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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].`
Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series.
Concept: Taylor’S Theorem (Statement Only)
Prove that `tan_1 x=x-x^3/3+x^5/5+.............`
Concept: Taylor’S Theorem (Statement Only)
Show that `sin(e^x-1)=x^1+x^2/2-(5x^4)/24+`...................
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Find the maxima and minima of `x^3 y^2(1-x-y)`
Concept: Maxima and Minima of a Function of Two Independent Variables
Using Newton Raphson method solve 3x – cosx – 1 = 0. Correct upto 3 decimal places.
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
If `u=r^2cos2theta, v=r^2sin2theta. "find"(del(u,v))/(del(r,theta))`
Concept: Jacobian
Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.
Concept: Maxima and Minima of a Function of Two Independent Variables
Examine the function `f(x,y)=xy(3-x-y)` for extreme values & find maximum and minimum values of `f(x,y).`
Concept: Maxima and Minima of a Function of Two Independent Variables
Show that xcosecx = `1+x^2/6+(7x^4)/360+......`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
If y= cos (msin_1 x).Prove that `(1-x^2)y_n+2-(2n+1)xy_(n+1)+(m^2-n^2)y_n=0`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
If coshx = secθ prove that (i) x = log(secθ+tanθ). (ii) `θ=pi/2tan^-1(e^-x)`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Prove that `cos^-1tanh(log x)+ = π – 2(x-x^3/3+x^5/5.........)`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
If` y= e^2x sin x/2 cos x/2 sin3x. "find" y_n`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Evaluate `Lim _(x→0) (cot x)^sinx.`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Prove that log `[sin(x+iy)/sin(x-iy)]=2tan^-1 (cot x tanhy)`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
If 𝒚 satisfies the equation `(dy)/(dx)=x^2y-1` with `x_0=0, y_0=1` using Taylor’s Series Method find 𝒚 𝒂𝒕 𝒙= 𝟎.𝟏 (take h=0.1).
Concept: Taylor’S Series Method
`"If" sin^4θcos^3θ = acosθ + bcos3θ + ccos5θ + dcos7θ "then find" a,b,c,d.`
Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Use Taylor’s series method to find a solution of `(dy)/(dx) =1+y^2, y(0)=0` At x = 0.1 taking h=0.1 correct upto 3 decimal places.
Concept: Taylor’S Series Method
Use Taylor series method to find a solution of `dy/dx=xy+1,y(0)=0` X=0.2 taking h=0.1 correct upto 4 decimal places.
Concept: Taylor’S Series Method
