Compare over the interval $-2 \leq x \leq 0$?
A. The exponential function decays at one-half the rate of the quadratic function.
B. The exponential function decays at the same rate as the quadratic function.
C. The exponential function decays at two-thirds the rate of the quadratic function.
D. The exponential function decays at three-fourths the rate of the quadratic function.
Answer (1)
Define exponential f ( x ) = a e k x and quadratic g ( x ) = b x 2 + c x + d functions.
Calculate average rates of change over [ − 2 , 0 ] : 2 f ( 0 ) − f ( − 2 ) and 2 g ( 0 ) − g ( − 2 ) .
Use example values a = 1 , k = − 0.5 , b = 1 , c = − 1 , d = 1 to find the ratio of exponential to quadratic decay rates.
The ratio depends on the parameters, so without more information, we cannot choose a definitive answer from the given options. The calculated ratio is approximately 0.286 .
Explanation
Problem Analysis We are asked to compare the decay rate of an exponential function to that of a quadratic function over the interval − 2 \[ x ≤ 0 .
To do this, we will analyze the average rates of change of both types of functions over the given interval and compare them.
Defining the Functions Let's define a general exponential function as f ( x ) = a e k x and a general quadratic function as g ( x ) = b x 2 + c x + d , where a , b , c , d , and k are constants. For the exponential function to decay, k must be negative.
Average Rate of Change To compare their decay rates, we can calculate the average rate of change for each function over the interval [ − 2 , 0 ] . The average rate of change is given by 0 − ( − 2 ) f ( 0 ) − f ( − 2 ) for the exponential function and 0 − ( − 2 ) g ( 0 ) − g ( − 2 ) for the quadratic function.
Example Values Let's choose some example values for the constants to illustrate the comparison. Let a = 1 , k = − 0.5 , b = 1 , c = − 1 , and d = 1 . Then, f ( x ) = e − 0.5 x and g ( x ) = x 2 − x + 1 .
Calculating Average Rates Now, let's calculate the average rates of change with these values:
For the exponential function: f ( 0 ) = e − 0.5 ⋅ 0 = 1 f ( − 2 ) = e − 0.5 ⋅ ( − 2 ) = e 1 ≈ 2.718 Average rate of change = 2 1 − e ≈ 2 1 − 2.718 = 2 − 1.718 ≈ − 0.859
For the quadratic function: g ( 0 ) = 0 2 − 0 + 1 = 1 g ( − 2 ) = ( − 2 ) 2 − ( − 2 ) + 1 = 4 + 2 + 1 = 7 Average rate of change = 2 1 − 7 = 2 − 6 = − 3
Comparing the Rates Now, we compare the magnitudes of the average rates of change. The ratio of the exponential to quadratic average rate of change is − 3 − 0.859 ≈ 0.286 . This is approximately equal to 3.5 1 , which is roughly between 4 1 and 3 1 .
Conclusion Based on this example, the exponential function decays at approximately 0.286 times the rate of the quadratic function. This value is closest to one-third, but none of the given options exactly match this ratio. However, it's important to note that this ratio depends on the specific parameters chosen for the functions. Different values for a , b , c , d , and k will result in different ratios.
Final Answer Since the relationship between the decay rates depends on the specific parameters of the functions, we cannot determine a definitive answer without more information. However, based on our example, the exponential function decays at roughly one-third the rate of the quadratic function, which is not among the given options. The closest answer based on the calculated ratio is one-fourth, but it is not accurate.
Final Conclusion Given the results from the tool, the ratio of exponential to quadratic average rate of change is approximately 0.286, and the ratio of exponential to quadratic derivatives at x=-1 is approximately 0.275. These values are closest to one-fourth or one-third, but none of the given options exactly match these ratios. Therefore, we cannot determine a definitive answer without more information.
Examples
Understanding the decay rates of different functions is crucial in various fields. For instance, in environmental science, it helps model the degradation of pollutants over time. Exponential decay can represent the breakdown of a chemical compound, while quadratic decay might describe the decrease in population size due to resource depletion. Comparing these rates allows scientists to predict long-term environmental impacts and develop effective mitigation strategies. Similarly, in finance, understanding decay rates helps in modeling the depreciation of assets or the decline in investment value over time, aiding in financial planning and risk management.
