[email protected] MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 1 Bruce Mayer, PE Chabot...

[email protected] • MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics§5.3

FundamentalTheorem of

Calc



Review §

Any QUESTIONS About• §5.2 → AntiDerivatives by Substitution

Any QUESTIONS About HomeWork• §5.22 → HW-23

5.2

http://www.google.com/url?sa=i&rct=j&q=antiderivatives+by+substitution+of+variables&source=images&cd=&cad=rja&docid=PYc6lVBDVmgAOM&tbnid=QDsVQrde6i4JCM:&ved=0CAUQjRw&url=http://www.youtube.com/watch?v=cbg4wphqFCg&ei=aVHvUdyhLeS6iwKghoCoBw&bvm=bv.49641647,d.cGE&psig=AFQjCNElcZ28nupE8MOX9qDR2TUhGjGJ3A&ust=1374724707273581



§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



Area Under the Curve (AUC)

The AUC has many Applications in Business, Science, and Engineering

Calculation of Geographic Areas

River ChannelCross Section

Wind-ForceLoading



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)



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

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8Bruce May er, PE • 24JUul13



Example: Strip Sum GamePlan

x

y =

f(x)

= 2

x2MTH15 • Area by Strip Addition

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

x

y =

f(x)

= 2

x2MTH15 • Area by Strip Addition

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8Bruce May er, PE • 24JUul13Bruce May er • 24Jul13



MA

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% 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),



MA

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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])



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


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

x

y =

f(x)

= 2

x2


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8Bruce May er, PE • 24JUul13Bruce May er • 24Jul13



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



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



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 ∑



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,"



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


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

x

y =

f(x)

= 2

x2


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

x

y =

f(x)

= 2

x2


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8Bruce May er • 24Jul13Bruce May er • 24Jul13Bruce May er • 24Jul13

xy

= f(

x) =

2x2


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

xy

= f(

x) =

2x2


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8Bruce May er • 24Jul13Bruce May er • 24Jul13

Twenty Strips Fifty Strips

dxxfxxfn

abxf

b

a

n

kk

n

n

kk

n

11

limlim



MA

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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)



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



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

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



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



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

http://www.google.com/url?sa=i&rct=j&q=fundamental%20theorem%20of%20calculus%20proof&source=images&cd=&cad=rja&docid=Mr1Js8sc0khjWM&tbnid=H0Ll7sjiIX8SMM:&ved=0CAUQjRw&url=http://memegenerator.net/instance/16780267&ei=70HwUZTdDIqyiQLSyIHYBA&bvm=bv.49784469,d.cGE&psig=AFQjCNGZpfcYxoh1wdy2QEylz3xL_ljlLA&ust=1374786380619984



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



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



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



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



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



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



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



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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



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



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 '

http://www.google.com/url?sa=i&rct=j&q=Net%20Change&source=images&cd=&cad=rja&docid=KI0zKMDgbie0YM&tbnid=5KUmCPx146HceM:&ved=0CAUQjRw&url=https://www.facebook.com/pages/Net-Change-Week/129987970408330&ei=1F3wUYrzNcKfiALru4C4BQ&bvm=bv.49784469,d.cGE&psig=AFQjCNEh7RuLQM2T0OPjQSY0y7iwU3tB6A&ust=1374793454755652



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




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] →




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$



WhiteBoard Work

Problems From §5.3• P74 → Water Consumption• P80 → Distance Traveled

http://www.google.com/url?sa=i&rct=j&q=water%20consumption&source=images&cd=&cad=rja&docid=ZjHdHJNMj8OmoM&tbnid=NeansUAcJmb1LM:&ved=0CAUQjRw&url=http://dejazu.blogspot.com/2011_08_01_archive.html&ei=f2PwUfiiJ6S6iwLMxIGICw&bvm=bv.49784469,d.cGE&psig=AFQjCNE8N3vmiZmk3eB-u-_3Xd-qKJGK1g&ust=1374794966706245

http://www.google.com/url?sa=i&rct=j&q=speed%20time%20graph&source=images&cd=&cad=rja&docid=rOsB6R_JlgUlzM&tbnid=ald--Mksg89fTM:&ved=0CAUQjRw&url=http://www.tutorvista.com/content/science/science-i/motion/velocity-time-graph.php&ei=N2TwUfHPGuTuigLm1IGgBA&bvm=bv.49784469,d.cGE&psig=AFQjCNFt2VnIv67P8EGQvz3kz7lWMclRYA&ust=1374795129596344



All Done for Today

StudentsShould Calc

dxx

xxx

1

0

232527 226652

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=wSGtOAIrRxIoIM&tbnid=l4Q1f5xFVNRraM:&ved=0CAUQjRw&url=http://www.cantonschools.org/~lforastiere/ap%20calculus?Templates=Printable&ei=2TLcUY-gOa2UjALp4IHgDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765



All Done for Today

FundamentalTheorem

Part-1

http://www.google.com/url?sa=i&rct=j&q=fundamental%20theorem%20of%20calculus&source=images&cd=&cad=rja&docid=sKaGyWM0SXwdJM&tbnid=uwKff_S3x5QWXM:&ved=0CAUQjRw&url=http://www.cafepress.com/+fundamental-theorem-of-calculus+coasters&ei=y2bwUa3ZDKSkiQLqmICgAw&bvm=bv.49784469,d.cGE&psig=AFQjCNHHPywVmexMF9irs0PYeaKXp9LmJQ&ust=1374795666928589



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



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



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




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




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.



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=derivative+power+rule&source=images&cd=&cad=rja&docid=NjbfKj_JZBSzwM&tbnid=jJkxz7X095T-NM:&ved=0CAUQjRw&url=http://mathworld.wolfram.com/Derivative.html&ei=TfPWUe_nA6akiQKZ64DIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=wSGtOAIrRxIoIM&tbnid=l4Q1f5xFVNRraM:&ved=0CAUQjRw&url=http://www.cantonschools.org/~lforastiere/ap%20calculus?Templates=Printable&ei=2TLcUY-gOa2UjALp4IHgDw&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=nuuUeFKHM5pqjM&tbnid=x7raEhpCGJ8edM:&ved=0CAUQjRw&url=http://designatedderiver.wikispaces.com/Games+and+fun+stuff&ei=IzPcUYPxBaOLiwLjs4GYAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

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[email protected] MTH15_Lec-24_sec_5-3_Fundamental_Theorem.pptx 1 Bruce Mayer, PE Chabot...

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