=
=
A Discussio=
n in
Intermediate Calculus
=
AVOID TO USE SPHERICAL COORDINATE =
SYSTEM
IN PARAMETRIC REPRESENTATION FOR DOUBE INTEGRALS
Thomas Nguy=
en
=
Abstract:=
One common technique that is used to evaluate integral=
s is
changing coordinate systems. As a principle, students are familiar with usi=
ng
Polar coordinate for double integrals (two-dimensional) and
Cylindrical/Spherical coordinate for triple integrals (three-dimensional).
However, James Stewart, in his Calculus book (Early Transcendentals
Multivariable 5e), applied several times the spherical coordinate system to
demonstrate the use the parametric representation for calculation of double
integrals. One particular demonstration is example 5 in pages 1117-1118. Th=
is
is not an appropriate way to demonstrate the use of parametric representati=
on.
It might cause confusing for students not only by inappropriate coordinate
system choice but also by difficulty of interpretation of the transformatio=
n:
dA --> dФ.dθ (*). A sug=
gested
solution will be proposed.
Problem: =
Find the flux of vector field F(x,y,z) =3D z.i + y.j + x.k across the unit sphere:
x2=
+ y2
+ z2 =3D1.
&n=
bsp;
Stew=
art’s solution:
&n=
bsp;
Using the parametric representation
r(Ф,θ) =3D sinФ
cosθ i + sinФ
sinθ j + cosФ k
0 ≤ Ф ≤ л 0 =
8804;
θ ≤ 2л
&nb=
sp;
We have<=
o:p>
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
And &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&=
nbsp;
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
Therefor=
e
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
and by formula 9, the flux is
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
=
(*)
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&nb=
sp;
&nb=
sp;
&nb=
sp;
Suggested solution:
Of course the best solution for this problem is using =
The
Divergence Theorem
&nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
However, we want to use parametric representation to s=
olve
it. Therefore I am going to solve this problem by using Polar coordinate sy=
stem
(2-D). I prefer to rewrite equation 8 from page 1116 (James Stewart’s Calculus 5e) as the following:
<=
!--[if gte mso 9]>
Apply this formula for our problem: <=
!--[if gte mso 9]>
and F =3D <z,y,x>
By symmetry, we can compute the flux across semi sphere first (we will double
answer later), therefore from:
<=
!--[if gte mso 9]>
We have =
<=
!--[if gte mso 9]>
&nbs=
p; &=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
Using Polar parametric representation: <=
!--[if gte mso 9]>
---&g=
t; <=
!--[if gte mso 9]>
where 0 ≤ r ≤ 1 and 0 ≤ θ=
≤
2л
&nb=
sp;
&nb=
sp;
Rewrite<=
o:p>
&nb=
sp;
&=
nbsp; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
Let u =
=3D 1- r2
&nb=
sp;
&nb=
sp;
&nb=
sp;
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&nb=
sp;
Hence to=
tal flux
crosses entire of the sphere is: <=
!--[if gte mso 9]>
&nb=
sp;
&nb=
sp;
&nb=
sp;
Conclusion:&=
nbsp;
&nb=
sp;
Although=
the
problem asks for the flux across a space surface S (3-D) and vector field F=
is
given with three components x, y, and z; however, because the problem’=
;s domain
is transferred from S to a plane surface D; furthermore, because of constra=
in x2+y2+z2=3D1,
we can consider this problem as a problem with two variables x and y.
Therefore, it makes sense for students to use Polar in double integral and =
to
avoid a confusing setting: dA=
=3D
dФ.dθ (*)
of using sphere parametric representation. We can use Polar instead =
of
Sphere for example 10 in page 1104, example 2 in page 1112, and example 5 i=
n page 1117 with the same technique that has shown above=
.
&nb=
sp;
San Diego, November 28th, 2006
&nb=
sp;
=
o:p>
Second discussion
A DISCUSSION IN POLAR COORDINATE S=
YSTEM
Thomas Nguyen
Abstract:
Isaac Newton is a gre=
at
scientist. He had invented many useful things for life. Polar coordinate sy=
stem
is one of his inventions (written 1671). Polar coordinate system is very
helpful in many cases. It is used in many fields, including mathematics,
physics, engineering, navigation, and robotics. However, I am still wonderi=
ng
why when he invented this coordinate system, he set =
not =
(zero or positive
numbers).
I prefer the following
setting:
<=
span
style=3D'font-size:14.0pt'>
w=
here: =
r ≥ 0
and =
<=
!--[if gte mso 9]>
There are two main re=
asons
for my claim:
-&nb=
sp;
When we con=
vert
from Cartesian to Polar, we do not need to worry about choosing r to be positive or negative.=
-&nb=
sp;
We do not h=
ave to
worry about whether we were missing solutions or not.
Let consider one exam=
ple
(Example 3 in page 671, Early Transcendentals Multivariable Calculus 5e of
James Stewart)
Problem:
Represent the point w=
ith Cartesian
coordinate system (1, -1) in Polar coordinate system.
Stewart=
’s
solution:
If we choose r to be
positive, the Equation 2 =
=
=
give:
<=
!--[if gte mso 9]>
<=
!--[if gte mso 9]>
Since the point lies =
in the
fourth quadrant, we can choose: <=
!--[if gte mso 9]>
Thus one possible ans=
wer
is <=
!--[if gte mso 9]>
and
another is <=
!--[if gte mso 9]>
Discussion:
<=
/span>
First,
as you can see the author first has to make the statement: “If we choose r to be positive, ...”; Studen=
ts can
wonder such as “Why we choose r
to be positive?” or “What if r
to be chosen negative?”
In
negative case of r, there are two more basic possible solutions, which are:=
&=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
and <=
!--[if gte mso 9]>
These two additional
solutions came up because of the rule
=
( i.e. r c=
an be
negative or positive). It seems like we put ourself into troubles (ex:
forgetting solutions sometimes or some students might be lost some points i=
f in
their solutions, they declair these two new solutions!). Why we don<=
span
lang=3DJA style=3D'font-size:14.0pt;font-family:"Arial Unicode MS";mso-asci=
i-font-family:
"VNI Times";mso-hansi-font-family:"VNI Times";mso-bidi-font-family:"Times N=
ew Roman";
mso-fareast-language:JA'>’t s=
et r to be positive because angle=
θ can cover all positions in the plane.
When we solve the
equation: <=
!--[if gte mso 9]>
we just hav=
e to
look at where the point located in Cartesian as James Stewart did and we ha=
ve
to write only one general solution:
=
&nb=
sp; =
<=
!--[if gte mso 9]>
In this case, we will=
not to
worry about all solutions with negative r. Life would be easier then for stud=
ents.
I
have tried all times to apply this setting into other applications of Polar
coordinate system (and Cylindrical coordinate
system-3D) such as solving double, triple integrals without any problems. I=
t is
very convinient to use this setting to determine is in which direction ( =
52;)
and how far (r) to move for a robot. It makes sense in avitation, too; for
example, an aircraft traveling 5 nautical miles due East will be travelling=
5
units at heading 90. However, I am worrying that perhaps in some cases we n=
eed
the setting r to be negative to=
cover
all possible options in some special problems. That’s why I really
appreciate it if someone can explain why Newton
made such that setting.
<=
/span>
Conclusion<=
/b>:
I
am always looking for simpler/better methods/techniques to solve problems. =
This
is one of my own experiences that have helped me went through my studying. =
San Diego=
, December 15=
th,
2006
=
o:p>
A Discussion in Geometry=
b>
DISCUSS ON OPTIMIZATION PROBLEMS IN
GEOMETRY
Thomas Nguyen
Abstract:
&nbs=
p; Calculus
is a very powerful tool to solve optimization problems in Geometry. Quite
commonly, students have been learned and paid respect to Calculus much more
than Algebra. They usually think Intermediate Algebra or College Algebra is
“elementary school” stuff.
&nbs=
p; Using
Algebra to solve optimization problems in Geometry has not been seen much in
college. In this article, I am going to demonstrate the beauty of some of
Algebra’s tools (Cauchy’s Formula, Ploting technique, etc). I t=
hink
after reading this article, students will have a new look on Algebra.
Example 1: (Example 6, page 958, Early Transcendental Multivariable Calculus 5e., James Stewart)
This =
example
demonstrates how to use partial derivatives to solve optimization problems<=
/i>.
&nbs=
p; A
rectangular box without a lid is to be made from 12 m2 of cardbo=
ard.
Find the maximum volume of such a box.
Book’s Solution: Let the length, width, and height of the b=
ox
(in meters) be x, y, and z, as shown in Figure 10. Then the volume of the b=
ox
is V =3D xyz
=
&nb=
sp; =
=
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
Figure 10
&nbs=
p; We
can express V as a function of just two variables x and y by using the fact=
that
the area of the four sides and the bottom of the box is
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp;
2xz + 2yz + xy =3D 12 (m2)
&nbs=
p; Solving
this equation for
z, we get z =3D=
(12
– xy)/[2(x+y)], so the expression for V becomes
&nbs=
p; &=
nbsp; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
&nbs=
p; We
compute the partial derivatives:
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nbs=
p; If V is =
maximum,
then <=
!--[if gte mso 9]>
but x =3D 0 or y =3D 0 gives V =3D 0, so we must solve the equa=
tions
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nbs=
p; There
imply that x2
=3D y2 and so x =3D y. (Note that x and y must both be positive in this
problem.) If we put&nb=
sp;
x =3D y in eith=
er
equation we get
&nbs=
p; 12
– 3x2 =3D 0 , which gives x =3D 2, y =3D 2, and z =3D (12 – 2.2)/[2(2+2)] =
=3D 1.
&nbs=
p; We
could use the Second Derivatives Test to show that this gives a local maxim=
um
of V, or we could simply argue from the physical nature of this problem that
these must be an absolute maximum volume, which has to occur at a critical
point of V, so it must occur when =
span>x
=3D 2, y =3D 2, and z =3D 1. Then V =3D 2.2.1 =3D 4 (m3<=
/sup>), so
the maximum volume of the box is 4m3.
Discussion:
As you can see in this solution, the author o=
nly
shows details in finding critical points, how to prove these critical values
are absolute maximum values is just a suggestion. Although partial derivati=
ves
can solve this problem, their solution is “pretty” tough,
isn’t it?
Before showing Algebra’s solution, I’d like to introduce one=
of
Algebra’s tools that I am going to use it to solve Example 1.
Cauchy’s Formula: =
span>The geometric mean is smaller than the arithmetic mean=
.
(That’s it. Very neat!)
&nbs=
p; We
can express it for three variables as follows:
For any a > 0 , b > 0, and c =
>
0, we have <=
!--[if gte mso 9]>
with
equality in the case a
=3D b =3D c.
Solving Example by using Cauchy’s Formula:
Call x, y, and z are three sides of the bo=
x as
shown in Figure 10. Because they are 3 sides of the box, they must be posit=
ive
(i.e x > 0, y > 0=
, and
z > 0 ---> Let&n=
bsp;
a =3D (2xz) > 0, =
span>b =3D
(2yz) > 0, and c =3D (xy) > 0
----> a, b, and c satisfies Cauchy’s condition, therefore, =
we
have:
&nbs=
p; <=
!--[if gte mso 9]>
=
The right side is always smaller or equal to the left side. The equa=
lity
case happens only when a =3D b =3D c.
That means when 2xz =3D 2y=
z =3D
xy -----> x =3D y =3D 2z &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; (1)
In this case, the right side will become <=
!--[if gte mso 9]>
=
<=
!--[if gte mso 9]>
&=
nbsp; &=
nbsp; (2)
From (2), we can say that the volume V is =
maximum, when condition (1) is
satisfied, but: a + b +=
c =3D
2xz + 2yz + xy =3D 12 m2 (3)
Therefore, by using (1) and (3) we can eas=
ily
have: x =3D y
=3D 2 and=
z =3D1 ------> Vmax=
=3D xyz =3D
2.2.1 =3D 4 m3.
Note: Now you have seen the beauty of
Cauchy’s Formula (one of Algebra’s tool). In this solution, we =
know
very clear that x =3D y =3D 2 and z =3D 1 are absolute critical values, and=
V =3D 4 m3
is the absolute maximum value that it can have (Cauchy’s equality
satisfied). The solution is short, compact, and clear by comparison to
Calculus’s partial derivatives tool.
Example 2: (Example 1, page=
332,
Early Transcendentals Single Variable Calculus 5e, James Stewart)
A farmer has 2400 ft fencing and wants to fence off a rectangular field =
that
borders a straight river. He needs no fence along the river. What are the
dimensions of the field that has the largest area?
&n=
bsp;  =
;
Book’s Solution: We
wish to maximize the area A of the rectangle. Let a and=
b be the depth and width of the rectangle (in feet). Then we express A in term of x and y:
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; A
=3D xy
We want to express A as a function of just one variable, so we eliminate=
y
by expressing it in terms of x. To do this we use the given information that
the total length of the fencing is 2400 ft. Thus
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; 2x
+ y =3D 2400
From this equation we have y =3D 2400 ̵=
1; 2x,
which gives
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; A
=3D x(2400 – 2x) =3D 2400x – 2x=
2
Note that x ≥ 0 and x ≤ 1200 (otherwise<=
span
style=3D'mso-spacerun:yes'> A < 0). So the function =
that
we wish to maximize is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; A(x)
=3D 2400x – 2x2
0 ≤ x ≤ 1200
The derivative is
A’(x) =3D 2400 – 4x, so to find the critical numbers we =
solve
the equation 2400
– 4x =3D 0
Which give x =3D 600=
. The
maximum value of A must occur either at this critical number or at an endpo=
int
of the interval. Since=
A(0) =3D 0, A(600) =3D 720,000
and A(1200) =3D =
0, the
Closed Interval Method gives the maximum value as A(600) =3D 720,000.
[Alternately, we could have observed that A”(x) =3D - 4 < 0 for=
all
x, so A is always concave downward and the local maximum at x =3D 600 must =
be an
absolute maximum.]
Algebraic Solution: Let x =3D 2a and y =3D b (x > 0=
, y > 0
satisfied Cauchy’s condition). Apply Cauchy’s Formula for two
variables:
&nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
Hence the area of the field A will be maxi=
mized
only when equality’s hold, i.e. x =3D y or 2a =3D b =
(1)=
o:p>
We have given information: 2a +=
b =3D
2400 =
&nb=
sp;
(2)
From (1) and (2), we have:
a =3D 600 and b =3D 1200 ------> Amax =3D 72=
0,000 ft2.
*******************
Example 3: (Example 5, page 336, Early
Transcendentals Single Variable Calculus 5e, James Stewart)
&nbs=
p; Find
the area of the largest rectangle that can be inscribed in a semicircle of
radius r.
Book’s Solution:
Let’s take the semicircle to be the upper half of the circle: x2 + y2 =3D =
r2
with center the origin. Then the word inscribed means that the rectangle has
two vertices on the semicircle and two vertices on the x-axis as shown in
Figure 9.
Let (x,y) be the vertex that lies in the first
quadrant. Then the rectangle has sides or lenghs 2x
and y, so its area is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; A
=3D 2xy
To eliminatey we use the fact that (x,y) lies=
on
the circle, so <=
!--[if gte mso 9]>
. Thus <=
!--[if gte mso 9]>
The domain of this function is 0 ≤ x ≤r. =
Its
derivative is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
Which is 0
when 2x2=
sup> =3D
r2, that is, <=
!--[if gte mso 9]>
(sinc=
e x ≥ 0). This value of x give=
s a
maximum value of A since A(0) =3D 0 and A(r) =3D=
0.
Therefore, the area of the largest inscribed rectangle is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
&nbs=
p; &=
nbsp;
=
&nb=
sp; =
&nb=
sp;
Figure 10.
Trigonometric Solution(special): A simpler solution is possible if =
we
think of using an angle as a variable. Let θ be the angle shown in Fig=
ure
10. Then the area of the rectangle is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
Wi know that sin(2θ) has a maximum value=
of 1
and it occurs when 2θ =3D
л/2.
So A(θ) has a maximum value of r2=
sup>
and it occurs when θ =3D
л/4
Algebraic Solution: We have x > =
0, y
> 0 ---> x2 > 0 and y2 > 0 satified
Cauchy’s condition. Apply Cauchy’s Formula for two variables
&nbs=
p; &=
nbsp; &nbs=
p; <=
!--[if gte mso 9]>
=3D half of
rectangle area.
If this area is maximized, then entire of
rectangle area will be maximized. This happens when equality’s hold:<=
span
style=3D'mso-spacerun:yes'> x2 =3D y2 or x =3D y (1)=
span>
Furthermore, we have x2 + y2 =3D r2 (Pythagorean Theorem) (2). <=
o:p>
From (1) and (2), we have <=
!--[if gte mso 9]>
and Amax<=
/sub>
=3D 2xmaxymax =3D r2.
***************
Example 4: (Example 2, page 333, Early Transc=
endentals
Single Variable Calculus 5e, James Stewart)
&nbs=
p; A
cylindrical can is to be made to hold 1 L of oil. Find the dimensions that =
will
minimize the cost of the metal to manufacture the can.
&nbs=
p; &=
nbsp; <=
![endif]>
Book’s Solution: &nbs=
p;
Diagram as in Figure 3, where r is the radius and h the height (both=
in
centimeters). In order to minimize the cost of the metal, we minimize the t=
otal
surface area of the cylinder (top, bottom, and sides). We see that the sides
are made from a rectangular sheet with dimension 2л.r and h. So the
surface area is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; A
=3D 2л.r2 + 2л.r.h
To eliminate h we use the fact that the volume is given as 1 L, where we
take to be 1000 cm3. Thus: л.r2<=
/sup>.h
=3D 1000
Which gives: h =3D 1000/(л.r2). Substitution of this into the
expression for A gives
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
Therefore, the function that we want to minimize is
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
To find the critical numbers, we differentiate
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
Then A’(r) =3D 0
when лr2 =3D 500, so the only critical number
is <=
!--[if gte mso 9]>
Since the domain of A is (0,∞), we
can’t use the argument of Example 1 concerning endpoints. But we can
observe that A’(r) < 0 for <=
!--[if gte mso 9]>
and
A’(r) > 0 for <=
!--[if gte mso 9]>
, so =
A is
decreasing for all r to the left of the critical number and increasing for =
all
r to the right. Thus <=
!--[if gte mso 9]>
must=
span>
give rise to an absolute minimum.
[Alternately, we could argue that A(r)--> =
∞
as r --> 0+ and=
A(r)
--> ∞ as r --> =
8734;,
so there must be a minimum value of A(r), which must occur at the critical
number]. The value of h corresponding to <=
!--[if gte mso 9]>
is
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
<=
!--[if gte mso 9]>
Thus, to minimize the cost of the can, the radius should be <=
!--[if gte mso 9]>
cm an=
d the
height should be equal to twice the radius, namely, the diameter.
The total surface will be
=
<=
!--[if gte mso 9]>
Discussion: This solution i=
s also
pretty tough for most of students at this level, especially the argument at=
the
end of the solution (argue that the value <=
!--[if gte mso 9]>
is the
minimum value). Let’s see how Algebra deals with this problem
&nbs=
p; &=
nbsp; &nbs=
p;
Algebraic solution : =
Let R be
the radius, h be the height, S be the total surface, V be the volue of the =
cylinder
can. We have
&nbs=
p; S
=3D 2* лR2 + 2лRh (1) and V =3D лR2h
=3D 1000 cm3
(2)
Solve h from (2) and substitude h into (1),
which gives =
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
=
(3)
Plot (3) in Cartesian coordinate system, w=
e have
&n=
bsp;  =
; &=
nbsp; =
o:p>
Note: R =3D (1.7* pi) is equivalent to <=
!--[if gte mso 9]>
. This
is the critical value for minimum value of total surface area.=
span>
We can have <=
!--[if gte mso 9]>
and S =
8776;
554 m2 the s=
ame as
Calculus solution (we don’t need derivatives).
***************
From now on, I just show problems and Algebraic solutio=
ns
only.
**************
Example 5: If the length of the diagonal of a
rectangular box must be L, what if the largest possible volume?
Algebraic Solution: Let x, y, and z be the width, length, and hei=
ght
of the rectangular box with diagonal’s length L. Set a =3D x2, b
=3D y2, and c =3D z2. Then a > 0, b > 0, and c =
> 0
satisfied Cauchy’s condition. Apply Cauchy’s Formula for 3
variables:
&nbs=
p; &=
nbsp; <=
!--[if gte mso 9]>
The right side is
<=
!--[if gte mso 9]>
Similarly, Volume=
V is
maximized only when equality’s hold. That means:
&nbs=
p; &=
nbsp; &nbs=
p; a
=3D b =3D c
-----------> =
x2
=3D y2 =3D z2 (1)
Furthermore, we have given information: x2 + =
y2
+ z2 =3D L2 (2)
From (1) , (2) and condition: x > 0, y > 0, z > 0, we
have: =
<=
!--[if gte mso 9]>
*****************
Example 6: Find the dimensions of a rectangul=
ar box
of maximum volume such that the sum of the lengths of its 12 edges is a
constant c.
Algebraic Solution: Setting 3 dimensions as Example 4. Then the s=
um
of all edges is: =
4(x + y + z) =3D c  =
;
(1)
Similar argument in Example 4, the volume of the rectangular box is
maximized only when: =
x =3D y =3D z &nbs=
p; &=
nbsp;
(2)
Solve (1) & (2), we have
x =3D y =3D z =3D c/12 hence Vmax
=3D (c/12)3
******************
Example 7: Find the dimensions of the rectang=
le of
largest area that has its base on the x-axis and its other two vertices abo=
ve
the x-axis and lying on the parabola:
y =3D 8 – x2.
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp;
Algebraic Solution: Let x
be the width and y be the length of a half of inscribed rectangle (they are
also two coordinates of point P).
We have A =3D
2* x.y =
(1) Furthermore, because P belong=
s to
the parabola, so we have y =
=3D 8
– x2. (2)<=
/p>
Substitute (2) into (1), which gives =
A =3D 16x – 2x3. =
(3)
Note: Since the rect=
angle
must be inscribed
---> Conditi=
ons
for x and y: =
<=
!--[if gte mso 9]>
Plot (3) in the Cartesian coordinate system with interval for x is <=
!--[if gte mso 9]>
, we got the following graph:
 =
; &n=
bsp;  =
; &n=
bsp; &=
nbsp; =
o:p>
We can see that xmax
=3D sqrt(8/3) ;=
therefore
y =3D 8 – x2 =3D 8 – (8/3) =3D 16/3 and<=
span
style=3D'mso-spacerun:yes'>
Amax =
=3D2xmaxy
≈ 17.4 (we don’t =
need
derivatives)
*******************
Example 8: A rectangular bu=
ilding
is being designed to minimize heat loss. The east and west walls lose heat =
at a
rate of 10 units/m2 per day, the north and south walls at a rate=
of 8
units/m2 per day, the floor at a rate of 1 unit/m2 per
day, and the roof at a rate of 5 units/m2 per day. Each wall mus=
t be
at least 30 m long, the height must be at least 4 m, and the volume must be
exactly 4000 m3. Find the dimensions that minimize heat loss?
Algebraic Solution:
&nbs=
p; &=
nbsp;
Let L, W, and H be the length, the width, and the height of the building,
then boudary conditions are: W ≥ 30 ; From V =3D LWH =3D 4000 m3=
sup> (1)
---> H =3D 4000/(WL) ≥ 4 so 30 ≤=
L ≤ 1000/W (*) We can plot these conditions as fo=
llows:
&n=
bsp;  =
;
Total heat loss is: &n=
bsp;
T =3D (2WH)*10 + (2LH)*8 + (WL)*1 + (WL)*5 =3D 20WH + 16LH + 6WL (2)<=
/p>
Let a =3D 20WH;&nbs=
p;
b =3D 16LH; c =3D
6WL; a, b, c >=
0
satisfied Cauchy’s Formula, hence
&nbs=
p; =
=
(3)
The left side (total heat loss T) will be minimized only when
equality’s hold; i.e. <=
span
class=3DGramE>only when
a =3D b =3D c &nbs=
p;
or &nb=
sp;
20WH =3D 16LH =3D 6WL <=
/span>(4)
By solving (1) & (4), we got a critical point: =
=
=
Heat loss in this case is about 9,396 units (this is the minimum heat lo=
ss
that we might have)
(No derivatives needed!)
This critical point is the only one we can have, but it’s not satisfied condition (*); However, thi=
s is
a bounday value problem (an absolute max/min problem), therefore we continue
checking the boundary to find the minimum heat loss which satisfied conditi=
on
(*). Solve the boundary value problem, we have:
=
=
On B1 Tmin(W,L)
=3D Tmin(30, 30) =3D 10=
,200
units (minimum heat loss)
On B2 Tmin(W,L) =3D Tmin(100/3, 30) =
8776;
10,533 units
On B3 Tmin(W,L) =3D Tmin(30, 30) =3D 10,200 units (minimum heat loss)=
=
=3D=3D=3D=3D=3D> Dimensions=
of the building that minimized heat loss (10,200 units) are: L
=3D W =3D 30 m and =
H =3D 40/9 ≈ 4.44 m
Note: Because of boundary conditions,
dimensions (W, L, H) =3D (30, 30, 4.44) become the chosen dimensions with t=
he
minimum heat loss of 10,200 units. If there were no boundary condition, then
the chosen dimensions would be (W, L, H) =3D ( 20.43,
25.54, 7.66) with the minimum heat loss of 9, 396 units.
***************
&nbs=
p; &=
nbsp; San Diego, Decemb=
er 18th,
2006
=
o:p>
A Discussion in Algebra
DISCUSS ON MASTER PRODUCT AND SUM=
b>
Thomas Nguyen
(This article was certified by
Register of Copyright, USA
in January 2007)
Abs=
tract: &n=
bsp;  =
; &n=
bsp;  =
; &n=
bsp;  =
;
In Algebra, when dealing with trinomials in general form ax2 + bx + c there is a very efficient factoring
method called “Master Product and Sum”. The main focus of this method is t=
rying
to split the middle term (bx) into two parts ( bx =3D n1x
+ n2x ), then using
grouping method to factor again.
In general, there are 4 steps as the following:
-
Step 1: Form the product (ac)
-
Step 2: Find a pair of numbers whose product=
is
(ac) and whose sum is (b)
-
Step 3: Rewrite the trinomial to be factored=
so
that the middle term (bx) is written as the sum of two terms whose coeffici=
ents
are the two numbers found in step 2
-
Step 4: Continue to factor by grouping metho=
d
Example: Factor 3x2 – 10x &=
#8211;
8
Solution: The above trinomi=
al has
the form ax2=
+ bx
+ c, wher=
e a =3D 3, b =3D -10, and
c =3D - 8
-
Step 1:&nbs=
p;
The product (ac) =3D 3. (-8) =3D - 24
-
Step 2: We need to find two numbers whose
product is – 24 and whose sum is – 10, Let’s list all the
pairs of number whose product is – 24 to find the pair whose sum is
– 10
Product =3D -24
|
Sum of two numbers =3D -10
|
1(-24) =3D -24
-1(24) =3D -24
2(-12) =3D -24
-2(12) =3D -24
3(-8) =3D -24
-3(8) =3D -24
4(-6) =3D -24
-4(6) =3D -24
|
1+(-24) =3D -23
-1+(24) =3D 23
2+(-12) =3D -10 <----=
- here
-2+(12) =3D 10
3+(-8) =3D -5
-3+(8) =3D 5
4+(-6) =3D -2
-4+(6) =3D 2
|
&nbs=
p; &=
nbsp; As
you can see, of all the pairs of numbers whose product is – 24, only 2
and – 12 have a sum of – 10
-
Step 3:&nbs=
p;
We now rewrite our original trinomial so the middle term -10x is wri=
tten
as the sum of – 12x and 2x (splitting step)
3x2 – 10x – 8 =3D 3x<=
sup>2
– 12x + 2x – 8
-
Step 4: Factoring by grouping method, we hav=
e
3x2 – 12x + 2x – 8 =
=3D (3x2
– 12x) + (2x – 8)
=
&nb=
sp; =3D =
3x(x
– 4) + 2(x – 4)
=
&nb=
sp; =3D =
(x
– 4).(3x + 2)
You can check this answer by multiplying (x
– 4) and (3x + 2) to get (3x2 – 10x – 8)
Discussion:
I can have a similar method to factor trinomial, but after step 1 and 2,
students can write out the answer in general form. Then they usually need o=
ne
more step to rewrite the answer in the simplest form. My method is actually=
not
save much space in the solution space; however, it helps students to avoid =
the
splitting step and the grouping step which are pretty “tough” f=
or
most of them at this level. Because they have a tough time to deal with rew=
rite
two splitting terms and decide which term will go with what in order to do
grouping.
My Method:
-
Step 1: Same as Step 1 above
-
Step 2: Same as Step 2 above
-
Step 3: After Step 2, we found two numbers n=
1
and n2 that satisfied the product and sum condition. Now, we just
write out the answer in the following form
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
=
&nb=
sp; =
=
&nb=
sp; =
&nb=
sp; =
&nb=
sp; =
(1) &=
nbsp; &nbs=
p;
Redo above example: After Step 2 (with table), we have=
found
two numbers 2 and -12 that satisfied the product and sum condition. =
Now
let’s try to write out the answer:
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; =
&nbs=
p; Of
course, you will ask me “Can you prove your formula?”
Proof of formula (1):
&nbs=
p; As
we know, two solutions of the general trinomial are: =
&nbs=
p; therefore &=
nbsp; &nbs=
p; =
&=
nbsp; &nbs=
p; =
&=
nbsp; &nbs=
p; (2)
&nbs=
p; and &=
nbsp; &nbs=
p; &=
nbsp; =
&=
nbsp; &nbs=
p; (3)
&nbs=
p; There is a theorem that states (2) & (3) of two
solutions of a quadratic equation, but it does not have any relations to n<=
sub>1
and n2
The next simple tr=
ick is
very important in order to find the formula (1).
Multiply both sides of (2) by a =
a(x1 + x2) =3D - b =
or =
(- ax1 ) + (- ax2 )=
=3D
b =
&nb=
sp; =
&nb=
sp;
(4)
&nbs=
p; Multiply
both sides of (3) by a2 =
a2(x1*x2) =3D a2 (c/a) or
(- ax1)*(- ax2) =3D ac =
&nb=
sp; =
&nb=
sp;
(5)
&nbs=
p; From
(4) and (5), we can see that (- ax1) and (- ax2) are =
the
two numbers that satisfied the product and sum condition (product is ac and=
sum
is b).
&nbs=
p; Hence =
&=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; (6)
&nbs=
p; We
all know that we can write the trinomial in the form: =
=
=
&nb=
sp; (7)=
p>
&nbs=
p; Solve
x1 and x2 from (6), and plug them into (7): &=
nbsp;
&nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; &nbs=
p; &=
nbsp; =
That is the formula (1).
Conclusion: Trying to find simpler
methods/techniques to solve problems is one of my hobbies. I think this is a
better way to do “Master Product and Sum” and it should be a
benefit to students. If anybody has seen the formula (1) before (i.e. someo=
ne
else did create this formula before me), please let me know.
&nbs=
p; &=
nbsp; San Diego, Fall 2004
&nb=
sp;
AN EXAMPLE NEEDED TO BE MODIFIED
IN JAMES STEWART'S CALCULUS BOOK
Thomas Nguyen
Example 7 (page 812, Early Transcendental Multivariable Calculus 5e -
James Stewart - ISBN: 0-534-41778-7)
A crate is hauled 8 (m) up a ramp under a constant force of 200 (N)
applied at an angle of 25o
to the ramp. Find the work done.
James Stewart's solution:
If F and D are the force and displacement vectors, as
pictured in Figure 7, then the work done is
&n=
bsp;  =
;
W =3D F. D =3D |F|.|D|. cos(<=
/span>α) =3D (200)(8)cos(25o) ≈ 1450 Nm =3D 1450 J
=
p>
&nb=
sp; =
&nb=
sp;
Figure 7
Suggested Modification:
There are two ways to modify this problem:
- First, easiest way, to keep James' solution<=
/u>,
we should change the question: "A crate is =
hauled
8 m up a ramp under a constant force of 200 N applied at an angle of =
25o
to the ramp. Find the work done by this
force (F force)"
However, in Physics, this is not a "normal" wa=
y to
have a question like this. We have to change problem a little bit: no more =
up
ramp, instead of that, the crate should be moved on horizontal platform as =
the
following figure:
<=
/span>
- Second, with the formal up ramp form (as in
Figure 7), the problem needs two more given parameters: the mass of the cra=
te
(m) and the deep of the ramp (angle "θ"). Then, in this case, the problem should be
modified as the following: "A crate is haul=
ed 8 m
up a ramp under a constant force of 200 N applied at an angle of =
25o
to the ramp. Find the work done on this
crate."
Solution, in this case, is modified, too. Work will be done by =
two force: F force
and gravitational force (g force).
<=
/span>
&n=
bsp;
W =3D F. D - [m.g.sin(θ)]. D &=
nbsp;
where "θ" =
is the
deep of the ramp.
<=
/span>
Last but not least, frictionless should be stated.
<=
/span>
&=
nbsp;
San Diego,
Tuesday August 15th, 2006
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