02x dx
Gives the area since 2x3 is nonnegative on [0, 2].
22
3 4
00
12
2x dx x 4 41 1
2 02 2
8 .sq units
Antiderivative Fund. Thm. of Calculus
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 22
Bruce Mayer, PE Chabot College Mathematics
Rules for Definite Integrals
1. Constant Rule: for any constant, k
2. Sum/DiffRule:
3. Zero WidthRule
4. DomainReversal Rule
Cxkdxk
0 dxxfa
a
4. ( ) ( ) ( ) ( )b b b
a a af x g x dx f x dx g x dx
2. ( ) ( ) b a
a bf x dx f x dx
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 23
Bruce Mayer, PE Chabot College Mathematics
Rules for Definite Integrals
5. SubDivision Rule, for (a<b<c)
5. ( ) ( ) ( )b c b
a a cf x dx f x dx f x dx
x
y
a b c
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 24
Bruce Mayer, PE Chabot College Mathematics
Example Eval Definite Integral
Find a Value for
The Reduction using the Term-by-Term rule
5
1
12 1x dx
x
2 25 ln 5 5 1 ln1 1
28 ln 5 26.39056
5125
1ln1
12 xxxdx
xx
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 25
Bruce Mayer, PE Chabot College Mathematics
Example Def Int by Substitution
Find:
Let: Then find dx(du) and u(x=0), and u(x=1)
2let 3u x x
dxxxxxxxx
x
1
0
2121
0
212 332332
323 3 3 222 xxxdx
du
dx
dxxu
dx
dxxu
41311
00300
3
2
2
2
xxux
32
321
32
x
dudx
x
dxx
dx
du
ClarifyLimits
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 26
Bruce Mayer, PE Chabot College Mathematics
Example Def Int by Substitution
SubOut x2+3x, and the Limits
Dividing out the 2x+3
Then
Thus Ans
x
3232332
4
0
211
0
212
x
duuxdxxxx
u
u
x
x
4
023
4
0
234
0
21
3
2
23
u
u
u
u
u
uu
uduu
3
16
1
8
3
264
2
24
3
204
3
2 32323
3
16332
1
0
212 xxx
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 27
Bruce Mayer, PE Chabot College Mathematics
The Average Value of a Function
At y = yavg there at EQUAL AREAS above & below the Avg-Line
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
x
y =
f(x)
MTH15 • Meaning of Avg
Bruce May er, PE • 24Jul13
Avg Line
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 28
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
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% Bruce Mayer, PE% MTH-15 • 24Jul13% Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m% Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E.% Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN% 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295%clc; clear% The Functionxmin = 0; xmax = 16;ymin = 0; ymax = 350;xct = 1000x = linspace(xmin,xmax,xct);y1 = .5*x.^3-9*x.^2+11*x+330;yavg = mean(y1)y2 = yavg*ones(1,xct)%%% Find Zerosplot(x,y1, x,y2, 'k','LineWidth', 2), axis([0 xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Meaning of Avg',]),... annotation('textbox',[.13 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24Jul13','FontSize',7)display('Showing 2Fcn Plot; hit ANY KEY to Continue')% "hold" = Retain current graph when adding new graphshold on%nct = 500xn = linspace(xmin, xmax, nct);fill([xn,fliplr(xn)],[.5*xn.^3-9*xn.^2+11*xn+330, fliplr(yavg*ones(1,nct))],'m'),gridplot(x,y1), grid
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 29
Bruce Mayer, PE Chabot College Mathematics
The Average Value of a Function
Mathematically - If f is integrable on [a, b], then the average value of f over [a, b] is
Example Find the Avg Value:
Use Average Definition:
1( )
b
af x dx
b a
3/ 2( ) over 0,9 .f x x
9 3/ 2
0
1
9 0x dx
9
5/ 2
0
1 2
9 5
x
5/ 229
45 54
5
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 30
Bruce Mayer, PE Chabot College Mathematics
Net Change
If the Rate of Change (RoC), dQ/dx = Q’(x) is continuous over the interval [a,b], then the NET CHANGE in Q(x) is Given by
dxxQdxdxdQaQbQb
a
b
a '
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 31
Bruce Mayer, PE Chabot College Mathematics
Example Find Net Change
A small importer of Gladiator merchandise has modeled her monthly profits since the company was created on January 1, 1997 by the formula
• Where–P ≡ $-Profit in 100’s of Dollars ($c or c-Notes)– t ≡ year of operation for the company
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 32
Bruce Mayer, PE Chabot College Mathematics
Example Find Net Change
What is the importer’s net change in profit between the beginning of the years 2000 and 2003?
SOLUTION: Recall t is in years after 1997, Thus• Year 2000 corresponds to t = 3 • Year 2003 corresponds to t = 6
Then in this case the Net Change in Profit over [3,6] →
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 33
Bruce Mayer, PE Chabot College Mathematics
Example Find Net Change
Thus Her monthly profits increased by about $1,354.50 between 2000 & 2003
632345 14667.8925.114.0 tttt
dttttttP 286.267.77.06
3
2346
3
c545.13$
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 34
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §5.3• P74 → Water Consumption• P80 → Distance Traveled
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 35
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
StudentsShould Calc
dxx
xxx
1
0
232527 226652
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 36
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
FundamentalTheorem
Part-1
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 37
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 38
Bruce Mayer, PE Chabot College Mathematics
a x
Let area under the
curve from a to x.
(“a” is a constant)
aA x
x h
aA x
Then:
a x aA x A x h A x h
x a aA x h A x h A x
xA x h
aA x h
Fundamental Theorem Proof
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 39
Bruce Mayer, PE Chabot College Mathematics
x x h
min f max f
The area of a rectangle drawn under the curve would be less than the actual area under the
curve.
The area of a rectangle drawn above the curve would be more than the actual area
under the curve.
short rectangle area under curve tall rectangle
min max a ah f A x h A x h f
h
min max a aA x h A x
f fh
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 40
Bruce Mayer, PE Chabot College Mathematics
min max a aA x h A x
f fh
As h gets smaller, min f and max f get closer together.
0
lim a a
h
A x h A xf x
h
This is the definition
of derivative!
a
dA x f x
dx
Take the anti-derivative of both sides to find an explicit formula
for area.
aA x F x c
aA a F a c
0 F a c
F a c initial value
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 41
Bruce Mayer, PE Chabot College Mathematics
min max a aA x h A x
f fh
As h gets smaller, min f and max f get closer together.
0
lim a a
h
A x h A xf x
h
a
dA x f x
dx
aA x F x c
aA a F a c
0 F a c
F a c aA x F x F a
Area under curve from a to x = antiderivative at x minus
antiderivative at a.
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 42
Bruce Mayer, PE Chabot College Mathematics
ConCavity Sign Chart
a b c
−−−−−−++++++ −−−−−− ++++++
x
ConCavityForm
d2f/dx2 Sign
Critical (Break)Points Inflection NO
InflectionInflection
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 43
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 44
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 45
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 46
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 47
Bruce Mayer, PE Chabot College Mathematics