Find Curve Given Slope, Point

Date: 8/29/96 at 23:8:33
From: Anonymous
Subject: Find Curve Given Slope, Point

Find the curve whose slope at the point (x,y) is 3x^2 if the curve 
passes throught the point (1,-1).

Have no idea where to begin!

Date: 8/30/96 at 11:22:43
From: Doctor Mike
Subject: Re: Find Curve Given Slope, Point

Hi Bill,
  
This is a Calculus problem.  One thing you learn how to do in calculus
is a process for finding the slope to the graph of a function.  If 
f(x) is a function you are graphing, then the "derivative of f", which 
is often written f'(x) , gives the slope of the original function.  
That is, the slope of the tangent line to the graph of f(x) at point 
(a,f(a)) is exactly f'(a). 
  
To get from f(x) to f'(x) is called finding the derivative, or
differentiation.  To get from f'(x) back to f(x) is called finding the
anti-derivative, or integration.
  
For your example y = f(x) is unknown, but f'(x)=3x^2 is given.  
Because I have had calculus, I can easily work out in my head that 
f(x) = x^3 + C where C is any arbitrary number.  Since you must have 
(x,y) = (1,-1) on the graph of f , f(1) = -1 must be true, which 
requires that C be -2. So, f(x) = x^3 - 2  or y = x^3 - 2 .
  
I hope this helps.  If you have not had Calculus and all this sounds
interesting, why not have a bash at it! 
 
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/