Question

\( \text { Rate of Change } \) Each of the following is the slope of a line representing daily revenue \(y\) in terms of time \(x\) in days. Use the slope to interpret any change in daily revenue for a one-day increase in time. (a) \(m=400\) (b) \(m=100\) (c) \(m=0\)

Short answer

(a) For m=400, a one day increase in time leads to a daily revenue increase of $400. (b) For m=100, a one day increase in time results in a daily revenue increase of $100. (c) For m=0, the daily revenue stays the same, regardless of time passed.

Step-by-step solution

Interpretation of Slope 'm=400'

If m=400, this means that for each day that passes, the daily revenue increases by $400. Thus, for every one day increase in time, the daily revenue goes up by $400.

Interpretation of Slope 'm=100'

If m=100, it suggests that for each additional day, the daily revenue increases by $100. Hence, for a one day increase in time, the daily revenue increases by $100.

Interpretation of Slope 'm=0'

If m=0, this implies that the daily revenue remains the same regardless of how much time passes. Therefore, a one day increase in time does not produce any change in the daily revenue.

Key concepts

Understanding the Slope of a Line

The concept of the 'slope of a line' is a fundamental idea in calculus and algebra that explains how one variable changes in relation to another. Imagine walking up a hill; the steeper the hill, the harder you have to work to go up it. In math, the 'slope' is a numeric measure of this steepness, often represented as 'm' in equations such as y = mx + b. The slope tells us the rate at which the dependent variable (y) changes for a unit change in the independent variable (x).

When analyzing the slope, we look at it in the form of a ratio. For a line, it's the 'rise over run', or the change in y divided by the change in x. In a practical scenario, like our textbook exercise, the slope represents the change in daily revenue (rise) over a one-day period (run). A positive slope, like 400 or 100, signifies an increase in revenue per day, while a slope of 0 indicates no change.

Common Mistakes to Avoid

• Don't confuse a steeper slope with just a higher value of 'y'. It's about how quickly 'y' is changing relative to 'x'.
• Remember that a negative slope signals a decrease, whereas the exercise deals with positive slopes or no change (slope of 0).

Daily Revenue Interpretation

Interpreting daily revenue changes in terms of calculus means understanding the practical implications of the slope in a real-world context. From the exercise, when the slope is given as 'm=400', it indicates a robust growth: for every additional day in time, the company's revenue is increasing by \(400. Conversely, a gentler slope of 'm=100' still shows growth, though at a more moderate pace of \)100 per day.

An essential factor in analyzing daily revenue is recognizing how this information can help a business make financial forecasts and decisions. If the slope is zero (m=0), it tells us that the time does not affect the daily revenue; it's stable.

Why These Insights Matter

• Understanding daily revenue changes assists in budgeting and planning for growth or addressing challenges.
• The slope not only measures revenue changes but can also signal when a business model needs reevaluating.

Calculus Real-world Applications

Calculus, specifically the concept of 'rate of change', is not just an abstract mathematical idea; it has a multitude of applications in various fields. For example, in economics, the slope of a line, or the rate of change, helps understand market trends. In physics, it is essential in explaining velocities and acceleration. Similarly, in biology, calculus can be used to model population growth or the spread of diseases.

The interpretation of daily revenue change using calculus demonstrates how businesses can apply mathematical concepts to track performance and anticipate future trends, ensuring better strategic planning.

From Theory to Practice

• Engineering fields use calculus to design and optimize systems and structures.
• In finance, calculus helps to evaluate risk and determine pricing structures.
• Medicine uses differential equations, a part of calculus, for modeling the behavior of human organs or understanding the dynamics of diseases.