Math 2A  Vector Calculus  Chapter 11 Review Problems  Fall ‘07

1.      Level curves of a function   are shown below. Use the contour diagram to approximate  where  and  

2.      Consider  

a.       Show that this function satisfies the equation: .

b.      Draw overlaid graphs lines for , ,  and .  You can assume c = 1 or just scale the x-axis in terms of c.

c.       What effect does the parameter c have on the position of these graphs?

d.       

3.      Consider  

a.       Find  and  

4.      Estimate  for    numerically

5.      Consider the surface .  Show that the tangent plane at point of the surface passes through the origin.        

6.      A rectangular beam, supported at its two ends, will sag when subjected to a uniform load.  The amount of sag is modeled by the formula  where
p = the load in (Newtons/meter)
x = distance between the supports (in meters)
w = the width of the beam (in meters)
h = the height of the beam (in meters)
C = const characterized by the material of the beam

a.       Determine dS for a beam 4 meters long, 0.1 meters wide, and 0.2 meters high, subjected to a load of 100 N/m.   

b.      What conclusions can be drawn about the beam from the expression for dS?  That is, which variables increase or decrease the sag?

c.       For which variables is the sag most sensitive?  For which variables is it the least sensitive?

7.      The depth of a pond at (x, y) is given by h(x, y) =12.69  1.34x²  4.82y².

a.       If a boat at (0.23, 1.44) is sailing in the direction , is the water getting deeper or shallower?  At what rate?

b.      In what direction should the boat head to remain at a constant depth?

8.      Consider the surface  

a.       Find the equation of the of the tangent plane at ( 3,1,-2) in two different ways: first, by viewing the surface as the level surface of a function of 3 variables, (x, y, z); second, by viewing the surface as the graph of a function of two variables z = f(x, y).

b.      Find all points on the surface where a vector perpendicular to the surface is parallel to the x-y plane.

9.      If  f(x, y) is a function of x and y and g(u,v) = f( e^u sin v,e^u cos v),  find g_u(0, 0) given
f(0,0) = π/4, f_x(0,0) = 2,  f_x(0,1) = 4.2,  f_x(1,0) = 3.1,  f(0,1)  = sqrt(2),  f_y(0,0) =  π/2,
f_y(1,0) = 2.8,  f_y(0,1) = π².

10.  Consider the surface .

a.       Find an equation for the tangent plane at  in two different ways: first by by viewing the surface as the level surface of a function of 3 variables,  and then, second, by viewing the surface as the graph of a function of two variable .

b.      What is the maximum rate of change of z per horizontal change as a point moves along the surface from ?

11.  Consider the surface  

a.       Find the vector which is normal to the surface at (1,0,1).

b.      For what values of y is the normal to the surface parallel to the xy-plane?

c.       What is the instantaneous rate of change in z as a point moves along the surface from (1,0,1) in the direction of  (10,20,1) .

12.  The displacement of a string of length L = 34 inches at time t and a distance x from one endpoint is given by .

a.       Consider a point initially at the center of the string after 1/440 of a second. .  Approximate the displacement from this point to the nearby point .

13.  Suppose z = x³y² and x = e^u while y = ue^w.  Find z_w(1,1).

14.  Find the local max and min and saddle points of the function .
SOLN:  Start by looking for the critical points.  These are where  or
So critical points lie along the y axis at   or at one of the four points .  We try the second derivative test with ,   and  so that  and each of these four points is either a max or a min. Checking that   we conclude that  are local maxima, just as  are local minima.
As for the points on the y axis, since  we get an inconclusive D = 0.  This means we need to analyze the function in other ways.  One approach is to look at neighboring points   so that at  the neighboring points not on the axis are above 0 and at  the neighboring points are below so  is a local min and   is a local max.

15.  If R is the total resistance of three resistors, connected in parallel, with resistances R₁, R₂ and R₃, then  .  If the resistances are measured in ohms as R₁ = 25Ω, R₂ = 40Ω,  and R₃ = 50Ω, with possible error of 0.5% error in each case, estimate the maximum error in the calculated value of R.
SOLN:   A key observation here is that the percentage error given are  …aha!  So this is what you plug in for the “delta R’s.”
Take  so that
Note that this is less than the total error of any one resistor, the smallest of which is 0.005*25=0.125, so the error of the parallel resistors is less than the smallest error in any one resistor by itself.