https://file.aiquickdraw.com/mathbot/file/1726443391700ihjzbywc.jpg

Let's analyze the graph and questions step by step.

### Understanding the Graph
- The graph shows the number of farms in the US (\( N = f(t) \)) over time, where \( t \) represents the year.
- The x-axis (horizontal) represents the year, ranging from 1930 to 2020.
- The y-axis (vertical) represents the number of farms in millions.
- We can observe a decline in the number of farms from 1930 to around 1990, after which the number stabilizes.

### Question (a)
**Is \( f'(1970) \) positive or negative? What does this tell you about the number of farms?**

- **Answer:** \( f'(1970) \) is **negative**.
- **Explanation:** The derivative \( f'(1970) \) represents the rate of change of the number of farms in the year 1970. Since the graph shows a decreasing trend around 1970, \( f'(1970) \) is negative. This tells us that the number of farms was decreasing in 1970.

- **Next part:** That means the number of farms in the US was **decreasing** in 1970.

### Question (b)
**Which is more negative: \( f'(1960) \) or \( f'(1980) \)?**

- **Answer:** \( f'(1960) \) is **more negative** than \( f'(1980) \).
- **Explanation:** In 1960, the graph shows a steeper decline compared to 1980, indicating that the rate of decrease of the number of farms was greater in 1960 than in 1980. Therefore, \( f'(1960) \) is more negative.

### Question (c)
**Is \( f'(2000) \) positive, negative, or zero? What does this tell you about the number of farms?**

- **Answer:** \( f'(2000) \) is **zero**.
- **Explanation:** Around the year 2000, the graph levels off, indicating that the number of farms is neither increasing nor decreasing significantly. This means that the rate of change of the number of farms is zero.
  
- **Next part:** That means the number of farms in the US was **steady** in 2000.

Would you like to explore any specific part of these questions in more detail?

### Related Questions:
1. How can the derivative \( f'(t) \) help us understand changes in the number of farms over different decades?
2. What does a positive value of \( f'(t) \) indicate in the context of this graph?
3. How would you describe the trend in the number of farms between 1930 and 1950?
4. What might be some reasons for the decline in the number of farms between 1930 and 1990?
5. If \( f'(t) = 0 \) represents a steady state, what would \( f''(t) \) (the second derivative) tell us about the concavity of the graph?

### Tip:
To understand the behavior of a function over time, analyzing the sign and magnitude of its derivative can provide insights into the rate and direction of change.

The figure shows the number of farms in the US from 1930 to 2020 as a function of time. It asks to analyze the rate of change of the number of farms in 1970, compare rates in 1960 and 1980, and determine the rate of change in 2000.

Learn how to interpret the derivative of a function showing the number of farms in the US from 1930 to 2020. Analyze the rate of change in 1970, compare the rates between 1960 and 1980, and determine the trend in 2000 using calculus concepts. Suitable for advanced high school students studying calculus.

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