University:
Whitman College
Course:
MATH-225 | Calculus III
Academic year:

2024
Views:

152

Pages:

3
Author:

Araceli Cuevas

EXAMPLE In this example, we will see that it is possible for the directional derivative to exist in all possible directions, but still the original function does not have to be differentiable (or even continuous!). The function of interest is: 2 x y/(x4 + y 2 ) if (x, y) 6= (0, 0) f (x, y) = 0 if (x, y) = (0, 0) Here is a series of short exercises (solved) that will lead us through the discussion: 1. Show that f is not continuous at the origin by showing that the limit does not exist. SOLUTION: For this function, if we move to the origin along any line y = mx, then x2 mx mx3 mx 0 = lim = lim 2 = 2 =0 4 2 2 2 2 2 2 x→0 x (x + m ) x→0 x + m (x,mx)→(0,0) x + m x m lim 1 But, along the curve y = x2 , we have: (x,x x2 x2 x4 1 = lim = 4 4 4 x→0 2x )→(0,0) x + x 2 lim 2 Conclusion: The limit does not exist at the origin, so this function is not continuous at the origin (and so it is also not differentiable at the origin). 2. Use the definition of the directional derivative to show that the directional derivative exists for all directions u = hu1 , u2 i. f (0 + hu1 , 0 + hu2 ) − f (0, 0) h→0 h h2 u21 hu2 −0 h3 u21 u2 h4 u41 +h2 u22 = lim = lim 2 h→0 h · h2 (h2 u4 h→0 h 1 + u2 ) u2 u2 = lim 2 41 h→0 h u1 + u2 2 Du f (0, 0) = lim I need to pause here for a moment. What happens in the direction u = h1, 0i (or more directly, if u2 = 0)? Then the limit becomes: Du f (0, 0) = lim h→0 0 h2 u41 = lim 0 = 0 h→0 And in fact, we have just found that fx (0, 0) = 0. Now, if u2 6= 0, we can continue where we left off: u21 u2 u21 u2 u21 = = 2 h→0 hu4 u22 u2 1 + u2 Du f (0, 0) = lim So, for example, if u = h0, 1i, then we get Du f (0, 0) = fy (0, 0) = 0. In the direction, say of √12 h1, 1i, then 1 Du f (0, 0) = √ 2 In general, we have shown that the directional derivative exists in all directions. 2 3. Show that Du f (0, 0) 6= ∇f (0, 0) · u SOLUTION: Since fx (0, 0) = fy (0, 0) = 0, then the dot product on the right side of the equation is zero for all directions u. However, we showed one direction where the directional derivative was not zero. CONCLUSION: Since f is not differentiable (or even continuous) at the origin, we cannot use the shortcut formula for computing the directional derivative. Final comment: These “bad” functions have been set up in a particular way so that they fail. You should rest assured that most functions you will use will be differentiable, and so the shortcut formula will work. But, these examples show us the differences between the derivative in one dimension and multiple dimensions. 3

Math 225 Example - Study Guide

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