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The problem in the image asks to find the rate of change of weight with respect to time, where the weight \( w(t) \) of a boy whose age is between 0 and 36 months is approximated by the function:

\[
w(t) = 7.1t + 1.78t^2 - 0.0085t^3 + 0.000865t^4
\]

To find the rate of change of weight with respect to time, \( w'(t) \), we need to take the derivative of \( w(t) \) with respect to \( t \). Let me compute that.

\[
w'(t) = \frac{d}{dt} \left( 7.1t + 1.78t^2 - 0.0085t^3 + 0.000865t^4 \right)
\]

Now, let me calculate the derivative step by step:

1. \( \frac{d}{dt}(7.1t) = 7.1 \)
2. \( \frac{d}{dt}(1.78t^2) = 2 \times 1.78t = 3.56t \)
3. \( \frac{d}{dt}(-0.0085t^3) = 3 \times -0.0085t^2 = -0.0255t^2 \)
4. \( \frac{d}{dt}(0.000865t^4) = 4 \times 0.000865t^3 = 0.00346t^3 \)

Thus, the derivative \( w'(t) \), which represents the rate of change of weight with respect to time, is:

\[
w'(t) = 7.1 + 3.56t - 0.0255t^2 + 0.00346t^3
\]

This is the rate of change function. You can substitute any value of \( t \) (in months) to find the rate of change of weight at that specific time.

Would you like to see further details on how to apply this? Or any clarifications?

Here are five related questions you might find interesting:
1. How do you interpret the rate of change in a real-life scenario for this function?
2. What does a negative value of \( w'(t) \) signify in terms of the boy's weight?
3. How can we find when the boy's weight is increasing the fastest?
4. At what age might the weight start to decrease based on this model?
5. How could you model a different growth pattern for boys using other mathematical functions?

**Tip**: To analyze a function’s behavior over time, critical points (where \( w'(t) = 0 \)) are helpful for understanding increasing and decreasing intervals.

The median weight of a boy whose age is between 0 and 36 months can be approximated by the function w(t) = 7.1t + 1.78t^2 - 0.0085t^3 + 0.000865t^4. Find the rate of change of weight with respect to time, w'(t).

Learn how to compute the rate of change of weight with respect to time for a polynomial function representing the median weight of a boy between 0 and 36 months. This problem involves applying differentiation techniques to find w'(t) for the given weight function.

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