Problem 68 - Calculus of a Single Variable

Question

Modeling Data The average typing speed \(S\) (in words per minute) of a typing student after \(t\) weeks of lessons is shown in the table. A model for the data is $$S=\frac{100 t^{2}}{65+t^{2}}, \quad t>0$$ (a) Use a graphing utility to plot the data and graph the model. (b) Use the second derivative to determine the concavity of \(S .\) Compare the result with the graph in part (a). (c) What is the sign of the first derivative for \(t>0 ?\) By combining this information with the concavity of the model, what inferences can be made about the typing speed as \(t\) increases?

Step-by-Step Solution

Answer

Using the derivatives and results from the graph, we can infer the typing speed's behavior over time. The specifics would depend on the outcome of calculations and the graph produced. The essence is understanding the relationship between the function's behavior and the student's typing speed: when the function increases, so does the speed, and the speed increases at a faster rate if the function is concave up and slower if it's concave down.

Step 1: Function Plotting

Utilize a graphing calculator or any graphing software to plot the given function \(S=\frac{100 t^{2}}{65+t^{2}}, \quad t>0\). Make sure to observe the curves and overall behavior of the function over time.

Step 2: Second Derivative Calculation

We need to compute the second derivative to determine the concavity of the function. Start by finding the first derivative using the quotient rule: \(S' = \frac{d}{dt} \left(\frac{100 t^{2}}{65+t^{2}}\right)\). Then, find the second derivative: \(S'' = \frac{d^{2}}{dt^{2}} S\). Analyze the sign of the second derivative which will tell you whether the function is concave up (if \(S'' > 0\)) or concave down (if \(S'' < 0\)).

Step 3: First Derivative Sign Analysis

Now, study the sign of the first derivative \(S'\). It'll help to understand if the function is increasing (if \(S' > 0\)) or decreasing (if \(S' < 0\)) over time. This information, combined with concavity, will allow you to infer about the typing speed as time increases.

Key Concepts

Second Derivative

In calculus, the second derivative of a function provides insight into the function's concavity. Concavity tells us how the function's slope behaves. The second derivative, denoted as \( S'' \), is essentially the derivative of the first derivative \( S' \).To find this, we first need to take the first derivative of the function using the rules of differentiation. Once we have \( S' \), we differentiate it once more to find \( S'' \). The sign of \( S'' \) plays a crucial role:

• If \( S'' > 0 \), the function is concave up, indicating that the graph of the function is curving upwards. This often suggests that the rate of change of the function is increasing.
• If \( S'' < 0 \), the function is concave down, suggesting that it is curving downwards, and the rate of change is decreasing.

Understanding concavity helps in visualizing the overall shape of the function's graph and is particularly useful in analyzing the behavior of functions over time, like the typing speed in this exercise.

First Derivative

The first derivative of a function is fundamental in understanding how the function behaves over its domain. It represents the rate of change or slope of the function at any given point.For the typing model \( S = \frac{100 t^2}{65 + t^2} \), the first derivative \( S' \) can be found using the quotient rule. This rule is key when dealing with ratios of functions. Once calculated, the sign of \( S' \) tells us whether:

• The function is increasing (\( S' > 0 \)) which implies that the typing speed is getting faster as weeks of lessons go on.
• The function is decreasing (\( S' < 0 \)) indicating a slowdown, although in this context, \( S' \) will likely be positive as typing speed is expected to improve over time.

Combining the knowledge of \( S' \) with the second derivative helps in drawing more detailed conclusions about the overall behavior of the typing speed as \( t \) increases.

Graphing Utility

A graphing utility is a tool, often available as a calculator or software, that plots the graph of a given function. Visualizing the function helps in understanding its shape, trends, and key characteristics such as maxima, minima, and points of inflection.To use a graphing utility effectively for the equation \( S = \frac{100 t^2}{65 + t^2} \) for \( t > 0 \):

• Input the function into the graphing tool. Ensure the range for \( t \) is set appropriately, considering the context of the problem (like weeks in this case).
• Observe the plotted graph. This will provide a visual confirmation of the function's behavior, aiding in the interpretation of calculated derivatives.
• Look for aspects like the steepness and direction of curves, which relates to the first derivative, as well as overall curvatures, which connect to the second derivative.

Using graphing utilities is invaluable for understanding complex functions beyond theoretical calculations, providing practical insights.

Concavity

Concavity refers to the direction in which a function curves. It is an important concept in calculus for understanding the nature of the rate at which a function's value changes.The sign of the second derivative informs us about a function's concavity:

• Concave up (\( S'' > 0 \)): The graph of the function is shaped like a bowl upwards. While this often indicates acceleration, it means the typing speed increases at an increasing rate in our case.
• Concave down (\( S'' < 0 \)): The function resembles a downward bowl-shaped graph. Here, it would mean that although typing speed increases, the rate of increase is slowing.

In the context of the typing speed exercise, understanding concavity helps infer how persistently or quickly typing speed penalties diminish or halt.
Combining this information with the first derivative gives a comprehensive view of changes in behavior over time, enhancing understanding of the situation.