Suppose we have a function
y = f(x). It has a derivative of "
2". Thus, by integrating "
2" we get
2x +C. The orginal function f(x) is represented by
y = 2x + C . This is a linear function with
constant slope of 2 and
Y-INTECEPT of C. Allowing C to take on each REAL NUMBER would create an
infinite number of lines that completely cover the xy-plane. If you happen to know that the orginal function y = f(x) passes thru a certain point say
(3,10) then "C" has unique value of
4. " y = 2 x + 4" is the only line with slope 2 that will pass thru the point (3,10)
In other words, the
INITIAL CONDITIONS OF (3,10) makes the equation "
y = 2 x + 4""TRUE"!
Thus, to find C we must have an INITIAL (CONDITION) POINT that f(x) passes through. Plugging the point into the equation with y and x and C will enable us to find the unique value for C that will allow the point to be on the graph of f(x).
HOW ABOUT ANOTHER EXAMPLE:
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