Problem 54 - Precalculus with Limits

Question

TYPING SPEED The average typing speed \(S\) (in words per minute) for a student after \(t\) weeks of lessons is given by $$ S = \dfrac{100t^2}{65 + t^2}, \quad t > 0\. $$ (a) What is the limit of \(S\) as \(t\) approaches infinity? (b) Use a graphing utility to graph the function and verify the result of part (a). (c) Explain the meaning of the limit in the context of the problem.

Step-by-Step Solution

Answer

The typing speed limit as time goes to infinity is 100 words per minute.

Step 1: Identify the function

Firstly, identify the function in the problem. Here, the function is \( S = \dfrac{100t^2}{65 + t^2} \)

Step 2: Find the limit

To find the limit of \(S\) as \(t\) approaches infinity, substitute \(t\) with infinity in the function. The limit of a fraction with \(t\) approaching infinity is determined by the highest power of \(t\) in the numerator and denominator. As both powers are the same, the limit of the function will equal the ratio of the coefficient of the highest power terms in the numerator and denominator. In this case, the coefficients are 100 and 1, so the limit is \(100/1=100\).

Step 3: Graph the function

Although this step can't be illustrated here, use a graphing calculator or online graphing utility to graph the function. The resulting plot will show that as \(t\) increases, the value \(S\) approaches 100, verifying the previous step.

Step 4: Interpret the limit

The limit of function \(S\) as \(t\) approaches infinity represents the maximum average typing speed a student can reach with endless weeks of lessons. In this context, it means that no matter how many weeks of lessons a student takes, their average typing speed will not exceed 100 words per minute.

Key Concepts

Rational Functions

Rational functions are expressions that are ratios of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are both polynomials. They are a fundamental type of mathematical function and appear in various real-world scenarios. For instance:

• The function given in the exercise, \( S = \dfrac{100t^2}{65 + t^2} \), is a rational function.
• The numerator is \( 100t^2 \) and the denominator is \( 65 + t^2 \).

One characteristic of rational functions is how they behave as their input becomes very large or very small. Since these functions can often simplify to constant values at extremes, they are particularly useful for modeling situations where outputs level off. In this exercise, the function describes typing speed leveling off with more practice time. Understanding the structure of rational functions helps in finding limits, analyzing behavior, and interpreting outcomes.

Asymptotic Behavior

The concept of asymptotic behavior is about understanding how functions behave as their inputs reach extreme values, such as approaching infinity. Asymptotes are lines that the graph of a function approaches but never actually touches.For the function \( S = \dfrac{100t^2}{65 + t^2} \):

• We look at \(t\) approaching infinity (i.e., as time goes on endlessly).
• In such cases, we use the highest power terms to determine the asymptotic behavior.
• As both numerator and denominator highest power term is \(t^2\), the ratio of their coefficients (100 and 1) gives us the function's horizontal asymptote.

So, the graph of \(S\) approaches the line \(y = 100\) as \(t\) gets very large. This tells us that with endless practice, a student's typing speed aims towards 100 words per minute. Recognizing this behavior allows us to assess long-term trends in real-world scenarios.

Graphical Analysis

Graphical analysis involves using a graph to visually represent and interpret functions like the one given in the exercise. It helps us verify findings like the limit calculated previously and better understand the behavior of functions.To analyze \( S = \dfrac{100t^2}{65 + t^2} \):

• Graphing the function, usually via a graphing calculator or software, shows a curve leveling towards a horizontal line.
• This line is at \(y = 100\), aligning with our asymptotic analysis.
• Visual observation confirms that after a certain number of weeks, the typing speed stabilizes and does not grow past 100 words per minute.

Graphical analysis is an essential practice because it provides a visual context that supports mathematical calculations. It offers an intuitive approach to understanding complex relationships and the concept of limits, as utilized in various domain analyses.