When distributing the marked test paper to students, one student, Carman, reminded me that there was a ‘question’ in the following question:
If , find at the point ().
Carman said, ‘the point does NOT lie on the curve.’
Good observation! I had to say thank you to him. Although I’m not the setter, I should bear the responsibility of checking the paper.
But a natural follow-up question turns up: what is the meaning of the number we are obtaining? Is the number meaningless or standing for something?
Let’s consider a simple example.
Now, we are free to put values of to obtain certain number, e.g. . Well, what is the meaning of this “12″?
Actually, we can easily solve the differential equation (by integration, say), and obtain
where is an arbitrary constant.
The equation is actually representing a family of curves.
Then, no matter which point we are considering, like (2,0), (2,10) or even (2,1997), we can find a ‘member’ in the family such that the point really lie on that member!
For (2,0), set , hence (2,0) lies on
For (2,10), set , hence (2,10) lies on
For (2,1997), set , hence (2,10) lies on
So, when we are finding , we are actually finding the slope of tangent at a point () to the curve , see the meaning?
Another example, say
Free to put values of and into the above, what does the answer stand for?
On solving the differential equation , yield
for any value of .
Student, you could see that it stands for a family of curves (in this case, circles), centred at .
Hence, if we ask to find the value say, we are actually calculating the slope of tangent to a circle which is centred at with radius .
Now, looking back to the original question,
DOES NOT IMPLY
but, as you can imagine,
(where is an arbitrary constant)
satisfies the differential equation
Now represents equation of family of curves.
It is easy to find a member in this family, such that it passes through (), just set
Hence, finding the value of at () can be interpreted as finding the slope of tangent to the curve
Apart from , will there be other equation(s) satisfying