Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Translate each of the following statements into mathematical language, sketch a plot of the appropriate function, and indicate the concavity. (a) The cost of a car continues to increase and at a faster and faster rate. (b) During the last 2 years, the United States has continued to cut its consumption of oil, but at a slower and slower rate. (c) World population continues to grow, but at a slower and slower rate. (d) The angle that the Leaning Tower of Pisa makes with the vertical is increasing more and more rapidly. (e) Upper Midwest firm's profit growth slows. (f) The XYZ Company has been losing money, but will soon turn this situation around.

Step-by-Step Solution

Answer

(a) \( C'(t)>0, C''(t)>0 \); (b) \( O'(t)<0, O''(t)>0 \); (c) \( P'(t)>0, P''(t)<0 \); (d) \( A'(t)>0, A''(t)>0 \); (e) \( G'(t)>0, G''(t)<0 \); (f) \( L'(t)>0, L''(t)>0 \).

Step 1: Translating the cost of a car

To express the statement about the cost of a car in mathematical language, we consider a function \( C(t) \) representing the cost of a car over time \( t \). The statement implies that \( C(t) \) is increasing and the rate of increase \( C'(t) \) is itself increasing. This means \( C'(t) > 0 \) and \( C''(t) > 0 \), signifying a positive first and second derivative, indicating an increasing cost and an acceleration in the increase.

Step 2: Plotting the cost of a car

For this statement, the graph of \( C(t) \) versus \( t \) would be an upward-curving shape, similar to an exponential or quadratic rise. The curve should reflect both an increase in cost and acceleration, showing positive concavity.

Step 3: Translating US oil consumption

The statement about US oil consumption can be modeled as \( O(t) \), where the consumption rate decreases over time. \( O(t) \) decreases with time \( t \) with a decreasing rate of reduction, implying \( O'(t) < 0 \) and \( O''(t) > 0 \), which indicates the rate of decline of oil consumption is slowing down.

Step 4: Plotting US oil consumption

The plot of \( O(t) \) vs. \( t \) would reflect a downward trend that starts steep, then becomes less steep over time, maintaining a positive concavity as the slow down in rate intensifies.

Step 5: Translating world population growth

Represent world population as \( P(t) \), growing with time but at a slowing rate, implying \( P(t) \) increases but \( P'(t) > 0 \) with \( P''(t) < 0 \). The positive first derivative indicates growth, but the negative second derivative shows the rate of growth is declining.

Step 6: Plotting world population growth

In the plot of \( P(t) \) versus \( t \), the curve rises but begins to flatten, reflecting a deceleration in growth, with negative concavity due to the decreasing rate of growth.

Step 7: Translating the Leaning Tower of Pisa

The angle of the tower \( A(t) \) increases over time more rapidly, implying \( A'(t) > 0 \) and \( A''(t) > 0 \). Both positive derivatives indicate that the angle is increasing and the rate of this increase is accelerating.

Step 8: Plotting the Leaning Tower of Pisa

The plot for the angle \( A(t) \) over time would show a rising curve that steepens, indicating rapid acceleration in increase, with a positive concavity.

Step 9: Translating firm's profit growth

Let \( G(t) \) represent profit growth, which slows over time, leading to \( G'(t) > 0 \) and \( G''(t) < 0 \). This signifies that while growth is positive, its acceleration is downward.

Step 10: Plotting the firm's profit growth

On the graph, \( G(t) \) vs. \( t \) would indicate a rise that begins to slow, showing initial steepness that flattens over time, with negative concavity from decelerating growth.

Step 11: Translating XYZ Company's loss

Assume the loss \( L(t) \) is decreasing, approaching a turnaround, indicating \( L(t) < 0 \), \( L'(t) > 0 \), and \( L''(t) > 0 \). The positive second derivative suggests acceleration towards neutral or positive.

Step 12: Plotting XYZ Company's loss

For \( L(t) \) vs. \( t \), the curve dips below zero, then trends upward as it approaches zero, illustrating positive concavity with accelerating improvement.

Key Concepts

First Derivative

The concept of the first derivative is central to understanding how a function behaves. The first derivative of a function, usually denoted as \( f'(x) \), provides the rate of change of the function concerning its input variable, \( x \). This means it tells us how the function value is changing at any given point.

• If \( f'(x) > 0 \), the function is increasing, indicating that as \( x \) increases, \( f(x) \) also increases.
• If \( f'(x) < 0 \), the function is decreasing, which means as \( x \) increases, \( f(x) \) decreases.
• If \( f'(x) = 0 \), the function could be at a local maximum, local minimum, or a point of inflection, depending on the behavior of the second derivative.

Understanding the first derivative is crucial for tasks like determining when costs for a car are rising (Step 1) or identifying when the rate of oil consumption is slowing down (Step 3). The sign and value of the first derivative give us insights into the overall trend of a function's graph.

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), provides information about the function's concavity and the rate of change of the first derivative. It's like the "acceleration" of the function. Understanding the second derivative helps to analyze whether the rate of change is speeding up or slowing down.

• If \( f''(x) > 0 \), the function is concave up, often described as having a 'smile' shape. This suggests the rate of growth is accelerating, as in the increasing cost of a car (Step 1) or the angle the Leaning Tower of Pisa is making (Step 7).
• If \( f''(x) < 0 \), the function is concave down, resembling a 'frown'. This indicates that the rate of increase is slowing, relating to the deceleration in world population growth (Step 5) or profit growth slowing down (Step 9).
• If \( f''(x) = 0 \), it suggests a possible change in concavity, known as an inflection point.

Understanding the second derivative allows us to predict not just whether a trend is going up or down, but also whether it's speeding up or slowing down.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios with mathematical language and structures. It simplifies complex systems, allowing us to predict and analyze behaviors using functions and their derivatives.

In this exercise, we translated everyday statements into mathematical expressions to understand and explore them through modeling. For instance:

• The increasing cost of a car is modeled as a function \( C(t) \) with both derivatives positive, expressing continuous growth and acceleration.
• US oil consumption, represented as \( O(t) \), used a negative first derivative to show a decrease in consumption, but a positive second derivative indicating a slowdown in the rate of reduction.

Mathematical modeling enables us to visualize trends and dynamics by using functions that replicate real-world behavior and then analyzing them through derivatives for deeper insights.

Function Plotting

Function plotting involves visually representing a function's behavior across a range of values. It is a powerful tool for understanding the effects of derivatives and the overall trends in a function's performance.

By plotting a function, we can instantly see key aspects such as:

• Increases and decreases in the function's value over time, highlighted by the slope of the plot.
• Concavity, which reveals acceleration or deceleration and is indicated by the way the curve bends.

For instance, plotting the function of car cost \( C(t) \) helps us visualize its ever-increasing nature. Similarly, the plot of \( P(t) \) for world population growth displays an upward curve that begins to flatten, indicating slowing growth as per its negative second derivative.

Plotting provides a visual context that complements mathematical calculations, making abstract concepts tangible and easier to grasp for learners.