Mathematics

Hints and Answers

4: 6.16
6(a) In R^2, a line parallel to the y-axis; in R^3, a plane parallel to the yz-plane.
6(b) Each equation represents a plane. The pair of equations represents the intersection of the two planes, which is a line.
10: (x-6)^2 + (y-5)^2 + (z+2)^2=7. The intersection with the xy plane is the set of points that satisfy the two equations (x-6)^2 + (y-5)^2 = 3 and z=0, a circle. The intersection with the xz plane is the set of points that satisfy the two equations (x-6)^2 + (z+2)^2 = -18 and z=0, which is the empty set. The intersection with the yz plane is also the empty set.
14: (x-1)^2 + (y+2)^2 + z^2 = 21/4. Center is (1,-2,0), radius is (square root of 21)/2.
30: 0 le x le 1, 0 le y le 2, 0 le z le 3.
34: (x-25/3)^2 + (y-1)^2 + (z+11/3)^2 = 332/9.

4(a) vector PR. (b) vector RS. (c) vector QP. (d) vector RQ.

16: Angles at P, Q, and R are 43 degrees, 58 degrees, and 79 degrees (approximately).

18(a) <-1,4,-1> or any scalar multiple thereof.
18(b) (3/2)(square root of 2)
26: Let a=vector(PQ), b=vector(PR), c=vector(PS). Evaluate det(a,b,c). It should be 0.

4: Parametric equations are x=2t, y=-t, z=3t.

8: Graph I.
16: Curve looks like z = cos x, but it is in the plane x=y.

2: [sqrt(5)*pi^2]/2
4: e^2
12(b) 1/(5t)
14(b) 2t/(2t^2+1)^2
30: a is near 0 where b is nearly straight: a is the curvature of b.

10: II is cone: circles are evenly spaced.
38: k=0: origin. k>0: ellipsoids centered at the origin, all the same shape, but get bigger as k gets bigger.

2(a) Continuous. (b) Can be discontinuous (vertical cliffs or overhangs). (c) Discontinuous. Cost jumps every eighth of a mile (or whatever).
7: Try x and y axes.
9: Try lines y=mx.
13: Try parabolas y=mx^2.
22: sin(y ln x), continuous on the set of (x,y) such that x>0.

28(a) An increase in temperature (if rainfall remains unchanged) would cause a decrease in wheat production. An increase in rainfall (if temperature remains unchanged) would cause an increase in wheat production.
28(b) - 1.1 units per year.

24: (cos theta, sin theta) with theta=0 or theta = 2 pi - cos^{-1}(4/5).
26: Deeper at a rate of 3.92 meters per meter.

22(a) V = (4/3)*pi*(r_2^2-r_1^2)^(3/2). (b) Since r_2^2=r_1^2+(h/2)^2, this simplifies to V=(pi/6)*h^3.

8: From the picture it appears that P is independent of y but increases when x increases: P_x > 0, P_y = 0. Also, it appears that Q is independent of x but increases when y increases: Q_x = 0, Q_y > 0. Of course, R = 0. From this you can show that div F > 0 and curl F = 0.
10(a) meaningless. (b) vector field. (c) scalar field. (d) vector field. (e) meaningless. (f) vector field. (g) scalar field. (h) meaningless. (i) vector field. (j) meaningless. (k) meaningless. (l) scalar field.

8: Requires integral tables or Maple, just set up.

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