Calculus Answers

fine laplace transforms of : (t-1) u(t-1)

[SADT8] If A

 and B

 are vector fields, prove the following:

∇(A⋅B)=(B⋅∇)A+(A⋅∇)B+B×(∇×A)+A×(∇×B).

A rectangular box whose volume is 32 is open at the top. If the surface of the area is 2( L + B )H +LB, where L L, B, H are length, breath and height respectively.

(A) Find the dimension of the box that may regure least material

(b) Investigate weather the dimension found require least material.

Determine the nature of the stationary value

X = t³ - 3t + ty²

Let r=xi^+yj^+zk^ and r=||r||. Show that:

∇(lnr)=r/r^2.

and

∇×((r^n)r)=0.

If A(u) is a differentiable vector function of u and ||A(u)||=1 , prove that dA/du is perpendicular to A .

Let E:=Exi+Eyj+Ezk  and H:=Hxi+Hyj+Hzk be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:

∇⋅E=0,∇⋅H=0,∇×E=−1/c. ∂H/∂t, ∇×H=1/c. ∂E/∂t

prove that E and H satisfy the equation

∇^2Ei =1/c^2. ∂^2Ei/∂t^2  and ∇^2Hi=1/c^2. ∂^2Hi/∂t^2

Here, i=x,y  or z.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇^V.

The gain of an amplifier is found to be G = 20 log(10Vout) Determine equations for: dG dVout d2G dVout?

Differentiate of the following functions with respect to x:

i) 𝑡𝑎𝑛𝑥∙ 𝑙𝑛(𝑠𝑖𝑛𝑥)

ii) √𝑐𝑜𝑡√𝑥

iii) 𝑒 ln (𝑡𝑎𝑛5𝑥)

iv) 𝑆𝑖𝑛2

{ln(𝑠𝑒𝑐𝑥)}

v) ln (𝑐𝑜𝑠𝑒𝑐𝑥)/𝑥