Euler's method, concavity and arc length

Name: ______________________ Date: ____________ Euler's method, concavity and arc length (9 points)

Graph

Let

y = f(x)

be the solution of the differential equation

\frac{dy}{dx} = 2x - y

with

f(0) = 1

. It is known that

f(x) > 2x - 2

for all

x

. The slope field of the differential equation is shown, with the point

(0, 1)

marked. A calculator is not allowed.

(a) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f(1)$ . Show the work that leads to your answer in a table with columns $x$ , $y$ , $\frac{dy}{dx}$ and $\Delta x$ . [3 points]
(b) Find $\frac{d^2y}{dx^2}$ in terms of $x$ and $y$ . The first step in part (a) gives $f(0.5) \approx 0.5$ . Is this an overestimate or an underestimate of $f(0.5)$ ? Explain your reasoning. [2 points]
(c) Write, but do not evaluate, an integral expression in terms of $x$ and $f(x)$ that gives the length of the graph of $f$ from $x = 0$ to $x = 1$ . [1 point]
(d) A different curve is the graph of $g(x) = \frac{x^3}{3} + \frac{1}{4x}$ for $1 \le x \le 2$ . Find the length of this curve. Show the work that leads to your answer. [3 points]

The question