Review MAT 235
Mohammad El Smaily
Department of Mathematics,
University of Toronto
February 21, 2014
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
1/9
Minima and Maxima, Critical Points, Lagrange Multipliers
Lagrange Multipliers
Lagrange Multipliers
Problem
Use Lagrange multipliers to find the triangle of largest area that can be
inscribed in a circle of radius r .
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
2/9
Minima and Maxima, Critical Points, Lagrange Multipliers
Lagrange Multipliers
Ideas
Our problem can be formulated precisely as: maximize
1 2
1
1
r sin θ1 + r 2 sin θ2 + r 2 sin θ3
2
2
2
subject to θ1 + θ2 + θ3 = 2π.
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
3/9
Minima and Maxima, Critical Points, Lagrange Multipliers
Second Derivative Test
Classify Critical Points
Problem
Find the critical points of f (x, y ) = 2x 2 − 2x 2 y + y 2 . Classify them as
minima, maxima, or saddle points.
Theorem (Second Derivative Test)
fxx fxy
Let D = det
= fxx fyy − fxy2 .
fyx fyy
1 If D > 0 and fxx > 0 at a critical point, then it is a minimum.
2
If D > 0 and fxx < 0 at a critical point, then it is a maximum.
3
If D < 0, at a critical point, it is a saddle point.
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
4/9
Chain Rule
Problem
Suppose f (x, y ) = g (x 2 + y 2 ) for some single-variable function g . Show
that the gradient of f at any point (x, y ) is always pointing toward or
away from the origin.
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
5/9
Chain Rule
Chain Rule and Partial Differential Equations (1)
Problem
Let u = u(x, y ) and let
x
y
= r cos θ
= r sin θ
Recall that ∆x,y u = ∂xx u + ∂yy u. Show that
∂2u
1 ∂ 2 u 1 ∂u
+
+
= ∆u.
∂r 2
r 2 ∂θ2
r ∂r
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
6/9
Chain Rule
Chain Rule and Partial Differential Equations (2)
Problem
Use the change of variables ξ = x + ct, η = x − ct (c is a fixed real
number) and v (ξ, η) = u(x, t) to show that the general solution of the
PDE
utt − c 2 uxx = 0
is given by
u(x, t) = f (x − ct) + g (x + ct)
for arbitrary functions f and g .
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
7/9
Change of Variables
Change of Variables (1)
Problem
Let E be the ellipsoid
x2 y2 z2
+ 2 + 2 = 1.
a2
b
c
Use the y
change of variables x = au, y = bv and z = cw to compute the
integral
1 dV
E
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
8/9
Change of Variables
Change of Variables (2)
Problem
Let Q be the quadrilateral in the xy -plane with vertices (1, 0), (4, 0), (0, 1)
and (0, 4). Evaluate
x 1
dA
x +y
Q
using the change of variables x = u − uv and y = uv .
M. El Smaily (Math. Toronto)
MAT 235 Review
February 21, 2014
9/9