Problem 42 - Applied Calculus

Question

A company's revenue from car sales, \(C\) (in thousands of dollars), is a function of advertising expenditure, \(a,\) in thousands of dollars, so \(C=f(a)\) (a) What does the company hope is true about the sign of \(f^{\prime} ?\) (b) What does the statement \(f^{\prime}(100)=2\) mean in practical terms? How about \(f^{\prime}(100)=0.5 ?\) (c) Suppose the company plans to spend about \(\$ 100,000\) on advertising. If \(f^{\prime}(100)=2,\) should the company spend more or less than \(\$ 100,000\) on advertising? What if \(f^{\prime}(100)=0.5 ?\)

Step-by-Step Solution

Answer

The company wants \( f'(a) > 0 \); \( f'(100)=2 \) means more gains, so spend more; \( f'(100)=0.5 \) means less gain, so spending more may not be wise.

Step 1: Understanding the Sign of Derivative

The company hopes that the sign of the derivative \( f'(a) \) is positive. This is because a positive derivative indicates that as advertising expenditure \( a \) increases, the company's revenue \( C \) also increases. In other words, more advertising should ideally result in higher car sales revenue.

Step 2: Interpreting the Derivative Value

When \( f'(100) = 2 \), it means that if the company increases its advertising expenditure by 1 unit (\\(1000), the revenue will increase by 2 units (\\)2000). Similarly, \( f'(100) = 0.5 \) means an increase in advertising by 1 unit (\\(1000) results in a revenue increase of only 0.5 units (\\)500). This describes the sensitivity of revenue to advertising at \( \$100,000 \) spent.

Step 3: Deciding on Advertising Expenditure

If \( f'(100) = 2 \), increasing advertising will likely lead to significantly higher revenue, so the company should consider spending more than \\(100,000 on advertising. On the other hand, if \( f'(100) = 0.5 \), the return on additional advertising is low, and it may not be worth spending more than \\)100,000.

Key Concepts

Revenue Function

In applied calculus, the concept of a revenue function is crucial in understanding how changes in one variable, like advertising, can influence another, such as revenue from car sales. Here, the revenue function is expressed as \( C = f(a) \), where \( C \) represents the revenue in thousands of dollars, and \( a \) stands for advertising expenditure also in thousands of dollars. This function helps businesses model and predict how different levels of advertising can affect their revenue.

The goal for a company is generally to optimize this revenue function. They aim to find the level of advertising spending that brings in the maximum revenue. Enterprises use historical data or market analysis to determine potential trends and project future revenues based on various advertising budgets.

Understanding the behavior of this function allows companies to make informed decisions and strategically allocate resources. This is not just about increasing spending recklessly but finding the sweet spot where expenditures lead to optimal gains.

Derivative Interpretation

The derivative of the revenue function, represented as \( f'(a) \), plays a pivotal role in understanding the relationship between advertising and revenue. This derivative gives the rate of change of the revenue with respect to advertising spending. In simple terms, it tells us how sensitive the revenue is to changes in advertising.

• A positive derivative, \( f'(a) > 0 \), indicates that increasing advertising spending will result in increased revenue.
• A larger positive value suggests a stronger relationship, implying that a small increase in advertising results in a significant increase in revenue.
• If the derivative is zero, \( f'(a) = 0 \), changes in advertising spending do not affect revenue.

When you see a statement like \( f'(100) = 2 \), it means that at \( \\(100,000 \) spent on advertising, every additional \( \\)1,000 \) leads to a \( \\(2,000 \) increase in revenue. Meanwhile, \( f'(100) = 0.5 \) signifies a weaker relationship, where the same increase in advertising only raises revenue by \( \\)500 \). This interpretation is critical for businesses to decide whether an increase in advertising budget is justified.

Advertising Expenditure Impact

When deciding on advertising expenditures, a company must consider the impact these costs will have on their overall revenue. Using derivative interpretation, companies understand how impactful additional spending could be at a point like \( \\(100,000 \) in advertising.

If the derivative \( f'(100) = 2 \), it suggests a strong return on investment from additional advertising. Companies are likely to benefit from spending beyond \( \\)100,000 \), as each extra \( \\(1,000 \) can yield \( \\)2,000 \) in new revenue. Hence, it is often wise to allocate more resources to advertising under these conditions.

Conversely, if \( f'(100) = 0.5 \), additional investments in advertising might not be lucrative. Here, spending another \( \\(1,000 \) would only increase revenue by \( \\)500 \). In this scenario, the company may choose to maintain or even reduce their advertising budget, focusing instead on other strategies to increase revenue. This kind of strategic thinking helps manage resources efficiently, ensuring that each dollar spent contributes positively to the company's financial goals.