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 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( eu sin v,eu cos v),
find gu(0, 0)
given
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 = eu while y =
uew. Find zw(1,1). 14. Find
the local max and min and saddle points of the function . 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.
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