Find the function f given that the slope of the tangent line to the graph of f at any point (x, f (x)) is f'(x) and that the graph of f passes through the given point.

Question

Quizplus · Accepted Answer

Let's integrate the given slope to obtain the function f: ∫f'(x) dx = ∫f(x) dx = ∫f(x) f'(x) dx = ∫f(x) dx + C where C is the constant of integration. Thus, we have that f(x) = Ce^(x^2/2) (since integrating f'(x) with respect to x gives us e^(x^2/2) up to a constant C.) Now, using the given point (0,1), we have 1 = f(0) = Ce^(0) = C, so f(x) = e^(x^2/2). Thus, the correct choice is D.

Find the Function F Given That the Slope of the Tangent