Advanced Math

Question : Find the Fourier series expansion of f(x)=-1 if -1<x<0 and 1 if 0<x<1 in the interval [-1 1] Solution: For the Given 2 periodic function \[f\left(x\right)=\ -1 -1<x<01…

Question : Verify the initial value theorem for the function f(t) =2e^(-3t) Solution : Given function \[𝒇(𝒕) = 𝟏 + 𝒆^{-𝒕}(𝒔𝒊𝒏𝒕 - 𝒄𝒐𝒔𝒕)\] First we shall verify initial value theorem…

Question : Verify the initial value theorem and Final value theorem for the following function\[𝒇(𝒕) = 𝟏 + 𝒆^{-𝒕}(𝒔𝒊𝒏𝒕 - 𝒄𝒐𝒔𝒕)\] Solution : Given function \[𝒇(𝒕) = 𝟏 + 𝒆^{-𝒕}(𝒔𝒊𝒏𝒕…

Question : Find the final value of the transfer function \[F\left(s\right)=\frac{5005}{s\left(s^2+2s+1001\right)} \] Solution : We shall Use final value theorem to find final value f(infinity) of the transfer function \[F\left(s\right)=\frac{5005}{s\left(s^2+2s+1001\right)}\]…

Question : Find the initial value of the transfer function \[F\left(s\right)=\frac{8}{s\left(s^2+2s+1\right)} \] Solution : We shall Use initial value theorem to find initial value f(0) of the transfer function \[F\left(s\right)=\frac{8}{s\left(s^2+2s+1\right)}\]…

Question : Prove that derivative of cos(x) is -sin(x) Solution : Let \[f\left(x\right)=\sin{\left(x\right)}\] By the definition of the derivative of a function \[\frac{df\left(x\right)}{dx}=limit_{∆x→0}\frac{f(x+∆x)-f(x)}{x+∆x-x}\] Using the definition of f(x) \[=limit_{∆x→0}\frac{cos(x+∆x)-cos(x)}{∆x}\] By…

Question : Show that the limit (xy+y^3)/(x^2+y^2) as (x y)->(0 0) does not exist \[limit_{\left(x,y\right)\rightarrow(0,0)}\frac{xy+y^3}{x^2+y^2} does \ not \ exist \] Solution : Let \[ L=limit_{\left(x,y\right)\rightarrow(0,0)}\frac{xy+y^3}{x^2+y^2}\] Limit along the path…

Question : Solve y′′ + y = sec x by variation of parameters. Solution : \[y^{\prime\prime}+y=\sec{\left(x\right)} \] (1) To find complementary solution of ode (1) we solve corresponding homogeneous eqn…

Question : Find real and imaginary parts of z^2 . Solution : Any complex number is represented as \[z=x+iy\] Where x , y are known as real and imaginary parts…