Problem 62 - Single Variable Calculus

Question

Global Warming The increase in carbon dioxide 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 2007 is approximated by $$ A(t)=0.010716 t^{2}+0.8212 t+313.4 $$ where \(t=1\) corresponds to the beginning of 1958 and \(1 \leq t \leq 50\) a. What can you say about the rate of change of the average amount of atmospheric \(\mathrm{CO}_{2}\) from 1958 through 2007 ? b. What can you say about the rate of the rate of change of the average amount of atmospheric \(\mathrm{CO}_{2}\) from 1958 through \(2007 ?\) Source: Scripos Institution of Oceanogranhv

Step-by-Step Solution

Answer

In conclusion: a. The average amount of atmospheric CO2 has increased from 1958 through 2007, as the first derivative, \(\frac{dA}{dt} = 2 \cdot 0.010716 t + 0.8212\), is a positive increasing function. b. The rate of change of the average amount of atmospheric CO2 is increasing over time, as the second derivative, \(\frac{d^2A}{dt^2} = 2\cdot 0.010716\), is a positive constant.

Step 1: Write down the given function

We are given the function that describes the average amount of atmospheric CO2 from 1958 through 2007: \(A(t) = 0.010716 t^2 + 0.8212 t + 313.4\) where \(t=1\) corresponds to the beginning of 1958 and \(1 \leq t \leq 50\). #Step 2: Find the first derivative of A(t)#

Step 2: Find the first derivative of A(t)

To find the rate of change of the average amount of atmospheric CO2, we need to find the first derivative of the function A(t) with respect to time t. \(\frac{dA}{dt} = \frac{d}{dt}(0.010716 t^2 + 0.8212 t + 313.4)\) Applying the power rule, \(\frac{dA}{dt} = 2\cdot0.010716 t + 0.8212\) #Step 3: Interpret the first derivative#

Step 3: Interpret the first derivative

The first derivative \(\frac{dA}{dt}\) represents the rate of change of the average amount of atmospheric CO2 with respect to time. Since the coefficient of the t term (2*0.010716) is positive, the function A(t) is increasing. This tells us that the average amount of atmospheric CO2 has increased between 1958 and 2007. #Step 4: Find the second derivative of A(t)#

Step 4: Find the second derivative of A(t)

To find the rate of the rate of change of the average amount of atmospheric CO2, we need to find the second derivative of A(t), which is the first derivative of the first derivative we found in step 2. \(\frac{d^2A}{dt^2}= \frac{d}{dt}(2\cdot 0.010716 t + 0.8212)\) Applying the power rule, \(\frac{d^2A}{dt^2} = 2\cdot 0.010716\) #Step 5: Interpret the second derivative#

Step 5: Interpret the second derivative

The second derivative \(\frac{d^2A}{dt^2}\) represents the rate of change of the rate of change of the average amount of atmospheric CO2 with respect to time. Since \(\frac{d^2A}{dt^2}\) is a positive constant (2 * 0.010716), the rate of change of the average amount of atmospheric CO2 is increasing over time. In conclusion, we can say that: a. The average amount of atmospheric CO2 has increased from 1958 through 2007. b. The rate of change of the average amount of atmospheric CO2 is increasing over time.

Key Concepts

Derivative

A derivative in calculus is a mathematical tool that helps us understand how a function is changing. More specifically, it tells us the rate at which one quantity changes with respect to another. Think of it as the "slope" at any given point on a curve.
To find a derivative, we use rules like the power rule. For example, if you have a function of the form \(f(x) = ax^n\), the derivative is found using the formula \(f'(x) = nax^{n-1}\).
The given function \(A(t) = 0.010716 t^2 + 0.8212 t + 313.4\) models the average amount of atmospheric CO2 over time. By differentiating this function with respect to \(t\), using the power rule, we get its first derivative: \(\frac{dA}{dt} = 2 \cdot 0.010716 t + 0.8212\). This derivative represents the rate of change of atmospheric CO2. It shows us how the CO2 concentration changes year by year. The positive coefficient of \(t\) indicates that the CO2 levels are increasing over time.

Rate of Change

The rate of change is essentially how fast a quantity is changing over a specific period. In practical terms, it is often referred to as velocity, growth rate, or speed, depending on the context.
In our context, the rate of change specifically refers to how the average amount of atmospheric CO2 is shifting over the years. By determining the rate of change, we can gain insights into environmental trends and support climate science.
The first derivative \(\frac{dA}{dt} = 2 \cdot 0.010716 t + 0.8212\) we calculated earlier directly reflects this rate of change. It mathematically determines how quickly CO2 concentrations are rising each year between 1958 and 2007. With a positive rate, we know there's a consistent increase in the amount of CO2, pointing to potential environmental impacts like enhanced global warming.

Second Derivative

The second derivative provides further insight into the behavior of a function. Instead of showing us just how fast something changes, it tells us if that rate of change is accelerating or decelerating. It's a rate of a rate, providing a deeper layer of analysis.
In math terms, finding the second derivative means taking the derivative of the first derivative. In our CO2 problem, this yields \(\frac{d^2A}{dt^2} = 2 \cdot 0.010716\), which is a constant value. This constancy suggests a uniform increase in the rate of change of atmospheric CO2.
Because the second derivative is positive, it signifies that the rate at which CO2 is increasing is itself increasing. This kind of analysis is crucial for environmental studies, as it tells us not just that CO2 levels are rising, but that they're rising faster and faster over time, which is a concerning trend for global climate health.