Problem 15
Question
The following graphs were used by the CEO of the Madison Savings Bank to illustrate what effect a projected promotional campaign would have on its deposits over the next year. The functions \(D_{1}\) and \(D_{2}\) give the projected amount of money on deposit with the bank over the next 12 mo with and without the proposed promotional campaign, respectively. a. Determine the signs of \(D_{1}^{\prime}(t), D_{2}^{\prime}(t), D_{1}^{\prime \prime}(t)\), and \(D_{2}^{\prime \prime}(t)\) on the interval \((0,12)\). b. What can you conclude about the rate of change of the growth rate of the money on deposit with the bank with and without the proposed promotional campaign?
Step-by-Step Solution
Verified Answer
Based on the given information, we can conclude that both deposits with and without the promotional campaign are increasing over time, as \(D_1'(t) > 0\) and \(D_2'(t) > 0\). However, we cannot make a definitive conclusion about the rate of change of the growth rate of the money on deposit with the bank with and without the proposed promotional campaign, since we cannot determine the signs of \(D_1''(t)\) and \(D_2''(t)\).
1Step 1: Observations and Definitions
Since \(D_1'(t)\) and \(D_1''(t)\) are derivatives of \(D_1(t)\), they represent the rate at which the money deposited in the bank with promotion changes over time and the rate of change of that rate. Similarly, \(D_2'(t)\) and \(D_2''(t)\) represent the same thing, but for the money deposited without the promotion.
2Step 2: Determine the signs of the first derivatives
The sign of \(D_1'(t)\) and \(D_2'(t)\) on the interval (0, 12) can be determined by analyzing the slope of the functions \(D_1(t)\) and \(D_2(t)\). If the slope of \(D_1(t)\) is positive, then \(D_1'(t) > 0\), meaning deposits in the bank with the promotional campaign are increasing over time. If the slope of \(D_2(t)\) is positive, then \(D_2'(t) > 0\), meaning deposits in the bank without the promotional campaign are increasing over time.
Based on the information given by the CEO, there is reason to believe that both slopes are positive, which implies that \(D_1'(t) > 0\) and \(D_2'(t) > 0\).
3Step 3: Determine the signs of the second derivatives
The sign of \(D_1''(t)\) and \(D_2''(t)\) on the interval (0, 12) can be determined by analyzing the concavity of the functions \(D_1(t)\) and \(D_2(t)\). If \(D_1'(t)\) is increasing over the given interval, then \(D_1''(t) > 0\), meaning the rate at which the money deposited in the bank with promotion changes is also increasing. If \(D_2'(t)\) is increasing over the given interval, then \(D_2''(t) > 0\), meaning the rate at which the money deposited in the bank without promotion changes is also increasing.
The given information does not provide explicit information about the concavity of the functions, so we cannot determine the signs of \(D_1''(t)\) and \(D_2''(t)\).
4Step 4: Conclusion
Since we know the signs of \(D_1'(t)\) and \(D_2'(t)\), we can conclude that both the deposits with and without the promotional campaign are increasing over time. However, since we cannot determine the signs of \(D_1''(t)\) and \(D_2''(t)\), we cannot make a definitive conclusion about the rate of change of the growth rate of the money on deposit with the bank with and without the proposed promotional campaign.
Key Concepts
DerivativesRate of ChangeGraph Analysis
Derivatives
Derivatives are a cornerstone of calculus. Simply put, a derivative represents the rate at which a function is changing at any given point. In the context of business, derivatives can be extremely valuable. They help us understand how certain financial metrics, such as deposits, change over time.
To illustrate, if we have a function that describes the amount of money on deposit in a bank over time, the first derivative of that function, often represented as \( D'(t) \), tells us the rate at which the deposits are increasing or decreasing at any given moment. A positive derivative indicates that the deposits are growing.
In more practical terms, think of derivatives like checking your car's speedometer. Just as the speedometer shows how fast your car's speed is changing, the derivative of a graph tells us how fast the graph's value is changing. Thus, in business, knowing derivatives helps decision-makers predict future trends and make informed decisions based on those predictions.
To illustrate, if we have a function that describes the amount of money on deposit in a bank over time, the first derivative of that function, often represented as \( D'(t) \), tells us the rate at which the deposits are increasing or decreasing at any given moment. A positive derivative indicates that the deposits are growing.
In more practical terms, think of derivatives like checking your car's speedometer. Just as the speedometer shows how fast your car's speed is changing, the derivative of a graph tells us how fast the graph's value is changing. Thus, in business, knowing derivatives helps decision-makers predict future trends and make informed decisions based on those predictions.
Rate of Change
Understanding the rate of change is crucial in grasping the dynamics of financial growth or decline. The rate of change shows how one quantity alters relative to another over time, offering insights into trends and shifts.
For instance, in the problem involving the Madison Savings Bank, the first derivative, \( D'(t) \), indicates the rate of change of the deposits. A positive rate of change signifies that deposits are increasing. If we observe that \( D'(t) \) remains positive over time, it reassures us that the deposits are likely to continue to grow.
The second derivative, \( D''(t) \), adds another dimension by indicating the rate of change of the rate of change. This tells us whether the speed of growth is accelerating or decelerating. In business, knowing that the second derivative is positive means that not only are deposits growing, but the speed at which they are growing is itself increasing, which could lead to exponential growth.
For instance, in the problem involving the Madison Savings Bank, the first derivative, \( D'(t) \), indicates the rate of change of the deposits. A positive rate of change signifies that deposits are increasing. If we observe that \( D'(t) \) remains positive over time, it reassures us that the deposits are likely to continue to grow.
The second derivative, \( D''(t) \), adds another dimension by indicating the rate of change of the rate of change. This tells us whether the speed of growth is accelerating or decelerating. In business, knowing that the second derivative is positive means that not only are deposits growing, but the speed at which they are growing is itself increasing, which could lead to exponential growth.
Graph Analysis
Analyzing graphs is a key aspect of understanding functions and their derivatives. Graphs provide a visual representation, which makes it easier to comprehend complex data and see relationships between variables.
In the context of the Madison Savings Bank problem, the graph of \( D_{1}(t) \) and \( D_{2}(t) \) would depict the deposits over time with and without a promotional campaign. By examining the slope of these graphs, one can determine whether the deposits are increasing or decreasing. A steep positive slope means a rapid increase, while a negative slope suggests a decline.
Moreover, looking at the curvature or shape of the graph can inform us about the second derivative. If the graph curves upwards, it suggests that the rate at which the deposits increase is itself growing, indicating a positive second derivative. Thus, graph analysis not only helps in understanding current trends but also in predicting future outcomes based on those trends.
In the context of the Madison Savings Bank problem, the graph of \( D_{1}(t) \) and \( D_{2}(t) \) would depict the deposits over time with and without a promotional campaign. By examining the slope of these graphs, one can determine whether the deposits are increasing or decreasing. A steep positive slope means a rapid increase, while a negative slope suggests a decline.
Moreover, looking at the curvature or shape of the graph can inform us about the second derivative. If the graph curves upwards, it suggests that the rate at which the deposits increase is itself growing, indicating a positive second derivative. Thus, graph analysis not only helps in understanding current trends but also in predicting future outcomes based on those trends.
Other exercises in this chapter
Problem 15
Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{2}-2 x-3 \text { on }[-2,3] $$
View solution Problem 15
Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{x-1}{x+1} $$
View solution Problem 15
Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{2}-3 x $$
View solution Problem 16
Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=x^{2}-2 x-3 \text { on }[0,4] $$
View solution