Question

From the following plot of \(\dfrac{dy}{dx}\) versus \(x\), and if \(y(2) = 5\), the value of \(y(3)\) is . (rounded off to one decimal place)

Hint:

The plotted line gives \(dy/dx = x+1\). Integrate this from \(x=2\) to \(x=3\) and add the result to \(y(2)=5\).

Answer 1

Correct Answer: 8.5

A second way to solve this is to find the complete function $y(x)$ first, instead of only the area under the curve between $x=2$ and $x=3$.

From the plot, the four marked points $(0,1)$, $(1,2)$, $(2,3)$ and $(3,4)$ lie on one straight line, so:

\[ \frac{dy}{dx} = x + 1 \]

Integrating both sides with respect to $x$ gives the general solution, with an unknown constant $C$:

\[ y = \int (x+1)\,dx = \frac{x^2}{2} + x + C \]

Now use the given condition $y(2) = 5$ to pin down $C$:

\[ 5 = \frac{(2)^2}{2} + 2 + C = 2 + 2 + C = 4 + C \] \[ C = 1 \]

So the complete function is:

\[ y(x) = \frac{x^2}{2} + x + 1 \]

Now put $x = 3$ directly into this formula:

\[ y(3) = \frac{9}{2} + 3 + 1 = 4.5 + 3 + 1 = 8.5 \]

Let's summarize:

• The graph shows $dy/dx = x+1$, a straight line, so its integral $y(x)$ is a parabola with an unknown constant of integration.
• Fixing that constant with the given point $y(2)=5$ pins down the entire function, after which any value of $y$, not just $y(3)$, can be read off directly.

Both routes agree: $y(3) = 8.5$.