Gradients and Directional Derivatives
Reading assignment: 2.6
Recommended exercises: 2.6: 1, 3, 5, 8, 9, 11, 12, 13, 16, 18, 29, 33, 35
For each problem 1 – 3, in part a) find the gradient of the given function, and in part b) find the
directional derivative in the direction of the given v at the given point P.
1) f ( x, y)  e x sin y
P (0,  )
a)
b) v = i + j,
2) g ( x, y)  x 2 y 3
a)
b) v = 3i + 4 j, P(1, 1)
3) h( x, y, z ) 
a)
x y
z
b) v = 2i  j + 2k,
P (1,1, 1)
4) Find an equation for the tangent plane to the surface z = x3 – 3xy2 at an arbitrary point (a, b, c)
on the surface. This surface is called a monkey saddle. Perhaps if you graph it you'll see why!
5) Find a tangent plane to the surface which is defined implicitly by the equation xy2 + yz2 + zx2 = 3
at the point (1, -2, 1).
6) Consider the function f ( x, y, z )  ze x sin y . At the point ( 1,

,1) it has value
2
the change in this value if you move 0.1 units in the direction of u  (2,3, 6) .
1
. Estimate
e
7) Let f ( x, y)  4  x 2  y .
(a) Sketch and label the level curves z = 2, 1, 0,1, 2 . [Neatness and accuracy count]
(b) Find f (1,1)
(c) Find a unit vector u such that Du f (1,1)  0 .
(d) Sketch and label f (1,1) and u with initial point (1,1) (Sketch these vectors in the
coordinate plane from part (a)).
 3 4 
 4 3
8) Let u   ,  and v   ,  . Suppose that at some point p  ( x, y ) , Du f  6 and
5 5 
5 5
Dv f  17 .
a. Find f at p .
b. Show, in general, that f
2
  Du f    Dv f  whenever u and v are perpendicular.
2
2
9) Consider the sphere ( x  a)2  y 2  z 2  2 and the circular cone z 2  2 x 2  2 y 2 . Find a such that
the two surfaces are tangent. [Two surfaces are tangent at a common point if and only if they share a
common tangent plane at that point.]