Matlab project calculus 2

Currently, science is growing, with this development strength, the application of science and scientific patents in schools is very feasible and significant. Since the first year, Ho Chi Minh City University of Technology lecturers have helped technical students get acquainted with programming applications such as MATLAB. MATLAB is a rising and programming environment that allows calculating numbers with matrices, graphing functions or charting information, implementing algorithms, creating user interfaces, and linking to computer programs written in many other programming languages. MATLAB enables simulation of calculations with the Toolbox library, experimenting with many models in practice and techniques. With more than 40 years of establishment and development, MATLAB is an effective calculation tool to solve technical problems today with a relatively simple and universal design. Therefore, for the problems in Algebra, especially matrix problems, we can use MATLABs computing applications to solve most simply and understandably, helping us get acquainted and add more skills to use programs application for students.

HCMC University of Technology MATLAB Project

Department of Applied Mathematics Subject: Calculus 2

1 Introduction

Currently, science is growing, with this development strength, the application of science and scientific patents in schools is very feasible and significant Since the first year, Ho Chi Minh City University of Technology lecturers have helped technical students get acquainted with programming applications such as MATLAB MATLAB is a rising and programming environment that allows calculating numbers with matrices, graphing functions or charting information, implementing algorithms, creating user interfaces, and linking to computer programs written in many other programming languages MATLAB enables simulation of calculations with the Toolbox library, experimenting with many models in practice and techniques With more than 40 years of establishment and development, MATLAB is an effective calculation tool to solve technical problems today with a relatively simple and universal design Therefore, for the problems in Algebra, especially matrix problems, we can use MATLAB's computing applications to solve most simply and understandably, helping us get acquainted and add more skills to use programs application for students.

3 Solutions:

Question 1: Find the extreme values of the function f(x; y) = x2 + 2y2 on the circle x2 + y2 = 1 Sketch the given surface and show the extreme values.

fx = 2x ; fy = 4y, the only critical point is (0;0) and f(0;0) =0 To find the critical points on the boundary

value is f (0 ;± 1)=2and the minimun point is f (0 ;0 )=0

Question 2: Find the extreme values of the function f(x; y) = x2y on the curve x2 + 2y2 = 6: Sketch the given surface and show the extreme values.

¿ 1 d2L(P ¿¿ 3), d2L(P ¿¿ 4 )< 0 so P3; P4¿ ¿is local maximum.

¿−1 d2

L(P ¿¿ 5),d2L(P ¿¿ 6)>0 so P5 ; P6islocal minimum ¿ ¿

Question 3: Find the extreme values of the function f(x; y) = 6 - 5x - 4y

on the curve x2 + y2 = 9 Sketch the given surface and show the extreme values.

Question 4: Find the extreme values of the function f(x; y) = 1 - 4x - 8y

on the curve x2 + 8y2 = 8 Sketch the given surface and show the extreme values.

Solve equations ∇f= λ∇g and g(x,y)=8 using Lagrange multipliers

Therefore the maximum value of f(x,y) on the curve x2 + 8y2 = 8 is f(- 4 √ 3

3 ,- √ 3

3 ) = 1 + 8 √ 3 and the minimum value is f( 4 √ 3

3 , √ 3

3 ) = 1 - 8 √ 3

Question 5: Find the extreme values of the function f(x; y) = x2 + y2 +

xy on the curve x2 + 2y2 = 8 Sketch the given surface and show the extreme values

Consider Lagrange function:

L ( x , y , λ )=x2

+ y2+ xy−λ (1−x2−2 y2

) Find stationary points:

12 ) ;− √ 3− √ 3

12 )

< 0 Maximum point ((1− √ 3) √ 3+ √ 13

12 ; √ 3+ √ 13

12 )

> 0 Minimum point ((1− √ 3)(− √ 3+ √ 13

f ( x , y )=2 x2+12 xy+ y2

x2+ y2=25

fx = 4x+12y gx = 2x

fy = 12x+2y gy = 2y

fx = lamda.gx => 4x +12y = lamda.2x (1)

fy = lamda.gy => 12x +2y = lamda.2y (2)

3 ) is the minimum point

Question 8: Find the local maximum and minimum values and saddle points of f ( x , y )=x4

+ y4−2 x2

+ 4 xy−2 y2 Sketch the given surface and show

the extreme values.

∆ = fxx.fyy – fxy2 = 288x2y2 – 96x2 – 24y2 + 8

fxx fyy ∆ = fxx.fyy – fxy2

Therefore, the maximum value of f is f(0,0) = 0, and f( 1

2 ,1) = - 9

8 , f( 1

2 ,1) = 9

-8 , f( −1

2 ,1) = - 9

8 , f( −1

2 ,-1) = - 9

8 are the minimum point.

Question 10: Find the local maximum and minimum values and saddle points of f ( x , y )=(x2

− 2 y2) ex− y Sketch the given surface and show the

Compute: { A=fxx

' '

=(2+2 x ) ex− y+(2 x+x2−2 y2) ex− yB=f' ' xy= (−4 y ) ex− y+ ( 2 x+x2−2 y2) (−1) ex− y

So M(-2,0) is the minimum point

Question 13: Find the local maximum and minimum values and saddle points of f ( x , y )=(x2

+ y2)( e−x2

∆ >0, A <0 →(1 ;1)is a local minimum

Question 17: Find the local maximum and minimum values and saddle points of f ( x , y )=x3

∆4< 0∧So M2is a saddle point

Question 18: Find the surface area of the part of the sphere x2

+ y2+ z2=4

that lies inside the cylinder x2

+ y2=2 y Sketch the given surface

that lies above the cone z2

= x2+ y2 Sketch the given surface.

Since the part of the sphere x2+y2+z2=2 that lies above the cone z2=

x2+y2, can be divided into 2 parts, which have the same area, thus the required surface area is

Question 22: Evaluate the solid bounded by 2x + z = 2 and (x - 1)2 + y2

= z Sketch the given solid.

Let { x=rcosφ y=rsinφ | J | = r 0≤ φ ≤ 2 π 0 ≤ φ ≤ 1

( x−1)2+ y2≤ z ≤ 2−2 x (rcosφ−1)2+ rsinφ2≤ z ≤ 2−2 rcosφ

Question 23: Evaluate the volume of the solid bounded by y = x; y = 2x;

x = 1; z = x2 + y2; z = x2 + 2y2 Sketch the given solid.

+ y2=2 x Sketch the given solid.

Letting x = r cosφ, y = r sinφ,

= 3 π

2

Question 25: Evaluate the volume V of the solid Ω bounded by z = 1 – x2

– y2 and z = 0 Sketch the given solid.

The volume V of the solid Ω is

Question 26: Evaluate the volume of the solid bounded by y = 1 + x2; z

= 3x; y = 5; z = 0 (where x > 0) Sketch the given solid.

The solid Ω lies under the surface z=3 x and above z=0.

The projection D is bounded by y=1+x2, y=5, x=0

Let x=rcosφ, y=rsinφ: D= { (r , φ): 0 ≤r ≤ 1,0 ≤ φ ≤ 2 π }

The volume V of the solid Ω is

2 dφ=

π

8

Question 29: Evaluate the area of the region D bounded by y = (x+1)2; x

= y-y3; x = -1; y = -1 Sketch the given region.

The curve x = y – y3 goes through the origin and intersects the y-axis at the points (0;-1) and (0; 1)

The curve x = y – y3 intersects the straight line y = 1 and the curve x =

-1 + √ y at (0;-1) and (0; 1) respectively, since they satisfy the equations:

-1 + √ y = y – y3 => { x=0 y=1

The curve x = -1 + √ y intersects the x-y coordinate at (-1; 0) and (0; 1)

The curve x = -1 + √ y intersects the straight line x = -1 at (-1; 0)

Question 30: Evaluate ∬

D

2 xdA where D={( x , y ) :2 x ≤ x2

+ y2≤ 6 x , y ≤ x Sketch the given region.

2 x ≤ x2

+ y2≤6 x ↔ { ( x−1)2

+ y2≥1

( x−3)2+ y2≤ 9

Let x=rcosφ, y=rsinφ: 2 cosφ≤ r≤ 6 cosφ

y ≤ x ↔ rsinφ ≤ rcosφ where φ ∈ [ − π

3

3 − cosφ

2(2 cosφ)3

3 ) dφ ≈157.189

Question 31: Evaluate the area of the region D bounded by x + y2 =1; y

- x = 1; x = 0 Sketch the given region.

Question 32: Evaluate ∬

D

( x + y )dA where

D={( x , y ) :1≤ x2+ y2≤ 4 , x ≥ 0, y ≥ 0, y ≥ x } Sketch the given region.

By the spherical coordinate system, let { x=rcosφ y=rsinφ | J | = r

Let x=rcosφ, y=rsinφ: 1 ≤r ≤2

x

ydA where D is the region bounded by y2 = x;

x = 0; y = 1 Sketch the given region.

The region D is bounded by { 0 ≤ x ≤ y 0≤ y ≤12

Then I= ∫

0

1

¿ ¿.

Question 36: Evaluate ∬

D

xdA where D is a triangle OAB; O(0; 0); A(1; 1);

B(0; 1) Sketch the given region.

0

π / 2cosydy = [ − cosx ] π /2

Sketch the given region.

The region D is bounded by { 1≤ y ≤ 2 0 ≤ x ≤ 2

− 2)dy=12

Question 39: Evaluate ∭

Ω

xy z2dV ; where Ω is the rectangular box given

by Ω:0 ≤ x ≤1 ,−1≤ y ≤ 2, 0 ≤ z ≤3 Sketch the given solid.

( x +2 y)dV e; where Ω is the solid bounded by

x2≤ y ≤ x ,0 ≤ z ≤ x Sketch the given solid.

Question 41: Evaluate I= ∭

Ω

√ x2+ z2dV where Ω is the region bounded by

x2+ z2≤ y ≤ 4 Sketch the given solid.

Sketch the given solid.

The region D is bounded by x2

Question 44: Evaluate the triple integral with cylindrical coordinates

I= ∭

Ω

1

x2+ y2dV where Ω is the solid bounded by z = 0; z = y; x2 + y2 = 1

and y ≥ 0 Sketch the given solid.

( x2+ y2) dV where Ω is the solid hemisphere x2 + y2 + z2 ≤ 1; z ≥ 0:

Sketch the given solid.

Evaluate the triple intergral with spherical coordinates

The solid Ω in the spherical coordinate system is bounded by { 0 ≤θ ≤ π

( x2+ y2+ z2) dV where Ω lies between the spheres x2 + y2 + z2 = 1 and

x2 + y2 + z2 = 4 in the first octant Sketch the given solid.

Let x=rcosφ, y=rsinφ

In the first octant x ≥ 0∧ y ≥ 0 so0 ≤ φ ≤ π

+ z2dV where Ω is the solid that lies within the sphere x2 + y2 +

z2 = 4 and above the cone z= √ x2

+ y2.Sketch the given solid.

Letting x = ρsinθcosφ, y = ρsinθsinφ, z = ρ cosθ

0 ≤ z ≤ √ x2+ y2  z ≤ r

Converting from Cartesian coordinate system to spherical coordinate system:

{ x φ=tan2+ y2+−1z2( = y ρ2

x ) θ=tan−1

Question 51: Use spherical coordinates to find the volume of the solid that lies above the cone z= √ x2+ y2 and below the sphere x2 + y2 + z2 = z

Sketch the given solid.

4 MATLAB code questions:

Question 1: Find the extreme values of the function f(x; y) = x2 + 2y2 on the circle x2 + y2 = 1 Sketch the given surface and show the extreme values.

syms x y lamd real

% Sketch the given surface and show the extreme values

x=linspace(-10,10,30); y=linspace(-10,10,30); [x,y]=meshgrid(x,y);

syms x y lamd real

Question 3: Find the extreme values of the function f(x; y) = 6 - 5x - 4y

on the curve x2 + y2 = 9 Sketch the given surface and show the extreme values.

syms x y lamd real

Question 4: Find the extreme values of the function f(x; y) = 1 - 4x - 8y

on the curve x2 + 8y2 = 8 Sketch the given surface and show the extreme values.

syms x y lamd real

Question 5: Find the extreme values of the function f(x; y) = x2 + y2 +

xy on the curve x2 + 2y2 = 8 Sketch the given surface and show the extreme values

syms x y lamd real

Question 6: Find the extreme values of the function f(x; y) = 2x2 + 12xy + y2 on the curve x2 + 4y2 = 25 Sketch the given surface and show the extreme values

syms x y lamd real

Question 7: Find the extreme values of the function f ( x , y )=x2+ y2 on the plane x

2 +

y

3 =1 Sketch the given surface and show the extreme values.

syms x y lamd real

Question 8: Find the local maximum and minimum values and saddle

points of f ( x , y )=x4+ y4−2 x2+ 4 xy−2 y2 Sketch the given surface and show

the extreme values.

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 9: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 10: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 11: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 12: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 13: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 14: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 15: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 16: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 17: Find the local maximum and minimum values and saddle

if ((Deltai>0) && (A1>0))

sprintf('The point (%8.4f, %8.4f) is minimum

point',double(criticalx(i)),double(criticaly(i)))

end

if ((Deltai>0) && (A1<0))

sprintf('The point (%8.4f, %8.4f) is maximum

Question 18: Find the surface area of the part of the sphere x2

polarfun = @(phi,r) fun(r.*cos(phi),r.*sin(phi)).*r;

rmax = @(phi) 2*sin(phi);

q = integral2(polarfun,0,pi,0,rmax)

area=double(q);

fprintf('The surface area is %6.2f',area)

%Sketch the figure %Rotate to observe the surface

syms phi theta z2

x1 = 2*sin(phi)*cos(theta), y1 = 2*sin(phi)*sin(theta), z1 = 2*cos(phi)x2 = cos(theta), y2 = 1+sin(theta)

x2 = cos(theta), y2 = 1+sin(theta)

figure; rotate3d on

fsurf(x1, y1, z1, [0 pi 0 2*pi], 'MeshDensity', 20); hold on

x1 = 2*cos(theta), y1 = 2*sin(theta)

fplot3(x1, y1, sym(0), [0 2*pi], 'r')

s1 = sym(sqrt(4)), y1 = s1*cos(theta), z1 = s1*sin(theta)

fplot3(sym(0), y1, z1, [0 2*pi], 'm')

hold on;

fsurf(x2, y2, z2, [0 2*pi -3 3], 'MeshDensity', 12)

axis equal; alpha 0.2

xlabel('x'); ylabel('y'); zlabel('z')

Question 19: Find the surface area of the sphere x2

fprintf('The surface area is %6.2f',area)

%Sketch the figure %Rotate to observe the surface

syms phi theta

x = 3*sin(phi)*cos(theta), y = 3*sin(phi)*sin(theta), z = 3*cos(phi)

figure

fsurf(x, y, z, [0 pi 0 2*pi], 'MeshDensity', 20); hold on

x = 3*cos(theta), y = 2*sin(theta)

fplot3(x, y, sym(0), [0 2*pi], 'r')

s1 = sym(sqrt(9)), y = s1*cos(theta), z = s1*sin(theta)

fplot3(sym(0), y, z, [0 2*pi], 'm')

axis equal; alpha 0.2

xlabel('x'); ylabel('y'); zlabel('z')

Question 20: Find the surface area of the part of the sphere x2

+ y2+ z2

= 2

that lies above the cone z2

= x2+ y2 Sketch the given surface.

x1 = sqrt(2)*sin(phi)*cos(theta), y1 = sqrt(2)*sin(phi)*sin(theta), z1 = sqrt(2)*cos(phi)

eq = z^2 == x^2 + y^2

figure

fsurf(x1, y1, z1, [0 pi 0 2*pi], 'MeshDensity', 20); hold on

x1 = sqrt(2)*cos(theta), y1 = 2*sin(theta)

fplot3(x1, y1, sym(0), [0 2*pi], 'r')

s1 = sym(sqrt(2)), y1 = s1*cos(theta), z1 = s1*sin(theta)

fplot3(sym(0), y1, z1, [0 2*pi], 'm')

hold on;

fsurf(r*cos(theta), r*sin(theta), r, [-3 3 0 2*pi], 'MeshDensity', 16)axis equal;alpha 0.2

xlabel('x'); ylabel('y'); zlabel('z')

Question 21: Find the surface area of the part of the surface z = xy that

lies inside the cylinder x2+ y2 = 1 Sketch the given surface.

fun = @(x,y) sqrt(1+x.^2+y.^2);

polarfun = @(phi,r) fun(r.*cos(phi),r.*sin(phi)).*r;

q = integral2(polarfun,0,2*pi,0,1)

area=double(q);

fprintf('The surface area is %6.2f',area)

%Sketch the figure %Rotate to observe the surface

syms z theta

x = cos(theta), y = sin(theta)

figure

fsurf(x, y, z, [0 2*pi -5 5], 'MeshDensity', 12)

axis equal; alpha 0.2

xlabel('x'); ylabel('y'); zlabel('z')

hold on

x=linspace(-1,1,30); y=linspace(-5,5,30); [x,y]=meshgrid(x,y);

f=x.*y

mesh(x,y,f)