Question

the figure below shows n = f(t), the number of farms in the us¹ between 1930 and 2020 as a function of year, t. millions of farms 7 6 5 4 3 2 1 1930 1950 1970 1990 2010 t (year) (a) is f(1970) positive or negative? what does this tell you about the number of farms? f(1970) is that means that the number of farms in the us was in 1970. (b) which is more negative: f(1960) or f(1980)? (c) is f(2000) positive, negative or zero? what does this tell you about the number of farms? f(2000) is that means that the number of farms in the us in 2000. ¹www.nass.usda.gov, accessed march 10, 2020. etextbook and media save for later attempts: 0 of 15 used submit answer using multiple attempts will impact your score. 5% score reduction after attempt 5

Explanation:

Step1: Recall derivative meaning

The derivative $f^{\prime}(t)$ represents the rate - of - change of the function $N = f(t)$. If $f^{\prime}(t)>0$, the function is increasing; if $f^{\prime}(t)<0$, the function is decreasing.

Step2: Analyze $f^{\prime}(1970)$

Looking at the graph, the function $N = f(t)$ is decreasing at $t = 1970$. So $f^{\prime}(1970)$ is negative. This means the number of farms in the US was decreasing in 1970.

Step3: Compare $f^{\prime}(1960)$ and $f^{\prime}(1980)$

The slope of the tangent line to the curve at $t = 1960$ is steeper (more negative) than the slope of the tangent line at $t=1980$. So $f^{\prime}(1960)$ is more negative.

Step4: Analyze $f^{\prime}(2000)$

The function $N = f(t)$ is decreasing at $t = 2000$. So $f^{\prime}(2000)$ is negative, which means the number of farms in the US was decreasing in 2000.

Answer:

(a) negative; decreasing
(b) $f^{\prime}(1960)$
(c) negative; decreasing