The increase in carbon dioxide (CO 2 ) in the atmosphere is...

The increase in carbon dioxide (CO $_{2}$ ) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of $\mathrm{CO}_{2}$ in the atmosphere from 1958 through 2013 is approximated by $$A(t)=0.012414 t^{2}+0.7485 t+313.9 \quad(1 \leq t \leq 58)$$ where $A(t)$ is measured in parts per million volume (ppmv) and $t$ in years, with $t=1$ corresponding to 1958 . a. What can you say about the rate of change of the average amount of atmospheric $\mathrm{CO}_{2}$ from 1958 through 2013? b. What can you say about the rate of change of the rate of change of the average amount of atmospheric $\mathrm{CO}_{2}$ from 1958 through $2013 ?$

The increase in carbon dioxide (CO $_{2}$ ) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of $\mathrm{CO}_{2}$ in the atmosphere from 1958 through 2013 is approximated by
$$A(t)=0.012414 t^{2}+0.7485 t+313.9 \quad(1 \leq t \leq 58)$$
where $A(t)$ is measured in parts per million volume (ppmv) and $t$ in years, with $t=1$ corresponding to 1958 .
a. What can you say about the rate of change of the average amount of atmospheric $\mathrm{CO}_{2}$ from 1958 through 2013?
b. What can you say about the rate of change of the rate of change of the average amount of atmospheric $\mathrm{CO}_{2}$ from 1958 through $2013 ?$

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Key Concepts

Quadratic Function Modeling

A quadratic function is used to model the average amount of atmospheric CO?, where the presence of a squared term leads to a non-linear trend. The structure of a quadratic function ensures that its first derivative is linear and its second derivative is constant, thus naturally representing a process where not only the value increases over time but the pace of that increase accelerates in a uniform manner.

Rate of Change (First Derivative)

The rate of change of a function with respect to time is captured by its first derivative. It represents how quickly the amount of atmospheric CO? is increasing or decreasing over a given time period. In the context of the quadratic model, calculating the derivative gives a linear function that indicates the changing pace of CO? accumulation, revealing a trend that is not constant but varies over time.

Acceleration (Second Derivative)

The second derivative of a function measures the rate of change of its first derivative, essentially capturing the acceleration or deceleration of the process. In this scenario, it quantifies how the speed at which CO? is accumulating itself changes over time; a positive constant second derivative implies that the rate of increase (the first derivative) is growing steadily.

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