Calculus Answers

1) Find the limit or explain why does it not exist

i) lim┬(h→0)⁡〖((x+h)^²-x²)/h〗
ii) lim┬(x→0)⁡〖((x+h)^²-x²)/h〗
iii) lim┬(x→0)⁡〖(1/(2+x)-1/2)/x〗
iv) lim┬(x→0)⁡〖((2+x)^3-8)/x〗

2) Find lim lim┬(∆x→0)⁡〖(f(x+∆x)-f(x))/∆x〗 if

i) f(x) = 4/x
ii) f(x) = x²-4x

A transmission line with characteristic impedance 250 Ω is connected to a pure
resistive load of 1000 Ω through a λ/4 line segment. Calculate the characteristic
impedance of this segment to achieve perfect matching between the load and the
transmission line.

b) A surface wave with frequency ω = 3 × 10^(−5)
rad/ s
is propagating at a metal-free
space boundary. The plasma frequency of metal is 9 × 10^(15) rad and εr = 7. Estimate
k. If the amplitude of the wave is 10^(5)
Vm−1
at the surface, what will be its value at
the depth of 0.2 µm in the metal? Ignore the collision effect.

If y = e^{m tan^(-1)x},show that (1+x^2)y n+1 + ( 2nx-m)yn + n(n-1)=0

2 kg of water is heated from 0°C to 100°C and converted into steam at the same
temperature. Calculate the increase in entropy, given that specific heat of water is
4.18 × 103
J kg−1
K−1
and Latent heat of vaporisation is 2.27 × 107
J kg−1
.

2 kg of water is heated from 0°C to 100°C and converted into steam at the same
temperature. Calculate the increase in entropy, given that specific heat of water is
4.18 × 103
J kg−1
K−1
and Latent heat of vaporisation is 2.27 × 107
J kg−1
.

Solve the following ODE using power series method. (x+2)y"+xy'-y=0

Show that the function u(x,y)=arc tan(y/x) is a solution of 2 Dimensional Laplace equation

x = 2t – 1
y = t2 + 5, -4 ≤ t ≤ 4

Find the interval if following function for f(x)=e^x(2-3x-x^3)-6/e is decreasing