The American Court Reporting Institute finds that the...

The American Court Reporting Institute finds that the average student taking Advanced Machine Shorthand, an intensive 20-wk course, progresses according to the function $$ Q(t)=120\left(1-e^{-0.05 t}\right)+60 \quad(0 \leq t \leq 20) $$ where $Q(t)$ measures the number of words (per minute) of dictation that the student can take in machine shorthand after $t$ wk in the course. Sketch the graph of the function $Q$ and answer the following questions: a. What is the beginning shorthand speed for the average student in this course? b. What shorthand speed does the average student attain halfway through the course? c. How many words per minute can the average student take after completing this course?

The American Court Reporting Institute finds that the average student taking Advanced Machine Shorthand, an intensive 20-wk course, progresses according to the function
$$
Q(t)=120\left(1-e^{-0.05 t}\right)+60 \quad(0 \leq t \leq 20)
$$
where $Q(t)$ measures the number of words (per minute) of dictation that the student can take in machine shorthand after $t$ wk in the course. Sketch the graph of the function $Q$ and answer the following questions:
a. What is the beginning shorthand speed for the average student in this course?
b. What shorthand speed does the average student attain halfway through the course?
c. How many words per minute can the average student take after completing this course?

Key Concepts

Graphical Interpretation

Sketching the graph involves representing the curve’s initial rapid increase, its subsequent gradual approach to a plateau, and its overall behavior over time. This visual analysis aids in understanding the dynamics of the function and how performance improves over the duration of the course.

Asymptotic or Long-term Behavior

As time increases (toward t = 20 weeks or beyond), the exponential term e^(-0.05t) diminishes, causing the function to gradually approach an ultimate limit. This limit represents the maximum or steady-state performance level achievable by the student in the context of the learning process.

Function Evaluation at Specific Times

Evaluating the function at given time points (like t = 10 weeks for halfway through the course) is essential to understand progress at various stages. This practice involves substituting the specific time into the function to obtain accurate performance measurements at that moment.

Exponential Growth Function

This function models growth that starts rapidly and then slows as it approaches a limiting value. The equation Q(t) = 120(1 - e^(-0.05t)) + 60 combines an exponential rise with a constant offset, which is a common form used in modeling learning processes or other phenomena approaching saturation.

Initial Value

The initial value is found by evaluating the function at time t = 0. This step determines the starting point of the process, which in learning or performance improvement models, represents the baseline measure before any progress has been made.

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