Problem 92 - Calculus and its applications

Question

Suppose that you are given the task of learning \(100 \%\) of a block of knowledge. Human nature is such that we retain only a percentage \(P\) of knowledge \(t\) weeks after we have learned it. The \(E b b\) inghaus learning model asserts that \(P\) is given by $$ P(t)=Q+(100-Q) e^{-k t} $$ where \(Q\) is the percentage that we would never forget and \(k\) is a constant that depends on the knowledge learned. Suppose that \(Q=40\) and \(k=0.7\). a) Find the percentage retained after 0 weeks, 1 week, 2 weeks, 6 weeks, and 10 weeks. b) Find \(\lim _{t \rightarrow \infty} P(t)\). c) Sketch a graph of \(P\). d) Find the rate of change of \(P(t)\) with respect to time \(t\). e) Interpret the meaning of the derivative.

Step-by-Step Solution

Answer

a) 100, 69.80, 54.80, 40.89, 40.05; b) 40; c) Graph starts at 100, approaches 40; d) \(P'(t) = -42e^{-0.7t}\); e) Rate of forgetting decreases over time.

Step 1: Calculate percentage retained at specific weeks (Part a)

We need to calculate \(P(t)\) at different values of \(t\): 0, 1, 2, 6, and 10 weeks. Using the given formula:\[P(t)=Q+(100-Q) e^{-k t}\]Given, \(Q = 40\), \(k = 0.7\).1. **For \( t = 0 \):** \[ P(0) = 40 + (100 - 40) e^{-0.7 \times 0} = 40 + 60 \times 1 = 100 \]2. **For \( t = 1 \):** \[ P(1) = 40 + 60 e^{-0.7 \times 1} \] \( e^{-0.7} \approx 0.4966 \), so, \[ P(1) = 40 + 60 \times 0.4966 \approx 69.80 \]3. **For \( t = 2 \):** \[ P(2) = 40 + 60 e^{-0.7 \times 2} \] \( e^{-1.4} \approx 0.2466 \), so, \[ P(2) = 40 + 60 \times 0.2466 \approx 54.80 \]4. **For \( t = 6 \):** \[ P(6) = 40 + 60 e^{-0.7 \times 6} \] \( e^{-4.2} \approx 0.0149 \), so, \[ P(6) = 40 + 60 \times 0.0149 \approx 40.89 \]5. **For \( t = 10 \):** \[ P(10) = 40 + 60 e^{-0.7 \times 10} \] \( e^{-7} \approx 0.0009 \), so, \[ P(10) = 40 + 60 \times 0.0009 \approx 40.05 \]

Step 2: Find the limit as time approaches infinity (Part b)

To find \(\lim _{t \rightarrow \infty} P(t)\), we need to evaluate the behavior of the exponential term:\[ P(t) = Q + (100 - Q) e^{-k t} \]As \( t \rightarrow \infty \), \( e^{-k t} \rightarrow 0 \), meaning that:\[ P(t) \rightarrow Q = 40 \].Thus, \( \lim _{t \rightarrow \infty} P(t) = 40 \).

Step 3: Sketch the graph of \(P(t)\) (Part c)

The function \(P(t) = Q + (100 - Q) e^{-k t}\) is an exponential decay model with an asymptote at \(P = Q\). Start at \(P(0) = 100\) and decay towards \(P = 40\) over time.Plot the following points: - \((0, 100)\), - \((1, 69.8)\), - \((2, 54.8)\), - \((6, 40.89)\), - \((10, 40.05)\).Draw a curve that starts at the point (0, 100) and approaches the horizontal line \(P = 40\) as \(t\) increases.

Step 4: Determine the rate of change of \(P(t)\) (Part d)

Find the derivative of \(P(t)\) with respect to \(t\):\[ P(t) = Q + (100 - Q) e^{-k t} \]The derivative is:\[ P'(t) = \frac{d}{dt}\left(Q + (100 - Q) e^{-kt}\right) = -k(100-Q)e^{-kt} \]Substituting \(Q = 40\) and \(k = 0.7\):\[ P'(t) = -0.7 \times 60 \times e^{-0.7t} \]\[ P'(t) = -42e^{-0.7t} \].

Step 5: Interpret the meaning of the derivative (Part e)

The derivative \(P'(t) = -42e^{-0.7t}\) represents the rate at which knowledge retention decreases over time. At \(t = 0\), the rate of change is the steepest, and as \(t\) increases, the rate of forgetting decreases, approaching zero. This means we forget rapidly at first, but the rate of forgetting slows down over time.

Key Concepts

Exponential Decay

Exponential decay is a type of mathematical model that describes a process where a quantity diminishes rapidly at first and then slows down over time. In the Ebbinghaus Learning Model, the equation \( P(t) = Q + (100 - Q) e^{-kt} \) shows exponential decay in knowledge retention. The key thing in exponential decay is the presence of the exponential function \( e^{-kt} \), which causes the quantity (here, knowledge) to reduce quickly initially.

Some essential characteristics of exponential decay include:

• A rapid decrease initially following an exponential pattern.
• A leveling off towards a stable value, often called the asymptote.

In our model, this asymptote is the value \( Q \), which signifies the percentage of knowledge that will never be forgotten.

Ebbinghaus Learning Model

The Ebbinghaus Learning Model is a significant theoretical framework used to understand how humans forget information over time. This model emphasizes that knowledge retention follows a predictable pattern of rapid decline initially, which then slows down. It's captured by the equation \( P(t) = Q + (100 - Q) e^{-kt} \).

Key components of the Ebbinghaus model include:

• **Initial Knowledge Level:** At \( t = 0 \), full retention is typically achieved, and students remember 100% of what they've learned.
• **Forgetting Rate:** Characterized by the constant \( k \), this rate determines how fast the information is forgotten.
• **Non-Forgettable Knowledge:** \( Q \) represents the knowledge that will always be retained and understood, forming a lifelong memory.

This model helps in planning effective learning strategies, such as spaced repetition, to enhance long-term knowledge retention.

Derivative Interpretation

The derivative of a function gives insight into the rate of change of the function at any given point. For knowledge retention, the derivative \( P'(t) = -k(100 - Q)e^{-kt} \) shows the rate at which knowledge is being forgotten.

Understanding the derivative helps us see:

• **Initial Forgetting Rate:** At \( t = 0 \), the forgetting rate is maximal, and the derivative shows the steepest negative slope.
• **Decreasing Forgetting Speed:** As time progresses, \( e^{-kt} \) decreases further, meaning the absolute value of the derivative diminishes, indicating a slower rate of forgetting.

This interpretation stresses that although forgetting is inevitable, the most critical period for retention strategies is right after learning.

Knowledge Retention

Knowledge retention refers to the ability of a student to maintain information in memory over time. Within the context of exponential decay and the Ebbinghaus model, retention decreases with time unless actively maintained.

To boost knowledge retention, consider:

• **Spaced Repetition:** Revisiting learned material at increasing intervals to combat natural forgetting. This takes advantage of the slowing rate of forgetting by reinforcing knowledge before it is lost.
• **Active Engagement:** Engaging with material actively through quizzes or teaching others helps anchor information more robustly.

Strong retention strategies can significantly mitigate the natural decay of memory and help students maintain high levels of comprehension over extended periods.