Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 3
Due: at the end of the tutorial session Tuesday/Thursday, 9/11 February 2016
Name and student number:
1. Consider the function
f (x, y, z) =
p
4y 2 − sin(2x − 3z) ,
and the point P (0, −1, 0) .
(a) Find a unit vector in the direction in which f increases most rapidly at the point P .
(b) Sketch the projection of the vector onto the xz-plane
(c) Find a unit vector in the direction in which f decreases most rapidly at the point
P.
(d) Sketch the projection of the vector onto the yz-plane
(e) Find the rate of change of f at the point P in these directions.
Show the details of your work.
2. Let r =
p
x2 + y 2 .
(a) Show that
∇r =
r
,
r
where r = x i + y j .
(b) Show that
∇f (r) = f 0 (r)∇r =
3. Consider the surface
p
3
z = ln
f 0 (r)
r.
r
2x3 + y 2 + 2
3
(a) Find an equation for the tangent plane to the surface at the point P (2, −3, z0 ) where
z0 = f (2, −3).
(b) Find points of intersection of the tangent plane with the x-, y- and z-axes.
(c) Sketch the tangent plane.
(d) Find parametric equations for the normal line to the surface at the point P (2, −3, z0 ).
(e) Sketch the normal line to the surface at the point P (2, −3, z0 ).
Show the details of your work.
4. Consider the function
f (x, y) = 5 − x4 + 2x2 y − 2y 2 + 2y
Locate all relative maxima, relative minima, and saddle points, if any.
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