use of technology in the teaching of maxima and minima

1. 0011 0010 1010 1101 0001 0100 1011
1
2
4
SUBTOPIC: APPLICATIONS OF
MAXIMA AND MINIMA
PRESENTED BY :
HIMANI ASIJA
P.G.T. MATHEMATICS
DELHI PUBLIC SCHOOL, VASANT KUNJ

2. EFFECTIVENESS OF USE OF
TECHNOLOGY
0011 0010 1010 1101 0001 0100 1011
For the best results of use of technology in education, it is
required to
1
2
Carefully integrate technology and the content
Select some abstract topics in the subject (math) and use
technology as a medium of instruction
4
Focus on professional development of teachers (both
in service and pre service)
Make an intelligent choice on the kind of software to be
used

3. REFLECTIONS ON THE USE OF
0011 0010 TECHNOLOGY FROM AROUND
1010 1101 0001 0100 1011
THE WORLD
 Is highlighted in the “Principles and Standards of Mathematics”,
U.S.A.
1
2
 Ministry Of Education, France since 2004, for future teachers
4
 Ministry Of Education, Singapore from 2006 for senior classes
and from 2007 for primary classes.

4. Use of technology to make teaching of
calculus innovative
0011 0010 1010 1101 0001 0100 1011
Subtopic: Applications of Maxima and Minima
Problems discussed:
1
2
1. Kepler’s problem.
2. To find the minimum length a ladder through a vertical
4
fence of a given height and at a given distance from a wall.
3. Extension of the above problem with a circular fence.

5. Problem 1: Kepler’s problem
0011 0010 1010 1101 0001 0100 1011
Statement :
In 1612, the great scientist Johannnes Kepler was about to remarry
and he needed to buy wine for the celebration. The wine salesman
1
2
brought barrels of wine to Kepler’s house and explained how they
were priced: a rod was inserted into the barrel and diagonally
4
through a small hole in the top. When the rod was removed, the
length of the rod which was wet determined the price for the barrel.

6. 0011 0010 1010 1101 0001 0100 1011
R
1
2
4
H L

7. A MATHEMATICAL SOLUTION OF
KEPLER’S PROBLEM
0011 0010 1010 1101 0001 0100 1011
Length of the rod (L) = fixed, since Kepler wanted to keep his
bugdet fixed.
1
2
Now, volume of the cylinder is given by V= Π R2 H
Also, L2 = R2 + H2
SO, V = Π (L2 - H2 ) H
4
On finding the first derivative and equating it to zero,
V´(H) = Π (L2 - 3H2 )
V´(H) = 0
1
H= L (Omitting the negative sign as h is positive)
3
FIND V˝(H) AND OBSERVE IT TO BE NEGATIVE

8. Using G.S.P. finding the solution of the
problem
0011 0010 1010 1101 0001 0100 1011
1
2
4

9. Problem 2: Minimum height of ladder
through a fence
0011 0010 1010 1101 0001 0100 1011
Statement: To find the minimum length of a ladder
passing through a fence of given height at a given distance
from the wall.
WA
1
2
4
L L
(y) LADDER (L)
F
E
N
a
C
E
b
x

10. A MATHEMATICAL SOLUTION
0011 0010 1010 1101 0001 0100 1011
Assuming a=3 and b=3.
Using geometry, y 3
x x 3
9x2
1
2
l2 x2
( x 3) 2
On differentiation w.r.t. x, and equating the derivative to zero,
4
we obtain x=6 and y=6 and l=6√2
On finding second differential, at x=6, we observe that it is
positive and hence ensures a minimum length of the ladder.

11. Using G.S.P. finding the solution of the
problem
0011 0010 1010 1101 0001 0100 1011
1
2
4

12. Problem 3 : Extension of the above
problem with a circular fence
0011 0010 1010 1101 0001 0100 1011
Statement: To find the minimum length of a ladder passing
through a circular fence.
1
2
4
W LADDER
A
L
L
y
r=3
FENCE
x

13. A MATHEMATICAL SOLUTION
0011 0010 1010 1101 0001 0100 1011
Given circular fence of radius r
B
Minimize length of the ladder BE=L
C
Let OB=y and OE=x, then by geometry,
A
AB=BC=y-3 and DE=CE=x-3
1
2
O
So, we have L = x+y-6 and L2 = x2 +y2 D E
Giving us L= y 2 6 y 18
y 6
4
On finding dL/dy, and putting it =0, y=6 3 2
y= 6+3 2 Satisfies the given condn.(i.e. finding L , it comes out
to be positive, thus ensuring a minima.)

14. Using G.S.P. finding the solution of the
problem
0011 0010 1010 1101 0001 0100 1011
1
2
4

15. CONCLUSION AND
RECOMMENDATIONS
Technology emphasizes 1011
0011 0010 1010 1101 0001 0100discovery and minimizes lecturing and
rote learning.
Teacher student interaction increases
1
2
To make the use of technology most effective, it is important
To correctly choose the topics to be taught using technology
4
To train the present teachers to use the technological aids
To make it an integral part of the pre service teacher training
To change the assessment techniques
That educators and programmers work together
Last, but not the least, to make the teachers believe that technology
is not their replacement, but an aid for teaching.