Problem 38 - Single Variable Calculus

Question

Suppose that the total revenue realjzed by the Odyssey Travel Agency is \(R=f(x)\) thousand dollars if \(x\) thousand dollars are spent on advertising. a. What does $$\frac{f(b)-f(a)}{b-a} \quad 0

Step-by-Step Solution

Answer

a. The expression \(\frac{f(b)-f(a)}{b-a}\) measures the average rate of change of revenue with respect to advertising expense between \(a\) and \(b\) thousand dollars spent. The units are thousands of dollars per thousand dollars. b. \(f^{\prime}(x)\) measures the instantaneous rate of change of revenue with respect to advertising expense at \(x\) thousand dollars spent. The units are thousands of dollars per thousand dollars. c. The approximate change in revenue if Odyssey increases its advertising budget from \(\$20,000\) to \(\$21,000\) is \(\$3,000\).

Step 1: a. Analyzing the Expression and Units

The expression is: \[\frac{f(b)-f(a)}{b-a}, \quad 0

Step 2: b. Interpreting f'(x) and Units

The function f'(x) represents the derivative of the function f(x) with respect to the advertising expense (x). It measures the instantaneous rate of change of revenue with respect to the advertising expense. In other words, it tells us how much the revenue will change for a small change in the advertising budget around x thousand dollars. The units for this derivative will be the same as the units for the expression in part a: thousands of dollars / thousands of dollars.

Step 3: c. Approximate Change in Revenue for Increased Advertising Budget

We are given that f'(20) = 3. This means that when Odyssey spends 20 thousand dollars on advertising, the instantaneous rate of change in revenue with respect to advertising is 3 thousand dollars per thousand dollars spent on advertising. Now we are asked to find the approximate change in the revenue if the advertising budget is increased from 20,000 to 21,000 dollars. Using the given information about the derivative, we can approximate the change in revenue as follows: ΔR ≈ f'(20) × Δx Where ΔR represents the change in revenue and Δx is the change in advertising budget (in thousands of dollars). Since the budget is increased from 20,000 to 21,000 dollars, Δx = (21 - 20) = 1 thousand dollars. Now we can substitute the given value of f'(20) and the calculated value of Δx: ΔR ≈ 3 × 1 ΔR ≈ 3 Thus, the approximate change in revenue if Odyssey increases its advertising budget from 20,000 to 21,000 dollars is 3 thousand dollars.

Key Concepts

Instantaneous Rate of Change

Understanding the instantaneous rate of change is a bit like capturing a snapshot of an object's speed at a precise moment during its journey. Imagine you're looking at a car speeding past you on the highway; the rate at which it's moving right this second is its instantaneous rate of change of position.

In calculus, the instantaneous rate of change of a function is the limit of the average rate of change as the interval gets infinitesimally small. It tells us how quickly something is changing at exactly one point in time. This concept is foundational because it is what we build the idea of the derivative on, which we'll explore in more detail in the next section.

Derivative of a Function

Now, let's dive into the concept of a derivative of a function. The derivative is essentially a mathematical tool that helps us compute the instantaneous rate of change of a function with respect to its variable. In simpler terms, it's the 'speedometer' reading of a function at any given point.

For the function describing Odyssey's revenue, the derivative tells us how the company's revenue is responding to changes in the advertising expenditure at any given level of spending. We denote this derivative as f'(x), and it has profound implications across various fields, like physics for velocity and acceleration, or economics for marginal costs and benefits.

Approximating Change in Revenue

When it comes to business, knowing how a decision will affect profits is crucial. The ability to approximate the change in revenue based on certain variables, such as the amount of money spent on advertising, is invaluable. Using the derivative, we get a powerful estimate of how revenue will shift with a change in investment.

In the context of Odyssey Travel Agency, as they consider upping their advertising from \(20,000 to \)21,000, we use the derivative at \(20,000 (which, in this example, is 3) to approximate the effect on revenue—suggesting an increase of roughly \)3,000, assuming other factors remain constant. This approximation is immensely helpful for making informed business decisions regarding budget allocations.

Economic Interpretation of Derivatives

Within economics, derivatives take on a special meaning. They encapsulate the very essence of decision-making in business environments—weighing incremental changes and their outcomes. The economic interpretation of derivatives is the consideration of how small changes in one economic variable, like advertising expenditure, can impact another, such as revenue or profit.

For Odyssey Travel Agency, the derivative of the revenue function, f'(x), conveys the expected increase in revenue for each additional thousand dollars spent on advertising. It offers a snapshot of the effectiveness of advertising dollars, guiding business leaders in strategizing investments and predicting outcomes. It's this economic acumen—interpreting and applying derivatives—that can lead to more strategic and profitable decision-making.

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Key Concepts

Instantaneous Rate of Change

Understanding the instantaneous rate of change is a bit like capturing a snapshot of an object's speed at a precise moment during its journey. Imagine you're looking at a car speeding past you on the highway; the rate at which it's moving right this second is its instantaneous rate of change of position.

In calculus, the instantaneous rate of change of a function is the limit of the average rate of change as the interval gets infinitesimally small. It tells us how quickly something is changing at exactly one point in time. This concept is foundational because it is what we build the idea of the derivative on, which we'll explore in more detail in the next section.

Derivative of a Function

Now, let's dive into the concept of a derivative of a function. The derivative is essentially a mathematical tool that helps us compute the instantaneous rate of change of a function with respect to its variable. In simpler terms, it's the 'speedometer' reading of a function at any given point.

For the function describing Odyssey's revenue, the derivative tells us how the company's revenue is responding to changes in the advertising expenditure at any given level of spending. We denote this derivative as f'(x), and it has profound implications across various fields, like physics for velocity and acceleration, or economics for marginal costs and benefits.

Approximating Change in Revenue

When it comes to business, knowing how a decision will affect profits is crucial. The ability to approximate the change in revenue based on certain variables, such as the amount of money spent on advertising, is invaluable. Using the derivative, we get a powerful estimate of how revenue will shift with a change in investment.

In the context of Odyssey Travel Agency, as they consider upping their advertising from \(20,000 to \)21,000, we use the derivative at \(20,000 (which, in this example, is 3) to approximate the effect on revenue—suggesting an increase of roughly \)3,000, assuming other factors remain constant. This approximation is immensely helpful for making informed business decisions regarding budget allocations.

Economic Interpretation of Derivatives

Within economics, derivatives take on a special meaning. They encapsulate the very essence of decision-making in business environments—weighing incremental changes and their outcomes. The economic interpretation of derivatives is the consideration of how small changes in one economic variable, like advertising expenditure, can impact another, such as revenue or profit.

For Odyssey Travel Agency, the derivative of the revenue function, f'(x), conveys the expected increase in revenue for each additional thousand dollars spent on advertising. It offers a snapshot of the effectiveness of advertising dollars, guiding business leaders in strategizing investments and predicting outcomes. It's this economic acumen—interpreting and applying derivatives—that can lead to more strategic and profitable decision-making.