Problem 119 - College Algebra Essentials

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 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. Average score on original exam was 88. b. Average scores after 2,4,6,8,10 months and 1 year are approximately 70.79, 61.68, 56.59, 53.15, 50.31 and 48.05 respectively. The graph of function \(f(t)\) shows a decreasing trend implying that the students' recall ability of course content decreases over time due to the natural 'forgetting' curve.

Step 1: Calculating Average Scores

a. The average score on the original exam is \(f(0) = 88 − 15 \ln(0 + 1) = 88 − 15 \ln(1) = 88− 0 = 88\). b. By substituting the appropriate values of \(t\) in the function \(f(t)\), the average scores at different intervals can be calculated: After 2 months: \(f(2) = 88 - 15 \ln(2+1) = approx 70.794\), After 4 months: \(f(4) = 88 - 15 \ln(4+1) = approx 61.685\), After 6 months: \(f(6) = 88 - 15 \ln(6+1) = approx 56.594\), After 8 months: \(f(8) = 88 - 15 \ln(8+1) = approx 53.15\), After 10 months: \(f(10) = 88 - 15 \ln(10+1) = approx 50.312\), After 1 year : \(f(12) = 88 - 15 \ln(12+1) = approx 48.059\)

Step 2: Drawing the Graph and Infering Material Retention

c. The graph of the function should show a gradual decrease, as it indicates the average scores are decreasing over time. The graph indicates that the students' ability to recall course content decreases over time, which is natural, due to the 'Forgetting Curve'.

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical relationship that uses a logarithm to model a variety of phenomena, particularly those related to growth or decay. In our exercise, the average score of students over time is modeled by a logarithmic function:\[ f(t) = 88 - 15 \ln(t+1) \]The logarithm in this function, specifically the natural logarithm denoted by \( \ln \), helps to capture the rate at which the students forget information.

• The constant 88 represents the starting score, when no time has passed.
• The term \(-15 \ln(t+1)\) models the rate of decrease in scores as time increases.

Understanding this relationship helps in analyzing real-world scenarios where information retention fades over time.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. This idea is subtly captured in our original exercise through the logarithmic function.Though not strictly an exponential decay, the average score function \(f(t)\) illustrates how the logarithmic function can model similar fluid processes with diminishing returns.

• The scores start high and decrease steadily.
• In this model, the scores decrease less sharply as time goes on, indicating a gradual retention loss.

In this context, understanding how exponential-like decay works in a logarithmic model is key to interpreting time-based data like test scores or similar metrics.

Average Score Calculation

Average score calculation is central to analyzing the students' performance over time. By substituting different time values \(t\) into the function \(f(t)\), you can calculate the average score at various intervals.For example:

• For \(t=0\), the original score is \(f(0) = 88\).
• At \(t=2\) months, \(f(2) \approx 70.794\).
• At \(t=4\) months, \(f(4) \approx 61.685\).

By calculating these averages, you can efficiently track how well the information is retained after successive months. This process demonstrates the effect of time on learning retention. Simplifying this to clear calculations using logarithms and constants helps in quick and accurate analysis.

Graph Interpretation

Graph interpretation is a fundamental skill to visualize and understand mathematical functions' meaning and implications in real-world contexts. For the function \(f(t)\), plotting on a graph provides a clear picture of the students' memory retention.

• The graph displays a downward curve, illustrating the "Forgetting Curve," where scores decrease over months.
• Initial retention drops are more dramatic, but tend to slowly flatten over time, demonstrating diminishing returns in the rate of score decrease.

Interpreting such a graph gives insight into how quickly students forget material and allows educators to strategize about reinforcing learning over time. Graphs make abstract numbers and equations tangible, especially useful in educational settings.