Instructions: Do work on your own paper, and neatly copy your final answers to the front. Be
sure to include your scratch work for full credit.
1. Consider the function $f \in \mathbb{R}^2 \rightarrow \mathbb{R}^1$, with
$$f(x, y) = 60 - 2x^2 - 5y^2$$
Notice I have chosen an example where you can do all the mathematics by hand.
(a) Get Maple to form a 3D graph of $f$.
(b) Get Maple to form a 2D contour plot for $f$.
(c) Write down the (total) derivative of $f$, using the protocols of the lecture.
$$\frac{df}{d(x, y)}(x, y) = [-4x, -10y]$$
(d) An hiker is at the point [2, 3]' in input-space. What's his elevation at this point, if we interpret $f$ as elevation?
60
(e) Our hiker would like to go in the direction normally called North-West, which is pointed to by the vector [-1, +1]'. If he goes in this direction, what will be the slope of the path in front of him? Note: His direction vector is not of unit length. Our hiker has not had a course in Multivariate Calculus and does not know to specify directions only with vectors of unit length. You do the math that he isn't doing here.
Keep going!
