Slope field, tangent line and separation for dy/dx = x(4 - y)

The question

Name: ______________________   Date: ____________   Slope field, tangent line and separation for dy/dx = x(4 - y) (9 points)

1. (a) Consider the differential equation $\frac{dy}{dx} = x(4 - y)$. On the axes provided, sketch a slope field for the equation at the nine points shown. [1 point]

2. (b) Let $y = f(x)$ be the particular solution to the differential equation with $f(2) = 3$. Write an equation for the line tangent to the graph of $f$ at $x = 2$, and use it to approximate $f(2.1)$. [2 points]

3. (c) Find $\frac{d^2y}{dx^2}$ in terms of $x$ and $y$. Determine whether the approximation in part (b) is an overestimate or an underestimate of $f(2.1)$. Give a reason for your answer. [2 points]

4. (d) Find $y = f(x)$, the particular solution to the differential equation with the initial condition $f(2) = 3$. [4 points]