Solutions to Quiz 1

1. Decide if the following statements must be true, might be true or can not be true. The function z = f(x, y) is defined everywhere. [2 each]

a) The level curves corresponding to z = 1 and z = −1 cross at the origin.

Can’t be true. Contours of different height can’t cross.

b) The level curve z = 0 consists of the circle x^{2} + y^{2} = 5 and the circle x^{2} + y^{2} = 7, but no other points.

May be true. Example: z = (x^{2} + y^{2} − 5)(x^{2} + y^{2} − 7).

c) The level curve z = −1 consists of two lines which intersect at the origin.

May be true. Example: z = xy − 1.

d) If z = e^{−(x^{2}+y^{2})}, there is a level curve for every value of z.

Not true, since there is no contour for z = −1.

e) If z = e^{−(x^{2}+y^{2})}, there is a level curve through every point (x, y).

True, since each point (x, y) lies on the contour z = c for c = e^{−(x^{2}+y^{2})}.

2. Consider the contour map given at the bottom of the page which represents altitude in a mountain range.

a) Mark the highest point on the map with an x. How high is the point? Which side of the mountain is steepest? [3]

x at (3,6) has a height strictly between 1000 and 1100. The north side is steepest since there the contours are closest.

b) Suppose that there is a spring at the point (5, 5). Carefully draw the path that the creek coming from this spring will take. At what angle does the creek cross the contours? Why? [4]

Approximate path of the creek: (5,5) - (4.8,4) - (5.2,3) - (7,2.8) - (9,4) - (9.5,6) - (11,8). The path will cross the contours at a right angle, since water takes the path of least resistence, flowing downhill as quickly as possible.

c) Sketch a cross-section (of the elevation) of the mountain range with y = 5 held constant. Label your axis! [3]

Label x-axis 0 to 10 and z-axis 300-1,000. Approximate graph (0,480) - (2,870) - (3,750) - (5,750) - (5.5,800) - (6.5,920) - (8,700) - (9,380) - (10,520).