Question

Suppose that a point moves along some unknown curve y = f(x) in such a way that, at

each point (x, y) on the curve, the tangent line has slope x^2. Find an equation for the
curve, given that it passes through (2, 1).​

Answer 1

f(x) = x³/3 - 5/3

Explanation:

To get the equation of the curve, integrate the slope

f(x) = integral of x²

= (x^(2+1))/3 + c

f(x) = x³/3 + c

To find c, use the given point

1 = 2³/3 + c

c = 1 - 8/3 = -5/3

f(x) = x³/3 - 5/3