Question

1. A car is driving northwest at vv mph across a sloping plain whose height, in feet above sea level, at a point NN miles north and EE miles east of a city is given by

h(N,E)=3250+125N+75E.h(N,E)=3250+125N+75E.

(a) At what rate is the height above sea level changing with respect to distance in the direction the car is driving?

(b) Express the rate of change of the height of the car with respect to time in terms of vv.

2. Suppose that you are climbing a hill whose shape is given by z=993−0.05x2−0.07y2z=993−0.05x2−0.07y2, and that you are at the point (70,80,300).(70,80,300).

a) In which direction should you proceed initially in order to reach the top of the hill fastest?

b) If you climb in that direction, at what angle above the horizontal will you be climbing initially (radian measure)?

3. Consider a function f(x,y)f(x,y) at the point (6,6)(6,6).
At that point the function has directional derivatives:
385√385 in the direction (parallel to) 〈7,6〉〈7,6〉, and
685√685 in the direction (parallel to) 〈6,7〉〈6,7〉.
The gradient of ff at the point (6,6)(6,6) is

4. Consider the surface x=3y2+5z2−126x=3y2+5z2−126.

a) Find an equation of the tangent plane to the surface at the point (2,6,2)(2,6,2).

b) Find a vector equation of the normal line to the surface at (2,6,2)(2,6,2)
r(t)r(t) =

Answer 1