Finding the Constant of Integration in Calculus

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:







LINK: Back to Contents