RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
TUTORIAL-01
[SUCCESSIVE DIFFERENTIATION]
1) If , then P.T.
2) Find nth derivative of y =
3) Find yn if y=
4) If then show that yn = (-1)n-2 (n-2)![ -]
5) Find nth order derivative of
6) y= tan -1 [] P.T yn= (-1)n-1(n-1)! Sin( nθ ) sin(n) θ, where θ = tan -1 ()
7) U = eax cos(bx+c) then P.T. Un= (a2+b2)n/2 eax cos[bx + c + n tan-1(b/a)]
8) Find yn if y = e5x cosx cos3x
9) If y= xn logx then P.T yn+1 =
10) If U1/k + U-1/k = 2x then show that (x2-1)Un+2 + (2n+1)xUn+1+ (n2-k2) Un=0
11) If y = a cos (logx) + b sin (logx) then show that x2 yn+2 + (2n+1)x yn+1+ (n2+1)yn =0
12) If y = P.T (1-x2) yn -[2(n-1)x+1]yn-1- (n-1)(n-2)yn-2=0
13) If y= eacosx then show that (1-x2) yn+2 + (2n+1)x yn+1- (n2+a2)yn =0
14) If x = sinθ , y=sin(2θ) prove that (1-x2) yn+2 - (2n+1)x yn+1- (n2-4)yn =0
15) If x = sinh(y) prove that (1+x2) yn+2 + (2n+1)x yn+1+ n2yn =0
16) State Leibnitz’s Theorem and use it to find yn for y = e2x (x3+x+1)
17) If y = then prove that yn =[ log x- (1+++---+)]
18) U = (x2-1)n then prove that [(x2-1)Un] = n(n+1) Un where Un=U
19) y = tan-1[] then prove that (x2+a2) yn+2 + 2 (n+1)x yn+1+ n(n+1)yn =0
20) If u= sin [ log(1+2x+x2)] then prove that
(x+1)2Un+2 + (2n+1) (x+1)Un+1+ (n2+4) Un=0
TUTORIAL-02
[PARTIAL DIFFERENTIATION]
1) If then prove that and also find the value of
2) If u = log (x2+ y2 + z2) then prove that x = y = z
3) If xx yy zz = c, show that at x = y = z, = -[xlog(ex)] and also P.T
- 2xy + = at x = y= z
4) x = rcosθ , y = rsinθ then prove that + = [()+()]
5) If u= (8x2+ y2)[logx-logy] then find x+y
6) If u= sin[] then show that = -
7) Verify the Euler’s theorem for u = x2 tan -1 () - y2 tan -1 () where xy 0
and also show that =
8) If F(x,y) = ++ then show that x+y+2F(x,y) =0
9) If u= tan -1 [] Then prove that x2+ 2xy+ y2 = 2cos3u sinu
10) If u= then prove that
x2+ 2xy+ y2= tan u[]
11) If x = ecos(rsinθ) and y = esin(rsinθ) then prove that
= and = - Hence deduce that ++= 0
12) If z = f(u,v) where u = x2-2xy-y2 and v=y then show that
(x+y) + (x-y) = 0 is equivalent to =0
13) If logeθ = r-x where r2= x2+ y2 then show that =
14) If F=F(x,y) and If x+y = 2eθcos(Ф) , x-y= 2ieθsin(Ф) then prove that
15) State and Prove Euler’s Theorem on Homogeneous Functions of Three
Independent Variables.
16) If θ= te then find the value of ‘n’ for which
17) If u = f(r) and x = rcoθ , y = rsinθ , then P.T., where
18) If , and lx+my+nz=0 then prove that
TUTORIAL NO -03
[Applications of Partial Derivatives, Jacobian]
1] Show that the minimum value of u =xy+a(+) is 3a
2] Find the dimensions of the rectangular box with open top of maximum
capacity whose surface area is 432 cm2
3] Find maximum value of xm yn zp where x + y +z = a
4] Find the minimum and maximum distances from the origin to the
Curve 3x2+4xy+6y2=140
5] Divide 24 into three parts such that the continued product of the first,
square of the second and cube of the third may be maximum.
6] Determine the point on paraboloid z = x2+y2 which is closest to the
Point ( 3, -6, 4)
7] Find the minimum distance of any point from the origin on the plane
x +2y +3z = 14.
8] Find the minimum and maximum distance of the point (3,4,12) from
The Sphere x2 + y2+ z2 = 1
9] Examine the minimum and maximum on the surface
10] Examine the minimum and maximum on the surface
11] A rectangular box , open at top is to have a volume of 108 m3. Find
the dimensions of the box so that the surface area is minimum.
12] Find if
1) u = x siny, v = y sinx, 2)
3) , 4) u = x + y, y = uv.
13] Find = J and = J’ for
1) , 2)
14] Find
15] Find if u = 2axy, v = a (x2 + y2) and x = r cos ө , y = r sin ө.
16] find and
u = r sinө cosφ, v = r sinө sinφ , w = sinө cosφ
TUTORIAL NO -04
[Indeterminate Forms,Expansions of functions Fitting of curves]
A] Evaluate the following limits:
1) 2) 3) 4)
5) Find values of a, b, c such that
6) Find the constants a, b, c if
7) Find value of ‘p’ if is finite.
8) Evaluate 9) Evaluate
10) Find a, b if 11) Evaluate
12) Prove that 13) Evaluate
14) Evaluate 15) Evaluate
16) Evaluate 17) Evaluate
18) Evaluate
B]
1) Prove that ecosbx = 1+ ax +
2) Prove that
3) Show that log(secx)=
4) P.T sin[e-1]=
5) P.T. log[1+e]=
6) P.T. sinx = x+
7) S.T. cos[tanh(logx)]=
8) Expand in powers of x and Hence show that
9) Expand log[] upto term in x
10) If y = then prove that x=
11) Show that xcosecx =
12) Express 2x +3x-8x+7 in terms of (x-2) by using Taylor’s Theorem.
13) By Using Taylor’s Theorem arrange the function in Powers of x
[7+(x+2) +3(x+2)+(x+2)-(x+2)]
14) Calculate the value of to four decimal places using Taylor’s
Theorem
15) Use TSE to calculate approximately (63.7)
16) Use TSE to calculate approximately (2.98)
C) 1) Fit a straight line for given value.
x / 12 / 15 / 21 / 25y / 50 / 70 / 100 / 120
2) Fit a straight line for given value.
Production in ton / 8 / 10 / 12 / 10 / 16
Find expected production in 2021.
3) Fit a straight line for given value
x / 0 / 1 / 3 / 6 / 8Y / 1 / 3 / 2 / 5 / 4
Find y(15)
D] Fit a Parabolic equation for the data given below
1) Find y(10)
x / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9y / 2 / 6 / 7 / 8 / 10 / 11 / 11 / 10 / 9
2) Fit x=a+by+cy2
X / -3 / -2 / -1 / 0 / 1 / 2 / 3Y / 4.63 / 2.11 / 0.67 / 0.09 / 0.63 / 2.15 / 4.58
Find x(2)
3) Using least squarer method fit y=a+bx+cx2 for (-1,2) (0,0) (0,1) (1,2)
E) 1) Fit x=aby for
Y / 2 / 3 / 4 / 5 / 6X / 144 / 172.8 / 207.4 / 248.6 / 298.5
2) Fit y=abx for the data. Find y at 25.
X / 1 / 2 / 5 / 10 / 20 / 30 / 40 / 50Y / 98.2 / 91.7 / 81.3 / 64.0 / 36.4 / 32.6 / 17.1 / 11.3
3) Fit y=aebx for
X / 0 / 1 / 2 / 3Y / 0.5 / 3.69 / 27.3 / 201.7
TUTORIAL NO -05
[Matrices]
1) Find the rank of the matrix using Row-echelon form of following matrix.
i) ii)
2) Reduce the matrix to normal form and find it’s rank
i) ii)
3) Find the non-singular matrix P and Q such that PAQ is in normal form,
i) ii)
4) Solve the following equations
i) x+2y+3z=0; 2x+3y+z=0; 4x+5y+4z=0; x+2y-2z=0
ii) x+y+2z=0; x+3y+4z=0; x+2y+3z=0; 3x+4y+7z=0
iii) 2x-y+3z=0; 3x+2y+z=0; x-4y+5z=0
iv) 3x+y-5z=0; 5x+3y-6z=0; x+y-2z=0; x-5y+z=0
5) Examine whether the following vectors are linearly dependent or independent
i) [2,1,1] , [1,3,1], [1,2,-1]
ii) [3,1,-4], [2,2,-3], [0,-4,1]
6) Test the following equations for consistency
i) x+2y-z=1; x+y+2z=9; 2x+y-z=2
ii) x-3y-3z=-10; 3x+y-4z=0; 2x+5y+6z=13
iii) 6x+y+z=-4; 2x-3y-z=0; -x-7y-2z=7
7) Find the values of λ for which the system of equations x+y+4z=1;
X+2y-2z=1; λx+y+z=1.
8) Solve the following equations by Gauss elimination method;
i) x+y+z=2; 2x+2y-z=1; 3x+4y+z=9
ii) 2x+y+z=10; 3x+2y+3z=18; x+4y+9z=16
9) Solve the following equations by Gauss-Jorden method,
i) 3x+2y-2z=4; x-2y+3z=6; 2x+3y+4z=15
ii) X+2y+z=8; 2x+3y+4z=20; 4x+3y+2x=16
10) Solve the equations by using Gauss-Seidel method,
i) 15x+y+z=17; 2x+15y+z=18; x+2y+15z=18
ii) 10x+2y+z=9; 2x+20x-2z=-44; -2x+3y+10z=22
TUTORIAL NO -06
[Complex Numbers]
Part I
1) Express the complex numbers in the form x+iy
i) ii) iii)
2) Find modulus and amplitude of
i) ii) iii)
3) Prove that the statements ‘ Re(z)>0’ and are equivalent
4) Express in polar form i)1-cosα+isinα ii)
5) If α & β are roots of x2-2x+2=0 then show that
6) If z=cosθ + isinθ , show that
7) If z1, z2 are two complex numbers then show that
8) If x and y are real, solve the equation
9) If (1+cosθ+isinθ)(1+cos2θ=isin2θ)=u+iv prove that
i) ii)
10) If sinθ+sinФ=0 & cosθ+cosФ=0 then prove
cos2θ+cos2Ф= 2cos(Π+θ+Ф) & sin2θ+sin2Ф = 2 sin(Π+θ+Ф)
11) Use De`moivre`s Theorem to solve equation x-x+1=0
12) solve
13) Express cosθ in term of multiple of θ
14) Express sinθ in term of multiple of θ
15) Express in powers of sinθ only
16) Show that
17) Show that
18) Simplify
Part II
1] Find tanhx if 5sinhx-coshx=5
2] S.T = cosh6θ+sinh6θ
3] If y=log(tanx) P.T i) sinhny =
ii) 2coshnycosec2x = cosh(n+1)y+cosh(n-1)y
4] If u= log tan[] P.T i) tanh = tan ii) coshu = secθ
5] S.T
6] If tan(α+iβ)=x+iy show that i) x2+y2+2xcot(2α)=1
ii) x2+y2-2ycoth(2β)+1=0
7] P.T i) ii) sinh(tanx) = log tan[]
8] P.T sin(cosecθ)=
9] If log[ cos(x-iy)]= α+iβ , P.T α= and find β.
10] Separate into real & imaginary part of ii
11] S.T
12] P.T
13] S.T i) sin(log i)=1 ii) cos(log i)=0
14] If Then S.T (α+β)= e and tan()=
15] P.T
16] P.T , is it defined for all values of z
Tutorial 7
Topic: Scilab
//Programme for Crout's Method ( Decomposition Method )
//Name:
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
l(1,1)=A(1,1);
l(1,2)=0;
l(1,3)=0;
u(1,1)=1;
u(1,2)=A(1,2)/l(1,1);
u(1,3)=A(1,3)/l(1,1);
u(2,1)=0;
u(2,2)=1;
u(3,1)=0;
u(3,2)=0;
u(3,3)=1;
l(2,1)=A(2,1);
l(2,2)=A(2,2)-(l(2,1)*u(1,2));
l(2,3)=0;
l(3,1)=A(3,1);
u(2,3)=(A(2,3)-l(2,1)*u(1,3))/l(2,2);
l(3,2)=A(3,2)-l(3,1)*u(1,2);
l(3,3)=A(3,3)-(l(3,1)*u(1,3)+l(3,2)*u(2,3));
L=l;
U=u;
B=input('Enter 3*1 constants matrix B:')
v=L\B;
x=U\v;
disp('A')
disp(A)
disp('B')
disp(B)
disp('Lower triangular Matrix L is')
disp(L)
disp('Upper triangular Matrix U is')
disp(U)
disp('By Crout Method,')
disp('Ans')
disp(x)
Tutorial 8
Topic: Scilab
//Programme for Cuvrve fitting 1 (line)
//Name:
//Roll No.:
//Batch: And Div:
//Fitting a straight line y= a + bx
clc
x=input('Enter the values of x:')
y=input('Enter the values of y:')
n=length(x)
sx=sum(x)
sx2=sum(x^2)
sy=sum(y)
sxy=sum(x.*y)
A=[n sx;sx sx2]
B=[sy;sxy]
c=linsolve(A,-B)
disp(c)
//Programme for Curve fitting 2( Parabola)
//Name:
//Roll No.:
//Batch: And Div:
//Fitting a parabola y = a + bx +c x^2
clc
x=input('Enter the values of x:')
y=input('Enter the values of y:')
n=length(x)
sx=sum(x)
sx2=sum(x^2)
sy=sum(y)
sxy=sum(x.*y)
sx3=sum(x^3)
sx4=sum(x^4)
sx2y=sum((x^2).*y)
A=[n sx sx2;sx sx2 sx3; sx2 sx3 sx4]
Tutorial 9
Topic: Scilab
// Programme for Gauss Joran Method
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter 3*1 constants matrix B:')
A_Aug=[A,B]
a=rref(A_Aug)
disp('A')
disp(A)
disp('B')
disp(B)
disp('Agmented Matrix [A:B] is')
disp(A_Aug)
disp('By Gauss Jordan Ellimination,')
disp(a)
disp('Ans')
disp(a(1:3,4))
Tutorial 10
Topic: Scilab
//Programme for Gauss Jacobi Method
//Name:
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter matrix B:')
i=input('Enter initial values')
for j=1:7
x1=(B(1)-(A(1,2)*i(2))-(A(1,3)*i(3)))/A(1,1);
x2=(B(2)-(A(2,1)*i(1))-(A(2,3)*i(3)))/A(2,2);
x3=(B(3)-(A(3,1)*i(1))-(A(3,2)*i(2)))/A(3,3);
i(1)=x1;
i(2)=x2;
i(3)=x3;
b=i;
disp(b)
end
//Programme for Gauss Seidal Method
//Name:
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter matrix B:')
i=input('Enter initial values')
for j=1:5
x1=(B(1)-(A(1,2)*i(2))-(A(1,3)*i(3)))/A(1,1);
i(1)=x1;
x2=(B(2)-(A(2,1)*i(1))-(A(2,3)*i(3)))/A(2,2);
i(2)=x2;
x3=(B(3)-(A(3,1)*i(1))-(A(3,2)*i(2)))/A(3,3);
i(3)=x3;
b=i;
disp(b)
end
Applied Mathematics-I 9