Calculus Answers

trace the curve x(y2+4)=8. stating all the points used while doing so

Trace the curve x=a(@-Sin@), y=a(1-Cos@).State the properties used for tracing it.

determine f'(x) if f(X)=(x^3-2x+1)^5

Q. Show that the circular cylinder S={(x,y,z)∈R^3 |x^2+y^2=1} can be covered by a single surface patch and so a surface.

Q. The hyperboloid of one sheet is S={(x,y,z)∈R^3 |x^2+y^2-z^2=1} show that for every θ,the straight line (x-z)cosθ=(1-y)sinθ,
(x+z)sinθ=(1+y)cosθ
Is contained in S and that every point of hyperboloid lies on one of these deduce that S can be covered by a single surface patch, and hence is a surface.

f(x,y)=5xy-ln(xy)-5,check whether there exists a continuously differentiable function g defined by f(x,y)=0 in the neighbourhood of x= 3,such that g(3)=1/3.find g'(3) if exists.

Find dy/dx for the function- x=a cos^3 t, y=a sin^3 t

Using Stokes’ Theorem evaluate the line integral ∫ F.dl where F = < y, xz^3, -zy^3 >
and C is a circle x^2 + y^2 = 4, in the plane z = − 3 ?

1) Find the values of the following functions

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

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

b) R(x) = √3+x 4/(x+1)

i) R(0)
ii) R(6)
iii) R(-9)
iv) R(x+1)
vi) R(x⁴- 3)
v) R(1/x-1)