BE Electrical Engineering छमाही १ (इंजीनियरिंग) - University of Mumbai Important Questions for Applied Mathematics 1

Find the maximum and minimum values of `f(x,y)=x^3+3xy^2-15x^2-15y^2+72x`

If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z
Prove that

If tan(θ+iφ)=tanα+isecα
Prove that
1)`e^(2varphi)=cot(varphi/2)`
2) `2theta=npi+pi/2+alpha`

If `z=tan(y-ax)+(y-ax)^(3/2)` then show that `(del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)`

State Euler’s theorem on homogeneous function of two variables and if `u=(x+y)/(x^2+y^2)` then evaluate `x(delu)/(delx)+y(delu)/(dely`

If u =`f((y-x)/(xy),(z-x)/(xz)),` show that `x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=0`.

If `u=sin^(-1)((x+y)/(sqrtx+sqrty))`,Prove that

`x^2u_(x x)+2xyu_(xy)+y^2u_(yy)=(-sinu.cos2u)/(4cos^3u)`

State and Prove Euler’s Theorem for three variables.

If `u=e^(xyz)f((xy)/z)` where `f((xy)/z)` is an arbitrary function of `(xy)/z.`

Prove that: `x(delu)/(delx)+z(delu)/(delz)=y(delu)/(dely)+z(delu)/(delz)=2xyz.u`

If Z=f(x.y). x=r cos θ, y=r sinθ. prove that `((delz)/(delx))^2+((delz)/(dely))^2=((delz)/(delr))^2+1/r^2((delz)/(delθ))^2`

State and prove Euler’s Theorem for three variables.

If z = f (x, y) where x = e^u +e^-v, y = e^-u - e^v then prove that `(delz)/(delu)-(delz)/(delv)=x(delz)/(delx)-y(delz)/(dely).`

If U `=sin^(-1)[(x^(1/3)+y^(1/3))/(x^(1/2)+y^(1/2))]`prove that `x^2(del^2u)/(del^2x)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(del^2y)=(tanu)/144[tan^2"U"+13].`

Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series. 

Prove that `tan_1 x=x-x^3/3+x^5/5+.............`

Show that `sin(e^x-1)=x^1+x^2/2-(5x^4)/24+`...................

Find the maxima and minima of `x^3 y^2(1-x-y)`

Using Newton Raphson method solve 3x – cosx – 1 = 0. Correct upto 3 decimal places. 

If `u=r^2cos2theta, v=r^2sin2theta. "find"(del(u,v))/(del(r,theta))`

Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.