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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
Find maximum and minimum values of x3 +3xy2 -15x2-15y2+72x.
Concept: Maxima and Minima of a Function of Two Independent Variables
Expand 2 𝒙3 + 7 𝒙2 + 𝒙 – 6 in power of (𝒙 – 2) by using Taylors Theorem.
Concept: Taylor’S Series Method
Expand sec x by McLaurin’s theorem considering up to x4 term.
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)
Evaluat `lim_(x->0) (e^(2x)-(1+x)^2)/(xlog(1+x)`
Concept: L‐ Hospital Rule
Solve the following equation by Gauss-Seidel method upto four iterations
4x-2y-z=40, x-6y+2y=-28, x-2y+12z=-86.
Concept: Gauss Seidal Iteration Method
Solve the following system of equation by Gauss Siedal Method,20x+y-2z=17
3x+20y-z =-18
2x-3y+20z=𝟐𝟓
Concept: Gauss Seidal Iteration Method
Find the roots of the equation `x^4+x^3 -7x^2-x+5 = 0` which lies between 2 and 2.1 correct to 3 places of decimals using Regula Falsi method.
Concept: Regula – Falsi Equation
Evaluate : `lim_(x->0)((2x+1)/(x+1))^(1/x)`
Concept: L‐ Hospital Rule
