Department of Mathematical and Computational Sciences
National Institute of Technology Karnataka, Surathkal
Engineering Mathematics II - MA111
Even Semester (2016 - 2017)
Problem Sheet 6
Directional Derivatives, Gradient vectors & Tangent planes
1. Find the derivative of the function at Po in the direction of A.
√
y
xy
(a) h( x, y) = tan−1 ( x ) + 3 sin−1 ( 2 ), Po (1, 1), A = 3i-2j
(b) h( x, y, z) = cos xy + eyz + ln zx, Po (1, 0, 21 ), A = i+2j+2k
(c) f ( x, y.z) = x2 yz2 , Po 1, 2, 3, A = -2i+3j-6k
2. Find the directional derivative of the function f ( x, y) = x2 + 3xy2 at (1, 2) in the direction
towards origin.
3. Find the directions in which the functions increase and decrease most rapidly at Po . Then
find the derivative of the functions in these directions.
(a) f ( x, y, z) = ln xy + ln yz + ln xz, Po (1, 1, 1)
(b) f ( x, y) = x2 y + e xy sin y, Po (1, 0)
4. In what direction is the derivative of f ( x, y) = xy + y2 at P(3, 2) equal to zero?
5. Is there a direction u in which the rate of change of the temperature function
T ( x, y, z) = 2xy − yz is −30 c/ f t (temperature in degrees Celsius, distance in feet) at
P(1, −1, 1).
6. The temperature of a point in space is given by T ( x, y, z) = x2 + y2 − z. A mosquito
located at (1, 1, 2) desires to fly in such a direction that it will get warm as soon as possible.
In what direction should it move?
7. Prove the following
(a) ▽r n = nr n−2 r
(b) ▽ 1r =
−r
r3
(c) ▽ln |r| =
r
r2
where r = xi+yj+zk, |r| = r and n is an integer.
8. Find the directional derivative of f ( x, y, z) = 5x2 − y − 5y2 z + 2.5z2 x at the point P(1, 1, 1)
y −3
1
z
in the direction of the line x−
2 = −2 = 1 .
9. Find the directional derivative of f ( x, y, z) = x2 − y2 + 2z2 at the point P(1, 2, 3) in the
direction of the line PQ where Q is the point (5, 0, 4). Also, calculate the magnitude of
the maximum directional derivative.
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10. Find an equation for the plane tangent to the level surface f ( x, y, z) = c at the point Po .
Also, find parametric equations for the line that is normal to the surface at Po .
(a) x2 − y − 5z = 0, Po (2, −1, 1).
(b) x2 + y2 + z = 4, Po (1, 1, 2)
11. Find an equation for the plane tangent to the surface z = f ( x, y) at the given point.
(a) z = ln( x2 + y2 ), (0, 1, 0)
(b) z =
1
,
x 2 + y2
(1, 1, 12 )
12. Find parametric equations for the line that is tangent to the curve of intersection of the
surfaces at the given point.
(a) Surfaces: x2 + 2y + 2z = 4, y = 1, Point: (1, 1, 12 )
(b) Surfaces: x + y2 + z = 2, y = 1, Point: ( 12 , 1, 12 )
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