Question

2a. The two space curves

and

r1(t) = <?1 + 5t, 3 − t^2, 2 + t − t^3> and? r2(s)=< ?3s−2s^2,s+s^3 +s^4,s−s^2 +2s^3>?

both pass through the point P(1,3,2). Find the values of t and s at which the curves pass through this point.

2b. Find the tangent vectors to each curve at the point P (1, 3, 2).

2c. Suppose S is a surface which contains the point P (1, 3, 2), and both r1(t) and r2(s) lie in S. We don’t have an equation for S, but we can still find the equation of the tangent plane to surface S at the point P (1, 3, 2). Use your answers to 2b. to do so:

Find the equation of the tangent plane to S at point P. (Hint: the vectors from 2b. lie in the tangent plane.)

3.

3. The above contour map shows the island of Hawai’i. Suppose that the height above sea level of the island is given by a function z = f(x,y) where (0,0) is at the peak of Mauna Loa and x, y, and z are measured in feet.

3a. If (a, b) is at the point P , determine if each of the following is positive, negative or zero (approximately). Briefly explain your answers. (i) fx (a, b) (ii) fy (a, b) (iii) fxy (a, b) (iv) fxx (a, b)

3b. If Mauna Loa is at a height of 13, 678 feet above sea level, write down the equation of the tangent plane at the point (0, 0).

3c. Approximate the equation of the tangent plane at the point P . Be sure to convert kilometers to feet in your computations!

Answer 1

Diagram(countour map) for question 3?