1. **State the problem:** We are given gold production data $G = f(t)$ for years $t$ from 2014 to 2018 and asked to analyze the derivative $f'(t)$, which represents the rate of change of gold production over time. 2. **Recall the meaning of the derivative:** The derivative $f'(t)$ measures how fast the gold production changes with respect to time (years). If $f'(t)$ is positive, production is increasing; if negative, production is decreasing. 3. **Calculate approximate values of $f'(t)$ between years:** - Between 2014 and 2015: $$f'(2014.5) \approx \frac{3100 - 2990}{2015 - 2014} = \frac{110}{1} = 110$$ metric tons per year. - Between 2015 and 2016: $$f'(2015.5) \approx \frac{3110 - 3100}{2016 - 2015} = \frac{10}{1} = 10$$ metric tons per year. - Between 2016 and 2017: $$f'(2016.5) \approx \frac{3230 - 3110}{2017 - 2016} = \frac{120}{1} = 120$$ metric tons per year. - Between 2017 and 2018: $$f'(2017.5) \approx \frac{3300 - 3230}{2018 - 2017} = \frac{70}{1} = 70$$ metric tons per year. 4. **Answer part (a):** Since all $f'(t)$ values are positive, $f'(t)$ appears to be positive. This means gold production is increasing over these years. 5. **Answer part (b):** The greatest $f'(t)$ is 120 metric tons/year between 2016 and 2017. 6. **Answer part (c):** To estimate $f'(2018)$, we use the rate between 2017 and 2018: $$f'(2018) \approx 70$$ metric tons per year. Interpretation: In 2018, gold production was increasing at a rate of approximately 70 metric tons per year. **Final answers:** - (a) $f'(t)$ appears to be positive. This means that the production of gold is increasing. - (b) $f'(t)$ appears to be greatest between 2016 and 2017. - (c) $f'(2018) \approx 70$ metric tons per year. In 2018, gold production was increasing at a rate of 70 metric tons per year.