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To solve the problem based on the given graph, we need to estimate the rates of change at two specific times, \( t = 3 \) and \( t = 18 \), by computing the slopes of the tangent lines at those points. Additionally, we are asked to calculate the average rate of change over the first year (0 to 12 months).

### Steps:

1. **Find the rate of change at \( t = 3 \) months**:
   The slope of the tangent at \( t = 3 \) can be approximated by looking at the graph. From the image, the weight at \( t = 3 \) seems to be around 7.5 pounds, and the tangent line appears to rise 5 pounds over a span of 4 months. Therefore, the slope (rate of change) is:

   \[
   \text{slope at } t = 3 \approx \frac{5}{4} = 1.25 \, \text{lb/month}
   \]

2. **Find the rate of change at \( t = 18 \) months**:
   From the graph, at \( t = 18 \), the weight appears to be approximately 22.5 pounds, and the tangent line rises 3.5 pounds over a span of 6 months. Therefore, the slope (rate of change) is:

   \[
   \text{slope at } t = 18 \approx \frac{3.5}{6} = 0.58 \, \text{lb/month}
   \]

3. **Average rate of change over the first year (0 to 12 months)**:
   To find the average rate of change over the first year, we use the formula for the average rate of change:
   \[
   \text{Average rate of change} = \frac{f(12) - f(0)}{12 - 0}
   \]
   From the graph, the weight at \( t = 12 \) is about 20 pounds, and at \( t = 0 \), it is about 7.5 pounds. So,
   \[
   \text{Average rate of change} \approx \frac{20 - 7.5}{12} = \frac{12.5}{12} = 1.04 \, \text{lb/month}
   \]

### Final Answers:
- Rate of change at \( t = 3 \): \( 1.25 \, \text{lb/month} \)
- Rate of change at \( t = 18 \): \( 0.58 \, \text{lb/month} \)
- Average rate of change over the first year: \( 1.04 \, \text{lb/month} \)

Would you like more details on these calculations or have any questions?

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Here are 5 related questions to extend your understanding:
1. How would the rates of change differ if the infant's growth were linear?
2. How can you use limits to more precisely estimate the slope of a tangent line at a point?
3. What is the significance of the average rate of change versus the instantaneous rate of change?
4. How could you apply these rate of change concepts to real-world scenarios, like predicting future growth?
5. How would the graph change if the growth rate decreased after 24 months?

**Tip:** When interpreting graphs, pay attention to the steepness of the lines—the steeper the line, the greater the rate of change at that point.

By computing the slopes of the respective tangent lines, estimate the rate of change (in lb/month) of the average infant's weight when t = 3 and when t = 18. What is the average rate of change (in lb/month) in the average infant's weight over the first year after birth?

Learn how to estimate the rate of change in an infant's weight using tangent lines on a graph. We calculate the rate at specific points (t = 3 and t = 18) and find the average rate of change over the first year. Suitable for high school students studying derivatives and rates of change.

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