Calculus Answers

Determine the set of points at which the function is continuous
1) G(x,y)=tan^-1((x+y)^-2)
2) f(x,y,z)=√y÷(x^2-y^2+z^2)
3) f(x,y,z)=√z+y+z

4) f(x,y)={x^2y^3÷(2x^2+y^2 if (x,y)≠(0,0)
1 if (x,y)=(0,0)

5) f(x,y)={ xy÷(x^2+xy+y^2 if (x,y)≠(0,0)
0 if (x,y)=(0,0)

Determine the set of points at which the function is continuous.
1) F(x,y)=sin(xy)÷(e^x-y^2)
2) F(x,y)=x-y÷(1+x^2+y^2)
3) F(x,y)=arctan(x+√y)
4) F(x,y)=e^x^2y+√x+y^2
5) G(x,y)=ln(x^2+y^2-4)

If c∈Vn, show that the function f given by f(x)=c.x is continuous on R^n.

Show that the function f given by f(x)=|x| is continuous on R^n. [Hint: Consider |x-a|^2=(x-a).(x-a).]

find dy/dx and d^2y/dx^2 if y=1+x/1+x^2

Find dy/dx and d^2y/dx^2 if y=1+x/1+x^2

Exercise 25: Let f : [a, b] → R be continuous and ϕ : [a, b] → R be integrable and
non-negative, i.e. ϕ ≥ 0. Then there exists c ∈ (a, b) such that
the integral of f(x)ϕ(x)dx from a to b = f(c) the integral of ϕ(x)dx from a to b.

Exercise 25: Let f : [a, b] → R be continuous and ϕ : [a, b] → R be integrable and
non-negative, i.e. ϕ ≥ 0. Then there exists c ∈ (a, b) such that
the integral of f(x)ϕ(x)dx from a to b = f(c) the integral of ϕ(x)dx from a to b.

Determine the \\(curl F\\) at the point (2, 0, 3) given that \\(F=xz i+(2x^2-y)j-yz^2 k\\).

1. Change the order of the integration in the following double integral:
∫_(√3)^1dx ∫_(-√(4-x^2 ))^0ydy

2. Evaluate the following integral:

∫(3x-5)/√(9+6x-3x^2 ) dx