Calculus Answers

trace the curve y^2(1+x^2)=x^2(1-x^2) stating all the properties used

A closed cylindrical tank with radius r and height h has a surface area of 100 sq. units.
Express its volume in terms of r.

A tent in the shape of a pyramid with a square base is to be constructed from a piece of material having
a side of length 5 meters. In the base of the pyramid, let x be the distance from the center to a side (see
figure below). Find a mathematical model expressing the volume of the tent as a function of x.
(The volume of a pyramid is V = Bh, where V, B and h are the volume, base area and height
of the pyramid respectively).

Differentiate f (x) = (8x^3 +4x)(2x^2 +5) using product rule

Differentiate (x^5 +6)^5 using chain rule

Find all the first and second order partial derivatives of f(x,y)=5sin(2x+y)−10cos(x−y).

A. ∂f/∂x=fx=
B. ∂f/∂y=fy=
C. ∂^2f/(∂x^2)=fxx=
D. ∂^2f/(∂y^2)=fyy=
E. ∂^2f/(∂x∂y)=fyx=
F. ∂^2f/∂y∂x=fxy=

Find the first partial derivatives of f(x,y,z)=z arctan(y/x) at the point (4, 4, -4).

A. ∂f/∂x(4,4,−4)=
B. ∂f/∂y(4,4,−4)=
C. ∂f/∂z(4,4,−4)=

Find the limits, if they exist, or type DNE for any which do not exist.

lim(x,y)→(0,0) 5x^2/(x^2+5y^2)
1) Along the x-axis:

2) Along the y-axis:

3) Along the line y=mx :

4) The limit is:

Each of the following functions has a set on which it is continuous and that set has a boundary. Match the verbal description of this boundary with the function by putting the letter of the boundary to the left of the letter of the function.

1. f(x,y)=1/(4−x^2−y^2)
2. f(x,y,z)=xyz/(x^2+y^2−z)
3. f(x,y,z)=1/(x^2+y^2+z^2)
4. f(x,y,z)=z/(1−x^2−y^2)
5. f(x,y)=xlny

A. one point
B. a circular parabaloid
C. a circular cylinder
D. a straight line
E. a circle