Problem 88 - Precalculus Mathematics for Calculus

Question

Learning Curve A learning curve is a graph of a function \(P(t)\) that measures the performance of someone learning a skill as a function of the training time \(t\). At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value \(M\), the rate of learning decreases. It has been found that the function $$P(t)=M-C e^{-k t}$$ where \(k\) and \(C\) are positive constants and \(C

Step-by-Step Solution

Answer

(a) \( t = -\frac{1}{k} \ln \left( \frac{M - P}{C} \right) \) (b) 30 months (c) Graph: S-shaped curve approaching 20 ft.

Step 1: Rearrange given equation

Start with the given learning curve equation \( P(t) = M - Ce^{-kt} \). We need to solve for \( t \) as a function of \( P \). First, isolate the exponential term:\[ Ce^{-kt} = M - P \]

Step 2: Solve for the exponential term

Divide both sides by \( C \) to isolate the exponential expression:\[ e^{-kt} = \frac{M - P}{C} \]

Step 3: Apply the natural logarithm

Take the natural logarithm (ln) of both sides to solve for \( -kt \):\[ -kt = \ln \left( \frac{M - P}{C} \right) \]

Step 4: Solve for t

Divide by \( -k \) to get \( t \) as a function of \( P \):\[ t = -\frac{1}{k} \ln \left( \frac{M - P}{C} \right) \]

Step 5: Substitute known values for the pole-vaulter

Insert the known constants from part (b)'s pole-vaulter equation: \( M = 20 \), \( C = 14 \), and \( k = 0.024 \). Also, \( P = 12 \) when solving for \( t \):\[ t = -\frac{1}{0.024} \ln \left( \frac{20 - 12}{14} \right) \]

Step 6: Calculate time t

Calculate using the expression from Step 5:\[ t \approx -\frac{1}{0.024} \ln(0.5714) \approx 29.7299 \text{ months} \]

Step 7: Graph the learning curve

Using the function \( P(t) = 20 - 14 e^{-0.024t} \), graph the curve by plotting \( P(t) \) against \( t \) from 0 to a reasonable time span (e.g., 100 months). You should see an S-shaped curve that approaches \( M = 20 \) as \( t \) increases. This reflects the increase in performance over time, leveling off as it nears 20 feet.

Key Concepts

Function of Time

In the context of learning curves, understanding how time affects performance is crucial. The function of time in this scenario is beautifully expressed through the equation \(P(t) = M - Ce^{-kt}\). This equation represents how a learner's performance progresses over time.
The variable \(t\) stands for "time" and is the independent variable in our equation, determining how performance \(P(t)\) changes. As time moves forward, performance typically improves until it stabilizes at a maximum level \(M\).
There are several components steeped in the function of time within this equation:

• \(M\) is the maximum performance level that will eventually be achieved.
• \(C\) and \(k\) are constants that modify how rapidly performance changes.

The function of time gives us a clear visualization: initial performance may start low but will increase swiftly at first. Over time, the rate of learning will slow down as it approaches the maximum \(M\). This function can be adapted to model various learning scenarios, making it a powerful tool for predicting progress as training continues.

Exponential Function

The equation \(P(t) = M - Ce^{-kt}\) involves an exponential function \(e^{-kt}\), which plays a key role in mapping out the learning curve. The exponential function is characterized by its constant multiplicative rate of change, meaning that it grows or decays at a constant rate relative to its current value.
When constancy is geared towards learning, like in our equation, the negative exponent \(-kt\) signifies decay. This is because we're observing a decreasing function rather than an increasing one. This setup is particularly useful because:

• The rapid initial increase in performance is mirrored by the exponential decay of \(Ce^{-kt}\).
• The value approaches zero as \(t\) tends to infinity, allowing performance \(P(t)\) to inch closer and closer to \(M\).

The elegance of the exponential function lies in its ability to model real-world learning scenarios effectively. Initially, learners tend to absorb new skills quickly, followed by a plateau phase at an advanced level, which is precisely what \(Ce^{-kt}\) captures in the context of the learning curve.

Performance Level

Performance level \(P\) in the learning curve equation \(P(t) = M - Ce^{-kt}\) highlights how skill enhancement can be mathematically modeled. It is essentially the outcome—a measurable indicator of how well someone is doing after a period of learning.
The realistic portrayal of performance going from \(P(t)\) at \(t = 0\) to \(M\) over time encapsulates the journey:

• At \(t = 0\), the equation simplifies to \(P(0) = M - C\), denoting the starting performance.
• As \(t\) increases, \(Ce^{-kt}\) shrinks, which means \(P(t)\) grows closer to \(M\), the maximum achievable performance.

Whether for athletes, students, or professionals, this allows us to assess training effectiveness and predict the time needed to hit performance goals. Understanding this concept helps us relate abstract mathematical models to tangible accomplishments, and is instrumental for setting realistic ambitions and milestones in educational or skill-based pursuits.