# Questions in Section Calculus III: page 3

Use Lagrange multipliers to find the maximum and minimum values of f(x)=x+y+z along the intersection of the constraints x^2-y^2=z and x^2+z^2=4.

Find the partial derivative with respect to x for the expression x^2(y+1)^3=(y^2)/(x-1).

Consider the function below. By straightforward inspection (no differentiation is necessary), determine the coordinates (x,y) of the point of minimum.

Further, by using the standart procedure involving differentiation, determine the coordinates (x,y) of all critical points (you are NOT required to classify which point is minimum or maximum or saddle). Is the above mentioned point of minimum among them?

Find the partial derivatives (partialf)/(partialt) and (partialf)/(partials) of the function f(s,t)=e^(s/t)ln(s^2+t^2).

Evaluate the double integral iint(2y)/(1+x^2)dydx over the region shown in the Figure.

Find the equation for the tangent plane to the surface given by z=arctan(x+y^2)+tan(x/y) at the point with the coordinates x=0, y=1, z=pi/4.

Note: arctan means inverse tangent (same as tan^-1) and (arctanx)'=1/(1+x^2).

Calculate the integral I=int_A(x^2-y^2)dA, where A is the region between the circles of radius 1 and radius 2, axis x and the line y=x in the first quadrant. Make a sketch and shade the integration region. For your information: cos^2 theta-sin^2theta=cos(2theta).

Find a directional derivative of the function f(x,y)=(x^2-sqrt(y))ln[cos(x/y)]-sinx+sqrt(3)e^(y-1) in the direction of the vector a=5/2i+(5sqrt(3))/2j.

A particle moves along the line that starts at (1,1) and ends at (2,2), and is acted upon by the force F=3x^2i+3y^2j.

1. Calculate the work W done by the force on the particle using the definition of the work W=intF*dr.
2. Find the potential of the force F.
3. Calculate the work using the potential and compare to your answer in a).

Find the equation for the tangent plane to the surface given by z=e^-x+sin(y) at the point with the coordinates x=0,y=0.