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HW7, Q3
Contents
Function to maximize
Constraint
Lagrange multiplier expression
Solution
pkg load symbolic
% skip this on matlab
Function to maximize
cost is sum of the side areas plus half of the base area
syms l w h
cost = w*h + w*h + l*h + l*h + l*w/2
253 HW6 Q12
Contents
First partials
Find the critical points
2nd partials
Construct "D" for the 2nd derivative test
CP (-1, -3)
CP (-1, 3)
CP (0, -3)
CP (0, 3)
CP (1, -3)
CP (1, 3)
pkg load symbolic
% only for Octave, skip this line on Matlab
syms x y z
f
Homework 4, Sec 12.3, Q8, Q20
Contents
Q8
Q20
Q8
Find derivatives of
pkg load symbolic
syms x y
f = log(x*y)
f(x, y) = ln(xy) and evaluate at (x, y) = (2, 3). Setup:
% needed for Octave
% "log" is "ln" in Matlab/Octave
f = (sym) log(xy)
Now dierentiate:
f
MATH 253
Page 1 of 5
Student-No.:
Midterm 2
November 16, 2016
Duration: 50 minutes
This test has 4 questions on 5 pages, for a total of 40 points.
Read all the questions carefully before starting to work.
Give complete arguments and explanations for all
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SOLUTIONS TO ASSIGNMENT #7
y= x
x=1
1. The iterated integral I =
sin
x=0
(y 3 3 y )
dA for a regionR in the x, y plane.
2
sin
ble integral
y =0
(y 3 3 y )
dy dx is equal to the dou2
R
(a) Sketch R.
(b) Write the integral with the order of integration re
Math 253 Homework assignment 8 Solutions
1. Find the mass and centre of mass of the lamina that occupies the region bounded by
y = ex , y = 0, x = 0 and x = 1 and having density (x, y ) = y .
ex
1
Solution: Mass =
0
1
1
2
y dydx =
0
e2x dx =
0
12
(e 1) .
SOLUTIONS TO HOMEWORK ASSIGNMENT #9, Math 253
1. For each of the following regions E , express the triple integral
iterated integral in cartesian coordinates.
E
f (x, y, z ) dV as an
(a) E is the box [0, 2] [1, 1] [3, 5];
Solution:
2
1
5
f (x, y, z ) dV =
SOLUTIONS TO ASSIGNMENT #10, Math 253
1. Compute the total mass of the solid which is inside the sphere x2 + y 2 + z 2 = a2 and
c
.
outside the sphere x2 + y 2 + z 2 = b2 if the density is given by (x, y, z ) =
x2 + y 2 + z 2
Here a, b, c are positive con
Math 253 Homework assignment 6
1. Consider the integral
R
xy 2 dA, where A is the rectangle [0, 1] [0, 1].
(a) Calculate the Riemann sum corresponding to this integral, with the subdivision
corresponding to x = y = 0.2 and using the centre of each small r
SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 253
1. For what values of the constant k does the function f (x, y ) = kx3 + x2 + 2y 2 4x 4y
have
(a) no critical points;
(b) exactly one critical point;
(c) exactly two critical points?
Hint: Consider k = 0 and k
SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
1. Find the equation of a sphere if one of its diameters has end points (1, 0, 5) and
(5, 4, 7).
Solution:
The length of the diameter is (5 1)2 + (4 0)2 + (7 5)2 = 36 = 6, so the
radius is 3. The centre is at
SOLUTIONS TO HOMEWORK ASSIGNMENT 3, Math 253
1. Calculate the following limits, or discuss why they do not exist:
y
(a) lim 2
(x,y )0 x + y 2
Solution: The limit does not exist. If (x, y ) approaches (0, 0) along the x-axis, we have
y = 0, x = 0 and the e
SOLUTIONS TO HOMEWORK ASSIGNMENT #4, MATH 253
1. Prove that the following dierential equations are satised by the given functions:
2u 2u 2u
(a)
+ 2 + 2 = 0, where u = (x2 + y 2 + z 2 )1/2 .
2
x
y
z
w
w
w
(b) x
+y
+z
= 2w, where w = x2 + y 2 + z 2
x
y
z
So
SOLUTIONS TO HOMEWORK ASSIGNMENT #1
1. Sketch the curve r = 1 + cos , 0 2, and nd the area it encloses.
1
0.5
0
0.5
1
1.5
2
0.5
1
Figure 1: The curve r = 1 + cos, 0 2
The area is given by
1 2
1
(1 + cos )2 d =
2 =0
2
1
3
=
(2 + 0 + ) =
2
2
2
A=
(1 + 2 cos
MATH 253
Page 1 of 5
Student-No.:
Midterm 1
October 12, 2016
Duration: 50 minutes
This test has 4 questions on 5 pages, each worth 10 points, for a total of 40 points.
Read all the questions carefully before starting to work.
Give complete arguments and