Calculus Answers

Express the vector field vector F=xzi+(x^2+ y^2) j+( x/z)k in cylindrical polar coordinates.

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 × 10^3 J/kg/K and Latent heat of vaporisation is 2.27 × 10^7 J/kg .

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

Show that the function u(x,t)=e^(-6t)cos2x is a solution of the one dimensional heat equation

Evaluate integration of (t+1)^3e^t ,x=x to x=-1 is decreasing

Find the area enclosed by the curve r = a(1-cos theta)

If xsiny=sin(p+y), p belongs to R ,show that sinp.dy/dx + sin^2y=0

1) g(t) = t/2t +6

i) g(0)
ii) g(-3)
iii) g(10)
iv) g(x²)
v) g(t + h)
vi) g(t² - 3t + 1)

2) R(x) = √3+x- 4/x+1

i) R(0)
ii) R(6)
iii) R(-9)
iv) R(x+1)
v) R(x•4 - 3)
vi) R [ 1/x - 1]

1) f(x) = 3 - 5x - 2x²

i) f(4)
ii) f(0)
iii) f(-3)
iv) f(6-t)
v) f(7-4x)
vi) f(x+h)

A jogger runs from her home to a point A, which is 6 km away. For there 6 km, shebegins by running at a constant speed till she reaches a hilly portion 2 km from herhome. Here her speed slows down while she runs up the hill, which is a 1-km run.Then she speeds up while running down the hill.The last 2 km of the run are again atconstant speed. Draw a graph to show the jogger’s speed as a function of the distancefrom her home. Also find the range of this function.