Problem 99 - University Calculus: Early Transcendentals

Question

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x e^{-x} \quad \text { on } \quad[0,1]$$

Step-by-Step Solution

Answer

Plot, partition, calculate average, solve equation for \(x\).

Step 1: Plot the Function

First, we need to plot the function \( f(x) = x e^{-x} \) over the interval \([0, 1]\). Use a graphing tool in a Computer Algebra System (CAS) to visualize how the function behaves over this range.

Step 2: Partition the Interval

Next, partition the given interval \([0, 1]\) into subintervals. We need to do this for three different values of \( n \): 100, 200, and 1000. This means dividing the interval into \( n \) subintervals, each of length \( \frac{1}{n} \).

Step 3: Evaluate Function at Midpoints

For each partition value of \( n \), calculate the midpoint of each subinterval. If \( \text{mid}_i \) represents the midpoint of the \( i^{th} \) subinterval, it is given by \( \text{mid}_i = \frac{i - 0.5}{n} \). Evaluate \( f(\text{mid}_i) = \text{mid}_i e^{-\text{mid}_i} \) for all subintervals.

Step 4: Calculate Average Value

Compute the average value of the function values from Step 3. For a partition with \( n \) subintervals, the average is \( \text{average} = \frac{1}{n} \sum_{i=1}^{n} f(\text{mid}_i) \). Calculate this for each \( n = 100, 200, \) and \( 1000 \), but note that we mainly need the result from \( n = 1000 \) in the next step.

Step 5: Solve for x with f(x) = Average Value

Using the average value calculated in Step 4 for \( n = 1000 \), solve the equation \( f(x) = \text{average value} \) for \( x \). This involves finding \( x \) such that \( x e^{-x} = \text{average value} \). Use a numerical solver in the CAS to find the solution.

Key Concepts

Function Plotting

Plotting a function can give a visual representation of its behavior, helping us understand how it changes over a particular interval. In our example, we aim to plot the function \(f(x) = x e^{-x}\) over the interval \([0, 1]\). Using a Computer Algebra System (CAS), we can easily graph this function.
By plotting the function:

• We can determine key characteristics such as maxima, minima, and points of inflection within the interval.
• We see that \(f(x)\) starts at zero when \(x=0\) and tends to decrease as \(x\) increases due to the exponential component \(e^{-x}\).

The graphing tool visually aids in confirming these behaviors, thereby providing a comprehensive understanding of the function’s overall shape and tendency.

Numerical Integration

To approximate the integral of a function over a given interval, we can use numerical integration techniques by partitioning the interval into subintervals. For the function \(f(x) = x e^{-x}\) on the interval \([0, 1]\), we divide it into 100, 200, and 1000 subintervals.
Here's how it works:

• The length of each subinterval is \(\frac{1}{n}\), where \(n\) is the number of subintervals.
• For each subinterval, we compute the function value at the midpoint, which provides a better estimate for the area under the function curve as compared to evaluating endpoint values.

Calculating these midpoint values and then possibly summing them allows us to approximate the integral, which is essentially the total area under the curve. This is crucial for understanding how the function accumulates over an interval and helps in further calculations like finding the average value.

Average Value Calculation

The average value of a function over a given interval provides an insightful measure of its overall behavior. Specifically, it tells us the typical height or value of the function within that interval. To calculate this for \(f(x) = x e^{-x}\), after partitioning the interval into \(n\) subintervals, apply the formula:
\[\text{average} = \frac{1}{n} \sum_{i=1}^{n} f(\text{mid}_i),\]
where \(\text{mid}_i\) represents the midpoint of each subinterval. We calculate this for \(n=100, 200,\) and \(1000\), although we focus most on \(n=1000\) for precision.

• This calculation gives us a singular value representing the mean height of the function's graph over the interval \([0, 1]\).
• Such an average value is helpful in solving equations and comparing with other functions or data sets.

Equation Solving

Equation solving is the process of finding an unknown variable that satisfies a particular mathematical condition. In this context, you need to find an \(x\) such that \(f(x) = \text{average value}\), where the average value comes from our previous calculations for \(n=1000\).
To solve \(x e^{-x} = \text{average value}\), we can use a numerical solver in a CAS, as analytical solutions might not always be convenient or possible.

• Numerical solvers iterate through possible \(x\) values, refining their approach with each step toward finding a precise solution that satisfies the equation.
• This method is especially useful for complex equations that do not lend themselves to simple algebraic manipulation.

Ultimately, solving this equation helps in pinpointing specific values of \(x\) within \([0, 1]\) that equate our function to its computed average value, providing deeper insights into function behavior across the interval.