# Questions in Section Calculus III

How can I find the stationary points of a multivariable function? For example: f(x,y)= 3+ cos(x) sin(2y). I know that I've to set their partial derivatives to zero but how can I sovle the equation?

Thank you!

True or False?

A sphere with center P(x_0, y_0, z_0) and radius r consists of all points (x, y, z) that satisfy the inequality (x-x_0)^2+(y-y_0)^2+(z - z_0)^{2} <= r^2.

True or False?

If a point belongs to both the xy-plane and the xz-plane, then the point lies on the x-axis.

True or False?

The graph of x^2+y^2=1 in 3-space is a circle of radius 1 centered at the origin.

True or False?

The line through the points (-3, 8,-4) and (1, 0, 8) intersects the yz-plane at (0,2,5).

Find the parametric equation of the line through (0,3) that is parallel to the line x=-5+t, y=1-2t.

Find the point on the line segment joining P_1=(1, 4,-3) and P_2=(1, 5,-1) that is 2/3 of the way from P_1 to P_2.

We need to construct a water channel with aluminium material. We have a piece of rectangular aluminium plate with width 24 cm. As shown in Figure, the channel will be formed with a shape of isosceles trapezoid (with an open topside).

Evaluate the line integral int_C[(4y+5)dx+2xzdy+(yz-x)dz] along the following paths C:

1. C is defined parametrically by x=2t^2, y=6t, z=t^3 from t=0 to t=1.
2. C consists of straight lines from (0,0,0) to (0,1,0) then to (0,1,1) and then to (6,1,1).
3. C is given by the straight line joining (0,0,0) to (1,2,6).

For the double integral I=int_0^6 int_y^6 xy sqrt(1+x^4)dxdy:

1. sketch the region of integration;
2. reverse the order of integration and then evaluate the integral analytically.