Poster: SGM 2010

Excel calculus

Definite Integration

b
∫f(t)dt = ∑ [((f(tp) - f(tp-1))/2)*(tp - tp-1)]
a

Here f(t) is the function which is to be integrated.
f(tp) is the value of the function at p-th cell in excel.
The summation is to be performed between a & b cells in excel.

There can be two errors due to assumption of the height of the rectangle as ((f(tp) - f(tp-1))/2) for area calculation. The other alternative heights can be f(tp) or f(tp-1). Error analysis shows that as the number of rectangles (smaller steps) or the number of cells increases, the value of percentage error decreases exponentially.

Example:

Plots of integration and error analysis of the function f(t) = t²





:-)

Disadvantages of Low EMI - Longer Period loans

Low EMI - Longer Period loans enables the borrower to pay back loan amounts to the loan granting agencies (e.g. banks etc) slowly but it has a major disadvantage. The borrower often pays back a total interest almost equal or may be more than the loan amount. So, temptations of Low EMI - Longer Period loans can often end up in a disadvantageous position for the borrower.

This is illustrated in the following graph.



In the graph,

Blue line is the reduction of actual loan amount as the EMI is paid, Initial is the actual amount taken as loan.
Purple line is the cumulative amount of fixed EMI paid.
Red line is the summation of the actual loan taken and the monthly interest.
Green lines are period marks.

Reduction method has been assumed in this calculation where the interest is on the reduced loan amount, due to payment of EMI

Total loan and interest on it is re-payed when the blue line cuts x-axis or the purple line intersects the red line.

In this example the loan was borrowed for a maximum period of 15 years.

The graphs show that the EMI pays back all the amount within the stipulated time period. However, closer inspection of the red line reveals that the borrower pays back almost twice the amount he took as loan i.e. an interest almost equal to the loan.

An advantageous position is moderate EMI - shorter period loan where the borrower pays a much lower total amount (1.2 -1.4 times the loan amount depending on time and EMI).

:-)

Cycles in cycles

Cycles in Cycles




f(t) = [sin (t*180/A)]*[sin (t*180/B)]*[sin (t*180/D)]

where
A=0.18, B=180, C=1800

:-)

My latest artwork

A double periodic function




Function 1: [e^t]* [sin (t*pi/5)]
Light Blue Graph

Function 2: e^t
Dark Blue Graph

Function 3: 3 term moving average trend-line of Function 1
Orange Graph


The fun was with "t".

t = sin a
where a is 0.05, 0.10, 0.15, 0.20 ..............


A non-uniform periodic function





Function 1: [e^t]*[10^t]* [sin (t*pi/10)]
Light Blue Graph

Function 2: [e^t]*[10^t]
Dark Blue Graph

The fun was with "t".

t = Log 10 (a)
where a is 1, 2, 3, 4, ..............



Thanks to Microsoft Excel 2007 for drawing the graph.

:-)