MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §7.1 MultiVar Functions

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.3 → Improper Integrals  Any QUESTIONS About HomeWork §6.3 → HW

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 3 Bruce Mayer, PE Chabot College Mathematics §7.1 Learning Goals  Define and examine functions of two or more variables  Explore graphs and level curves of functions of two variables  Study the Cobb-Douglas production function, isoquants, and indifference curves in economics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 4 Bruce Mayer, PE Chabot College Mathematics Functions of 2+ Variables Functional Machinery

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 5 Bruce Mayer, PE Chabot College Mathematics 2Var Fcn → Unique Assignment  DEFINITION: A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y).  The set D is the domain of f and its range is the set of values that f takes on, that is,

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 6 Bruce Mayer, PE Chabot College Mathematics 2Var Fcn → Unique Assignment  We often write z=f (x, y) to make explicit the value taken on by f at the general point (x, y). The variables x and y are INdependent variables z is the DEpendent variable.  Note that Assignment is Unique; i.e., and Input Values, x & y, Will Produce exactly ONE value of z

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 7 Bruce Mayer, PE Chabot College Mathematics 2Var Fcn → Unique Assignment  YES, a 2Var Function  NO, NOT a 2Var Function NONunique Assignment

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example: f(x,y) Domain & Plot  Given 2Var Fcn: Find Function Domain Plot Over: x 0→10 & y 0→5  SOLUTION  For Domain description look for cases where any term is NOT Defined ln(x) not defined for x ≤ 0 No restriction on y as e y defined for all real numbers

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example: f(x,y) Domain & Plot  Thus the Domain  For the Plot, Make 3D “T-Table”

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example: f(x,y) Domain & Plot  The plot by MATLAB

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 11 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-15 13Jan14 % MTH15_Quick_3Var_3D_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = 0; xmax = 10; % BASE max & min2 ymin = 0; ymax = 5; NumPts = 40 % The GRID ************************************** xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts); [x,y]= meshgrid(xx,yy); % The FUNCTION*********************************** z = x.*exp(y) + log(x+1); % % the Plotting Range = 1.05*FcnRange zmin = min(min(z)); zmax = max(max(z)); % the Range Limits R = zmax - zmin; zmid = (zmax + zmin)/2; zpmin = zmid *R/2; zpmax = zmid *R/2; % % The ZERO Lines % zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green mesh(x,y,z, 'LineWidth', 2),grid, axis([xmin xmax ymin ymax zpmin zpmax]), grid, box,... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = f(x,y)'),... title(['\fontsize{16}MTH16 Bruce Mayer, PE',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7)

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Cobb-Douglas Model  The Cobb-Douglas productivity Model for a given factory says that the production P in a market, in units produced in a given time period, is a function of the labor L and capital K used in production Where –L measured in Worker-Hours –K measured in total-k$ –P measured in Units/Month

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Cobb-Douglas a)Find the production level when 30 workers are employed at full-time (8- hour days for 22 working days a month) and $100,000 (Capital Cost) of machinery are required. b) In order to produce 10,000 units each month using the same capital as in part (a), how many workers would need to be employed at full-time?

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Cobb-Douglas  SOLUTION a)Use the Model to Find P a  Thus There were about 5,876 units produced each month

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Cobb-Douglas a)Need to Increase the Production to Units/mon withOUT purchasing any more Machinery; i.e., K = $100k Solve the Cobb-Douglas Eqn for L

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Cobb-Douglas  The New WorkLoad  10,728 worker-hours (almost 61 full- time equivalent workers) would achieve the goal of 10,000 units each month

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 17 Bruce Mayer, PE Chabot College Mathematics Level Curves  Level Curves are described by 3D Lines of “Constant z”  The most common Level Curve, or IsoQuantity Line, plots are earth- elevation TopoGraphical Maps The Levels

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Surface & Contours  Level-Curves can be found by “Slicing” the Surface Plot  Surface Plots can be found by “Connecting” the IsoLevel Contour Plot Levels

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 19 Bruce Mayer, PE Chabot College Mathematics MATLAB Code % Bruce Mayer, PE % MTH-15 13Jan14 % MTH15_Quick_3Var_3D_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = -1; xmax = 1; % BASE max & min2 ymin = -1; ymax = 1; NumPts = 50 % The GRID ************************************** xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts); [x,y]= meshgrid(xx,yy); % The FUNCTION*********************************** n = 3 z = exp(-n*(x.^2 + y.^2)); % HyperBolic Paraboloid % the Plotting Range = 1.05*FcnRange zmin = min(min(z)); zmax = max(max(z)); % the Range Limits R = zmax - zmin; zmid = (zmax + zmin)/2; zpmin = zmid *R/2; zpmax = zmid *R/2; % % the Domain Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green mesh(x,y,z, 'LineWidth', 2),grid, axis([xmin xmax ymin ymax zpmin zpmax]), grid, box,... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = f(x,y)'),... title(['\fontsize{16}MTH16 Bruce Mayer, PE',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7) % display('Plot PAUSED - hit any key to continue') pause % contour3(x,y,z,20), grid on, axis([xmin xmax ymin ymax zpmin zpmax]), box,... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = f(x,y)'),... title(['\fontsize{16}MTH16 Bruce Mayer, PE',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7) hold on h = findobj('Type','patch'); set(h,'LineWidth',2) hold off

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 20 Bruce Mayer, PE Chabot College Mathematics Level Curves Quantified  A function, z = f(x,y), has level curves consisting of graphs in the xy-plane for fixed, or Constant, values of z.  In other words a level curve of f at C is all points that satisfy the equation

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  WindChill City  A common Math Model for the wind chill Factor is given by where V is the wind velocity in mph (V>5), and T is the temperature in °F  If the temperature remains fixed at 40 °F and the wind speed is 10 mph, what effect does doubling the wind speed have on wind chill?

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  WindChill City  SOLUTION  Need to calculate the difference between wind chill between conditions: T = 40 °F, V = 10 mph T = 40 °F, V = 20 mph  At 10 mph

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  WindChill City  Now at 20 mph  Thus Doubling the wind speed reduced wind chill by a little over three °F  Note that is was an IsoTemperature, or IsoThermal Situation

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  WindChill City  Find Answer on Contour plot

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 25 Bruce Mayer, PE Chabot College Mathematics Contour Plot MATLAB % Bruce Mayer, PE % MTH-16 13Jan14 % MTH15_Quick_3Var_3D_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = 20; xmax = 60; % °F ymin = 5; ymax = 25; % mph NumPts = 50 % The GRID ************************************** xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts); [x,y]= meshgrid(xx,yy); % The FUNCTION*********************************** z = *x *y.^ *x.*y.^0.16; % % the Plotting Range = 1.05*FcnRange zmin = min(min(z)); zmax = max(max(z)); % the Range Limits R = zmax - zmin; zmid = (zmax + zmin)/2; zpmin = zmid *R/2; zpmax = zmid *R/2; % % the Domain Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green [C,h]=contour(x,y,z,15), grid on, axis([xmin xmax ymin ymax]),... xlabel('\fontsize{14}T (°F)'), ylabel('\fontsize{14}V (mph)'), zlabel('\fontsize{14}WindChill (°F)'),... title(['\fontsize{16}MTH16 WindChill (°F)',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7) hold on set(h,'LineWidth',2) %[C,h] = contour(x,y,z); set(h,'ShowText','on','TextStep',get(h,'LevelStep')*2) hold off

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 26 Bruce Mayer, PE Chabot College Mathematics WindChill Surface Plotted

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 27 Bruce Mayer, PE Chabot College Mathematics Surface Plot MATLAB % Bruce Mayer, PE % MTH-16 13Jan14 % MTH15_Quick_3Var_3D_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = 20; xmax = 60; % °F ymin = 5; ymax = 30; % mph NumPts = 50 % The GRID ************************************** xx = linspace(xmin,xmax,NumPts); yy = linspace(ymin,ymax,NumPts); [x,y]= meshgrid(xx,yy); % The FUNCTION*********************************** z = *x *y.^ *x.*y.^0.16; % HyperBolic Paraboloid % the Plotting Range = 1.05*FcnRange zmin = min(min(z)); zmax = max(max(z)); % the Range Limits R = zmax - zmin; zmid = (zmax + zmin)/2; zpmin = zmid *R/2; zpmax = zmid *R/2; % % the Domain Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green mesh(x,y,z, 'LineWidth', 2),grid, axis([xmin xmax ymin ymax zpmin zpmax]), grid, box,... xlabel('\fontsize{14}T (°F)'), ylabel('\fontsize{14}V (mph)'), zlabel('\fontsize{14}WindChill (°F)'),... title(['\fontsize{16}MTH16 WindChill Function',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7) %

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 28 Bruce Mayer, PE Chabot College Mathematics WindChill Plot by MuPAD

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 29 Bruce Mayer, PE Chabot College Mathematics MuPAD code  Bruce Mayer, PE  MTH15 13Jan14  3D_3_Dimensional_Fcn_Plot_BMayer_1401.mn  Change BackGround: Scene3d → Style → BackGroundColor  WC := *x *y^ *x*y^0.16  plotfunc3d(WC, x = , y = 5..25, GridVisible = TRUE, LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm, AxesTitleFont = ["sans-serif", 24], TicksLabelFont=["sans-serif", 16]):

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 30 Bruce Mayer, PE Chabot College Mathematics 3D Graphing by Computer  Excel can make Nice 3D Plots for People who do not have MATLAB & MuPAD  To make an Excel Plot, first construct a Basal, or Base Plane  Next Enter “z” formula and calculate  Use Insert→Charts→Surface to Create the Chart  And then Fine Tune

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 31 Bruce Mayer, PE Chabot College Mathematics 3D Computer Graphing (Excel)  Basal Plane Detail for =-EXP(-3*(H$7^2/1.8+ $F9^2))

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 32 Bruce Mayer, PE Chabot College Mathematics Resulting 3D Excel Graph 3D_Surface_Plot_1401.xlsx

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 33 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 34 Bruce Mayer, PE Chabot College Mathematics Computer Graphics  FootBall  From the National FootBall League RuleBook: Rule 2 The Ball –Section 1  BALL DIMENSIONS  The Ball must be a “Wilson,” hand selected, bearing the signature of the Commissioner of the League, Roger Goodell.  The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder enclosed in a pebble grained, leather case (natural tan color) without corrugations of any kind. It shall have the form of a prolate spheroid and the size and weight shall be: long axis, 11 to 11 1/4 inches; long circumference, 28 to 28 ½ inches; short circumference, 21 to 21 1/4 inches; weight, 14 to 15 ounces.

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 35 Bruce Mayer, PE Chabot College Mathematics Computer Graphics  FootBall  A FootBall is an Ellipsoid of Revolution Using the RuleBook Specs:

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 36 Bruce Mayer, PE Chabot College Mathematics Computer Graphics  FootBall  MATLAB Makes Easy Vertical Revolution  The FootBall Ellipse-Equation:  Solving for y  FootBall Plots on Next Slide

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 37 Bruce Mayer, PE Chabot College Mathematics Computer Graphics  FootBall

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 38 Bruce Mayer, PE Chabot College Mathematics MATLAB FootBall Code % Bruce Mayer, PE % MTH-15 01Aug13 Rev 11Sep13 % MTH15_Quick_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The ELLIPSE ************************************** a = 5.5; b = 4.5; x = linspace(-a,a, 10000); y = sqrt(b^2-(x*b/a).^2); %*********************************** % The Domain Limits xmin = -0.05*b; xmax = 1.05*b; % the Plotting Range = 1.05*FcnRange ymin = min(y); ymax = max(y); % the Range Limits R = 2*a; ymid = 0; ypmin = ymid *R/2; ypmax = ymid *R/2 % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green plot(y,x, 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH16 FootBall',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd m','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) hold off % Bruce Mayer, PE % MTH-15 13Jan14 % MTH15_Quick_3Var_3D_Plot_BlueGreenBkGnd_ m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = -10; xmax = 10; % BASE max & min2 ymin = -10; ymax = 10; % The ELLIPSE ************************************** a = 5.5; b = 4.5; t = linspace(-a,a, 80); [x,y,Z] = cylinder(sqrt(b^2-(t*b/a).^2)); z = 2*a*Z % scale 0-1 to 0-2b %*********************************** % % the Domain Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue- Green mesh(x,y,z, 'LineWidth', 2),grid on, box,... xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y'), zlabel('\fontsize{14}z = f(x,y)'),... title(['\fontsize{16}MTH16 FootBall',]),... annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 3Var 3D Plot.m','FontSize',7) %

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 39 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.1 P47 → Skin Surface Area P48 → WindMills –Smith-Putnam WindMill  Circa 1941  Grandpa's Knob in Castleton, Vermont Castleton, Vermont P49 → Human Energy Expenditure 175 ft

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 40 Bruce Mayer, PE Chabot College Mathematics All Done for Today Spiraling FootBall

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 41 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 42 Bruce Mayer, PE Chabot College Mathematics P Skin Surface Area  Skin Area Formula based on easy to perform Measurements Where –S ≡ Surface Area in sq-meters –W ≡ Person’s mass in kg –H ≡ Height in CentiMeters (cm)  Make Contour Plot S(W,H), and find Height for W=18.37kg & S=0.048m2

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 43 Bruce Mayer, PE Chabot College Mathematics S(15.8kg,87.11cm)  By MuPad S := *(W^0.425)*(H^0.725) Sa = subs(S,W=15.83,H=87.11) –In Sq-Meters

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 44 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 45 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 46 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 47 Bruce Mayer, PE Chabot College Mathematics Basal Metabolism  The Harris-Benedict Power Eqns for Energy per Day in kgCalories Human Males Human Females –h ≡ hgt in cm, A ≡ in yrs, w ≡ weight in kg

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 48 Bruce Mayer, PE Chabot College Mathematics Basal Metabolism a)Find Ba := subs(Bm, w=90,h=190,A=22 b)Find Bb := subs(Bf, w=61,h=170,A=27)d

MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 49 Bruce Mayer, PE Chabot College Mathematics Basal Metabolism c)Find Ac := subs(Am, wm=85, hm=193, Bmm=2108) d)Find Ad := subs(Af, wf=67, hf=173, Bff=1504)