Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A typist's daily wages are determined by the number of words per minute he averages on his shift. Let \(D(w)\) be his daily earnings (in dollars) as a function of \(w\), the average number of words per minute he types. Suppose that yesterday he was paid \(\$ B\) for averaging \(C\) words per minute. Interpret each of the following equations or expressions in words. Your answer should be expressed in terms of pay and words per minute. (a) \(D^{-1}(70)=50\) (b) \(D(C+5)=1.1 B\) (c) \(D^{-1}(B+10)\)

Step-by-Step Solution

Answer

Equation (a) interprets that typist types at an average of 50 words per minutes if he earned 70 dollars. Equation (b) shows that the typist's wage increases by 10% if he increases his average typing speed by 5 wpm. Expression (c) tells how much faster the typist needs to type to earn $10 more than his previous day's wage.

Step 1: Interpretation of Equation (a)

(a) \(D^{-1}(70)=50\)\nThis says that, if the typist earned 70 dollars on a particular day, he must have typed at an average of 50 words per minute.

Step 2: Interpretation of Equation (b)

(b) \(D(C+5)=1.1B\)\nThis equation implies that if the typist increases his average wpm by 5 from a previous average of \(C\), his wage would be increased by 10% from the previous earning \(B\).

Step 3: Interpretation of Expression (c)

(c) \(D^{-1}(B+10)\)\nThis expression tells us the average number of words per minute the typist would need to type to earn \((B+10)\) dollars, where \(B\) is his previous day's earnings.

Key Concepts

Inverse Functions

Understanding inverse functions is crucial in various branches of mathematics, including calculus. In the context of our typist's wages example, the function D(w) represents daily earnings based on the average number of words per minute typed. When we talk about the inverse function, D^{-1}(y), it allows us to determine the words per minute given the earnings.

With the equation D^{-1}(70)=50, the inverse function reveals that if the typist earned $70, they averaged 50 words per minute. It's like reversing the process – instead of figuring out the pay from performance, we discover performance from pay. Inverse functions are critical in decoding relationships where we have the output but need to identify the corresponding input.

Rate of Change

The concept of the rate of change is pivotal in calculus and represents how one quantity changes in relation to another. For our typist, the equation D(C+5)=1.1B highlights a practical application of this concept. It indicates that when the average typing speed (words per minute) increases by 5 units, the earnings rise by 10%.

The rate of change here is not constant; it is proportional to the typist's performance. If we interpret this algebraically, we infer that for every additional word per minute, there is a consistent increase in the typist's earnings, reflecting a direct relationship between productivity and pay – a fundamental principle in employment metrics.

Real-world Applications of Functions

Functions are not abstract mathematical notions; they have tangible applications in real life. In the typist's context, the function D(w) assigns a dollar value to every average speed of typing measured in words per minute. This real-world application helps in pay scale structuring and can be applied to various performance-based earning models.

When we look at D^{-1}(B+10), it relates to finding out what increase in work rate is required to achieve a specific financial goal – something common in sales commissions, productivity bonuses, or other incentive-based roles. Such applications show the usefulness of functions in planning, analysis, and goal setting in a business context.

Algebraic Interpretation

The algebraic interpretation of functions and their equations allow us to make sense of mathematical relationships. Equations like D(C+5)=1.1B and expressions like D^{-1}(B+10) give us structured ways to see the relationship between different variables. In our example, algebra helps us understand how a typist's pay structure works and what factors influence earnings.

Through algebra, we can manipulate expressions to isolate variables and solve for unknowns, providing a way to quantify and optimize work performance and compensation. It shows us that mathematics is not just theoretical but a practical tool that can be used to navigate and solve real-life problems.