Partial Derivatives

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1. Consider the function 
        f(x,y)=(x2+y2-1)1/2
(2 points) The domain of f is
the entire plane the exterior of the unit circle (including the circle) the interior of the unit circle
(2 points) The range of f is the set of
all real numbers all numbers greater than or equal to 1 all numbers greater than or equal to 0
(2 points) The level curve f(x,y)=31/2 is
an ellipse a circle of radius 4 a circle of radius 2

2. Three resistors, x, y, z ohms, are connected in parallel to give a net resistance of w ohms, where 

        w=f(x,y,z)=(1/x+1/y+1/z)-1
If x=301, y=299, z=302, estimate w by replacing f(x,y,z) by the equation of its tangent plane at (x,y,z)=(300,300,300).
(6 points) The estimate for w is

3. Suppose f is a differentiable function with 

        fx(0,0,0)=3, fy(0,0,0)=7, fz(0,0,0)=-2 .
Let the function g be defined by 
        g(s,t)=f(s2-t2,4s-4t,5s-5).
(3 points) The partial derivative gs(1,1) is equal to
(3 points) The partial derivative gt(1,1) is equal to

4. Suppose that the temperature T (measured in degrees Celsius) at a point (x,y,z) in space is given by 

        T(x,y,z)=100-x2-y2-2z2,
where x,y,z is measured in centimeters.
(2 points) The rate of change of T at the point P(2,1,1) in the direction of <2/3,1/3,-2/3> is equal to
(3 points) One should move in the direction (unit vector) i + j + k from the point P in order to cool of most rapidly.
(2 points) The maximum rate of change of T at P is degree Celsius per centimeter.

5. The equation of the tangent plane to the sphere 

        x2+y2+z2=14
at the point (-1,3,2) is given by the equation
(6 points) x + y + z = 18

6. We want to determine the number of critical points of the function 

        f(x,y)=x2+3xy2+y2.
(4 points) The function f has critical points.
(4 points) The function f has local minima, local maxima and saddle points.

7. We want to find the extrema of the function 

        f(x,y)=y2-x2+x-xy+1
on the triangular domain with vertices at (0,0), (1,0), (0,1).
(4 points) The absolute maxium value is
(4 points) The absolute minimum value is

8. We apply the Lagrange multiplier method in order to find the extrema of the function 

        f(x,y)=x2-y2-y
subject to the constraint 
        x2+y2=1.
(3 points) We find critical points.
(2 points) The absolute maximum value is
(2 points) The absolute minimum value is

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