Problem 93 - Introductory and Intermediate Algebra for College Students 4th

Question

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score, to the nearest tenth, after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

Step-by-Step Solution

Answer

a. The average score on the original exam was 88. b. The average scores after 2, 4, 6, 8, 10 months and one year can be found by substituting the values into the function. c. The graph which is a gradually decreasing curve, shows that the average score decreases as time passes, demonstrating that the students are forgetting the material over time.

Step 1: Calculate the original exam score

To find the initial or original examination score, we should set \(t=0\) in the given function because \(t\) represents the months after the exam and at the time of the exam, \(t=0\). Following this, the calculation becomes: \(f(0) = 88 - 15 \ln(0+1) = 88\).

Step 2: Calculate the scores after two, four, six, eight, and ten months and one year

To find the average scores after two, four, six, eight, ten months, and one year, we substitute the corresponding values of \(t\) into \(f(t)\). Therefore, to find the score after two months we substitute \(t=2\) to get \(f(2) = 88 - 15 \ln(2+1)\), and follow a similar process for the other times.

Step 3: Sketch the graph and interpret it

Plot the function \(f(t) = 88 - 15 \ln(t+1)\) for the interval \(0 \leq t \leq 12\). The graph starts at (0,88) and decreases gradually as \(t\) increases. The decrease in value of \(f(t)\) as \(t\) increases shows that the average score decreases over time, reflecting that the students tend to forget some of the material as time passes after the exam.

Key Concepts

Exam Score Retention

Understanding exam score retention through mathematical modeling is crucial in evaluating how information is recalled over time. The given function,
\(f(t) = 88 - 15 \ln(t+1)\),
demonstrates the phenomenon where, as time progresses (\(t\) months after the exam), the knowledge students retain from their course decreases. In practical terms, this reveals that memory and retention are not static but diminish as time moves forward. This is a key concept in educational psychology, emphasizing the importance of ongoing study and review to maintain knowledge levels.

Graphing Logarithmic Functions

Graphing logarithmic functions like \(f(t) = 88 - 15 \ln(t+1)\) is a powerful visual aid in understanding how data changes over time.
To graph this function, plot a point for each calculated score at the corresponding time interval, connect these points smoothly, and observe that the graph decreases at a decelerating rate.
This is characteristic of logarithmic relationships, where initial changes are rapid, but the rate of change slows down. It provides a clear image to students that as time passes, the average scores tend to decrease in a specific pattern which reflects logarithmic decay.

Interpreting Function Models

The ability to interpret function models is essential in many fields, including education. By analyzing the given function,
\(f(t) = 88 - 15 \ln(t+1)\),
we can deduce quantitative information about the trends and patterns in exam score retention. For instance, by observing the slope and curvature of the graphed function, we recognise a decaying trend. This tells us that the average score decreases over time, and it quantifies this decrease, allowing educators and students to apprehend the natural decline of memory over time unless reinforced.

Logarithmic Decay

The concept of logarithmic decay is exemplified by the function given in our exercise.
In the context of exam score retention,
\(f(t) = 88 - 15 \ln(t+1)\)
is a model that showcases how academic information is forgotten. Mathematically, the negative coefficient in front of the logarithm (-15) suggests that there is a decrease in the average score. This decay is not linear; it is much faster immediately after learning (closer to the exam date) and slows down as time progresses, which mirrors how humans tend to preserve core knowledge over time while details get blurry.