Step 1: Read the straight line from the plot.
The graph plots \(\dfrac{dy}{dx}\) on the vertical axis against \(x\) on the horizontal axis. Reading the marked points, the line passes through \((0,1)\), \((1,2)\), \((2,3)\) and \((3,4)\). These four points are collinear, so \(\dfrac{dy}{dx}\) is a linear function of \(x\).
Step 2: Find the equation of this line.
The slope between any two of these points, say \((0,1)\) and \((3,4)\), is
\[
m = \frac{4-1}{3-0} = 1
\]
and the line meets the vertical axis at \(1\), so the intercept is \(1\). This gives
\[
\frac{dy}{dx} = x + 1
\]
Step 3: Integrate to get the change in \(y\) from \(x=2\) to \(x=3\).
Since \(dy/dx\) is known, \(y(3)-y(2)\) is the definite integral of \(dy/dx\) between these limits:
\[
y(3) - y(2) = \int_{2}^{3} (x+1)\,dx = \left[\frac{x^2}{2} + x\right]_{2}^{3}
\]
Evaluating the antiderivative at the two limits:
\[
\left(\frac{9}{2}+3\right) - \left(\frac{4}{2}+2\right) = (4.5 + 3) - (2 + 2) = 7.5 - 4 = 3.5
\]
Step 4: Add this change to the given value \(y(2)=5\).
\[
y(3) = y(2) + 3.5 = 5 + 3.5 = 8.5
\]
