Problem 30 - Applied Calculus

Question

After \(x\) practice sessions, a person can accomplish a task in \(f(x)=12 x^{-1 / 2}\) minutes. Find the average time required from the end of session 1 to the end of session 9 .

Step-by-Step Solution

Answer

The average time is approximately 6.27 minutes.

Step 1: Understand the Function

We are given the function \( f(x) = 12x^{-1/2} \), which represents the time in minutes it takes to complete a task after \( x \) practice sessions. The task is to find the average time required from the end of session 1 to the end of session 9.

Step 2: Calculate Individual Times

Calculate the time required after each practice session from 1 to 9 using the given function. - For session 1: \( f(1) = 12(1)^{-1/2} = 12 \) minutes.- For session 2: \( f(2) = 12(2)^{-1/2} = 12/\sqrt{2} \approx 8.49 \) minutes.- For session 3: \( f(3) = 12(3)^{-1/2} = 12/\sqrt{3} \approx 6.93 \) minutes.- For session 4: \( f(4) = 12(4)^{-1/2} = 12/2 = 6 \) minutes.- For session 5: \( f(5) = 12(5)^{-1/2} = 12/\sqrt{5} \approx 5.37 \) minutes.- For session 6: \( f(6) = 12(6)^{-1/2} = 12/\sqrt{6} \approx 4.90 \) minutes.- For session 7: \( f(7) = 12(7)^{-1/2} = 12/\sqrt{7} \approx 4.53 \) minutes.- For session 8: \( f(8) = 12(8)^{-1/2} = 12/\sqrt{8} = 3\sqrt{2} \approx 4.24 \) minutes.- For session 9: \( f(9) = 12(9)^{-1/2} = 12/3 = 4 \) minutes.

Step 3: Find Average Time

To find the average time from session 1 to session 9, add up all the time values calculated in the previous step and divide by the number of sessions. \[ \text{Average Time} = \frac{f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9)}{9} \]Plug in the values:\[ \text{Average Time} = \frac{12 + 8.49 + 6.93 + 6 + 5.37 + 4.90 + 4.53 + 4.24 + 4}{9} \] Calculate the sum: \[ 12 + 8.49 + 6.93 + 6 + 5.37 + 4.90 + 4.53 + 4.24 + 4 = 56.46 \]Divide by 9:\[ \text{Average Time} = \frac{56.46}{9} \approx 6.27 \] minutes.

Key Concepts

Calculus Application

Calculus plays a crucial role in understanding various real-world functions and changes, like how efficiency improves with practice.
In our exercise, we use calculus to analyze the function given as \( f(x) = 12x^{-1/2} \). This function signifies the time taken to complete a task after \( x \) practice sessions.
Here, the exponent \(-1/2\) indicates an inverse relationship: as the number of practice sessions increases, the time taken to complete the task decreases.

• This reflects a practical learning curve, where more practice leads to efficiency.
• The calculus concept of function evaluation lets us quantify this change across the sessions.

The task itself asks for the average time over nine sessions, highlighting another calculus concept: determining the rate of change. By evaluating the function at different points, we find the total time and average it, showing how calculus helps provide meaningful insights into dynamic situations.

Practice Sessions

Practice sessions are foundational in improving skills and efficiency. In mathematics, functions can model this improvement. In the given function \( f(x) = 12x^{-1/2} \), each session impacts the time taken to complete a task.
To make the most of practice sessions:

• Evaluate the function at each session. Notice how times decrease as sessions progress.
• From \(f(1)\) (12 minutes) down to \(f(9)\) (4 minutes), the pattern is clear.

Through exercises like this one, students learn that systematic practice leads to measurable progress. By breaking down session outcomes, you can visualize the benefits of repeated practice on efficiency. Thus, the exercise serves as a wonderful demonstration of how theoretical math concepts apply to tangible improvements in skill.

Function Evaluation

Function evaluation involves calculating the output of a function for specific inputs. In this scenario, we calculate times for each session from \( x = 1 \) to \( x = 9 \). This process shows how to apply a function to real numbers and interpret outcomes.
Steps to evaluate the function are simple:

• Substitute each \( x \) value into the function \( f(x) = 12x^{-1/2} \).
• Calculate the result, noting how it changes with each session.

Through evaluation, we see the function's behavior over a series of inputs. It crystalizes why the average calculation is meaningful: the varied session times reflect the decreasing impact of practice. Students learn the importance of precision in plugging values into functions and why these patterns matter in broader computational contexts.