Find the function f given that the slope of the tangent line to the graph of f at any point is and that the graph of f passes through the given point. ;

Question

Find the function f given that the slope of the tangent line to the graph of f at any point  is  and that the graph of f passes through the given point. ​  ;

Quizplus · Accepted Answer

Since the slope of the tangent line at any point is given to be 3x^2, we know that the derivative of f is 3x^2. Integrating 3x^2 gives x^3 + C, where C is the constant of integration. We can find the value of C by using the fact that the graph of f passes through the point (2, 4). Plugging in x = 2 and y = 4, we get 4 = 2^3 + C, which gives C = -4. Therefore, the function f is given by f(x) = x^3 - 4. Taking the derivative of f(x) gives f'(x) = 3x^2, which matches the given slope at any point.

Find the Function F Given That the Slope of the Tangent