School of Mathematics and Statistics
Carleton University
Math. 69.204B
ANSWERS TO TEST 3

• 1. [1.5 marks] An equation for the tangent plane to the level surface f(x,y,z)=4 of the function at the point P=(2,1,1) can be written as 4(x-2)+(y-1)-2(z-1)=0.

  Reason: , which is (4,1,-2) at P.

• 2. [1.5 marks] Suppose that the gradient of f(x,y) is given by and the location of a traveller at time t is given by the position vector . Then the rate of change of f at t according this traveller, namely , is .

  Reason: With x=t and , we have and . Hence

  

• 3. [1 mark] Let C be the intersection of the surface and the plane . Then the straight line tangent to the curve C at the point (1,1,2) can be written in parametric equations as x=1+t, y=1-3t, z=2-4t. (Hint: Find a normal vector to S and and a normal vector to . The tangent line is perpendicular to both normal vectors.)

  Reason: The gradient vector of at P=(1,1, 2) is

  

  which is normal to S at P. On the other hand, is normal to the plane . So the vector is tangent to the curve at P.

• 4. [1.5 marks] Suppose we have w=f(x,y,z) with , and . Also suppose , and z=st so that w can be regarded as a function of s and t. Then .

  Reason: By the chain rule, we have

  

• 5. [2 marks: 1 mark for the method; 0.5 mark for the critical point; 0.5 mark for the minimum value] Use the method of Lagrange multipliers to find the minimum value of the function f(x,y,z)=2x+4y+z subject to the constraint .

  Answer. The condition yields

  

  giving , x=-1, y=-4/2=-2 and . Thus the minimum value of f subject to g=0 is .

• 6. [2.5 marks: 1 mark for the correct method, 0.5 mark for the critical point, 1 mark for determining the nature of the critical point by the second derivative test] Find the (unique) critical point of the function
  . Determine whether it is a saddle point, or gives a local maximum of f, or gives a local minimum of f.

  Answer. Setting and , we have

  

  solving this linear system, we have x=-1, y=2, which gives us a critical point (-1, 2). At this point

  

  Hence (-1, 2) is a saddle point.

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