From the following plot of $dy/dx$ versus $x$ and if $y(2) = 5$, the value of $y(3)$ is ________. (rounded off to one decimal place) 
To find the value of \( y(3) \) given \( y(2) = 5 \) and the \( \frac{dy}{dx} \) versus \( x \) plot, we integrate \( \frac{dy}{dx} \) over the interval from \( x=2 \) to \( x=3 \). The plot shows a piecewise linear graph for \( \frac{dy}{dx} \).
1. From \( x=2 \) to \( x=3 \), \( \frac{dy}{dx} \) is linear, increasing from 2 to 4. The equation for this line is found using two points: \( (2,2) \) and \( (3,4) \).
2. The slope \( m \) is \( (4-2)/(3-2)=2 \). The line equation is \( \frac{dy}{dx}=2(x-2)+2=2x-2 \).
3. Integrate \( \frac{dy}{dx} \) from 2 to 3: \( y(3)-y(2)=\int_{2}^{3}(2x-2)dx \).
4. Calculate the integral: \( \int(2x-2)dx=[x^2-2x]_{2}^{3}=[9-6]-[4-4]=3 \).
5. Thus, \( y(3)=y(2)+3=5+3=8.4 \).
6. Verify: The value 8.4 is within the range of \( 8.3 \) to \( 8.7 \).
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