Let $f$ be a function with $f(1) = 4$ such that for all points $(x, y)$ on the graph of $f$ the slope is given by $\dfrac{3x^2 + 1}{2y}$. (a) Find the slope of the graph of $f$ at the point where $x = 1$. (b) Write an equation for the line tangent to the graph of $f$ at $x = 1$ and use it to approximate $f(1.2)$. (c) Find $f(x)$ by solving the separable differential equation $\dfrac{dy}{dx} = \dfrac{3x^2 + 1}{2y}$ with the initial condition $f(1) = 4$. (d) Use your solution from part (c) to find $f(1.2)$.
Let $f$ be a function with $f(1) = 4$ such that for all points $(x, y)$ on the graph of $f$ the slope is given by $\dfrac{3x^2 + 1}{2y}$.
(a) Find the slope of the graph of $f$ at the point where $x = 1$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = 1$ and use it to approximate $f(1.2)$.
(c) Find $f(x)$ by solving the separable differential equation $\dfrac{dy}{dx} = \dfrac{3x^2 + 1}{2y}$ with the initial condition $f(1) = 4$.
(d) Use your solution from part (c) to find $f(1.2)$.