Exercise 3 A friend in Calculus III approaches you for help relating to the concept of \"the directional derivative.\" You happen to have the graph of z = f (x, 3!) provided below in front of you at the time, so you use this graph to aid in your explanations, referencing the point (a, b), which is denoted by the red ball on the plot. 2 a) b) First your friend asks \"What exactly does the directional derivative f 11' (a, b) measure?\" How do you answer? Next they ask you what it means for the directional derivative to be positive at a point. Like a good mathematician, you value precision, so you start your explanation with: \"Given a locally planar (i. e., differentiable) function f, a point (a, b), and a vector if, then face, b) > 0 if...\" How do you complete the explanation? Finally, your friend asks: \"Given a locally planar function f and a point (a, b). is it always possible to find a direction i such that f (a, b) = 0'? If so, what can you say about such a direction IT? Exercise 12 Let f (x, y) be a differentiable function. Prove that V(f ( x, y ) g ( x, y) ) = f (x, y) Vg(x, y) + Vf(x, y)g(x,y). Here is a "mini lesson" on proving two things are equal: 1. Start with the left-hand side and rewrite it using relevant definitions (in this case the gradient). 2 If Step 1 doesn't lead you straight to the thing on the right-hand side, then start with the right-hand side and rewrite it using relevant definitions. With hope, the work you do in rewriting the right-hand side equals something you obtained while working on the left-hand side. They meet in the middle. 3. Hopefully, Steps 1 and 2 enable you to see how the two sides are equal. Starting with just one of the sides, show through a sequence of equalities that the left-hand side equals the right-hand side. The final step in your proof should be the equality. Never start a proof with the statement to be proved