480 CHAPTER 22. DERIVATIVE OF A FUNCTIONS OF MANY VARIABLES
22.7 Exercises1. Suppose f : U → Rq and let x ∈ U and v be a unit vector. Show that Dvf (x) =
Df (x)v. Recall that
Dvf (x)≡ limt→0
f (x+ tv)−f (x)
t.
2. Let f (x,y) ={
xysin( 1
x
)if x ̸= 0
0 if x = 0. Find where f is differentiable and compute the
derivative at all these points.
3. Let
f (x,y) ={
x if |y|> |x|−x if |y| ≤ |x| .
Show that f is continuous at (0,0) and that the partial derivatives exist at (0,0) butthe function is not differentiable at (0,0).
4. Let
f (x,y,z) =(
x2 siny+ z3
sin(x+ y)+ z3 cosx
).
Find Df (1,2,3).
5. Let
f (x,y,z) =(
x tany+ z3
cos(x+ y)+ z3 cosx
).
Find Df (x,y,z).
6. Let
f (x,y,z) =
xsiny+ z3
sin(x+ y)+ z3 cosxx5 + y2
.
Find Df (x,y,z).
7. Let
f (x,y) =
(x2−y4)2
(x2+y4)2 if (x,y) ̸= (0,0)
1 if (x,y) = (0,0).
Show that all directional derivatives of f exist at (0,0), and are all equal to zero butthe function is not even continuous at (0,0). Therefore, it is not differentiable. Why?
8. In the example of Problem 7 show that the partial derivatives exist but are not con-tinuous.
9. A certain building is shaped like the top half of the ellipsoid, x2
900 +y2
900 +z2
400 = 1determined by letting z ≥ 0. Here dimensions are measured in feet. The buildingneeds to be painted. The paint, when applied is about .005 feet thick. About howmany cubic feet of paint will be needed. Hint: This is going to replace the numbers,900 and 400 with slightly larger numbers when the ellipsoid is fattened slightly bythe paint. The volume of the top half of the ellipsoid, x2/a2+y2/b2+z2/c2 ≤ 1,z≥ 0is (2/3)πabc.