Question
A: The function is decreasing at a decreasing rate (that is, the changes are negative and growing in
size),
B: The function is decreasing at a constant rate.
C: The function is decreasing at an increasing rate (that is, the changes are negative but becoming
smaller in size).
D: The function is increasing at a decreasing rate (that is, the changes are positive but shrinking),
E: The function is increasing at a constant rate.
F: The function is increasing at an increasing rate.
size),
B: The function is decreasing at a constant rate.
C: The function is decreasing at an increasing rate (that is, the changes are negative but becoming
smaller in size).
D: The function is increasing at a decreasing rate (that is, the changes are positive but shrinking),
E: The function is increasing at a constant rate.
F: The function is increasing at an increasing rate.
Ask by Cook Barrett. in the United States
Jan 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
A function is decreasing at a decreasing rate if it’s going down but doing so more sharply. It’s decreasing at a constant rate if it’s going down uniformly. If it’s decreasing but the rate is slowing down, it’s decreasing at an increasing rate. If it’s increasing but the rate is slowing down, it’s increasing at a decreasing rate. A function is increasing at a constant rate if it’s going up uniformly, and increasing at an increasing rate if it’s going up more rapidly over time.
Solution
It looks like you’ve provided a set of descriptions labeled A through F, each characterizing how a function behaves in terms of increasing or decreasing rates. Understanding these descriptions involves analyzing the function’s first and second derivatives, which inform us about the function’s slope and concavity, respectively.
Here’s a breakdown of each option:
A: The function is decreasing at a decreasing rate (that is, the changes are negative and growing in size)
- First Derivative (
): Negative (since the function is decreasing). - Second Derivative (
): Negative (since the rate of decrease is itself decreasing, meaning is becoming more negative).
Interpretation: The function is not only going down, but it’s doing so more sharply over time.
B: The function is decreasing at a constant rate
- First Derivative (
): Negative and constant. - Second Derivative (
): Zero (since the rate of decrease doesn’t change).
Interpretation: The function decreases uniformly without any acceleration or deceleration.
Example: A linear function like
C: The function is decreasing at an increasing rate (that is, the changes are negative but becoming smaller in size)
- First Derivative (
): Negative but increasing (i.e., becoming less negative). - Second Derivative (
): Positive (since the rate of decrease is slowing down).
Interpretation: The function is still decreasing, but it’s doing so more gradually as time progresses.
D: The function is increasing at a decreasing rate (that is, the changes are positive but shrinking)
- First Derivative (
): Positive but decreasing. - Second Derivative (
): Negative (since the rate of increase is slowing down).
Interpretation: The function is going up, but the speed of its increase is reducing over time.
E: The function is increasing at a constant rate
- First Derivative (
): Positive and constant. - Second Derivative (
): Zero.
Interpretation: The function increases uniformly without any change in the rate of increase.
Example: A linear function like
F: The function is increasing at an increasing rate
- First Derivative (
): Positive and increasing. - Second Derivative (
): Positive (since the rate of increase is accelerating).
Interpretation: The function is not only increasing but doing so more rapidly over time.
Summary Table
| Option | Behavior Description |
|
|
Interpretation |
|---|---|---|---|---|
| A | Decreasing at a decreasing rate | Negative | Negative | Decreasing more sharply over time |
| B | Decreasing at a constant rate | Negative | Zero | Uniformly decreasing |
| C | Decreasing at an increasing rate | Negative | Positive | Decreasing but more gradually |
| D | Increasing at a decreasing rate | Positive | Negative | Increasing but the rate of increase is slowing down |
| E | Increasing at a constant rate | Positive | Zero | Uniformly increasing |
| F | Increasing at an increasing rate | Positive | Positive | Increasing more rapidly over time |
Visual Representation
To better understand these behaviors, consider the following general shapes based on concavity and monotonicity:
- Linear Functions (
): Correspond to Options B and E. - Concave Up (
): Correspond to Options A and F. - Concave Down (
): Correspond to Options C and D.
Real-World Examples
- Option B (Constant Decrease): A vehicle moving at a steady speed decelerates uniformly.
- Option F (Increasing Rate): Compound interest growing at an accelerating rate.
If you have a specific question or need further clarification on how these behaviors relate to particular functions or scenarios, feel free to ask!
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
Did you know that many mathematical functions can be described as either linear or nonlinear? In a linear function, the rate of change remains constant, much like riding a smooth escalator. Meanwhile, nonlinear functions can take you on a more twisty, turny ride, creating fascinating curves and variations in their slopes. This distinction can help you understand everything from economics to physics!
In real-world scenarios, many phenomena can be modeled using these types of functions. For instance, when examining the growth of a company’s revenue, you may find it increasing at an increasing rate if market conditions are favorable, resembling a rocket launch! Conversely, a product nearing the end of its life cycle might show a decreasing rate of revenue, indicating a gradual decline. Understanding these patterns can greatly aid in effective business forecasting and decision-making.
