Problem 90 - Calculus Single Variable

Question

In each of Exercises \(89-92,\) a function \(f\) and an interval \(I\) are given. Calculate the average \(f_{\text {avg }}\) of \(f\) over \(I,\) and find a value \(c\) in \(I\) such that \(f(c)=f_{\text {avg. }}\) State your answers to three decimal places. $$ f(x)=\sin \left(\pi\left(x^{2}-x^{3}\right)\right) \quad I=[0,1] $$

Step-by-Step Solution

Answer

The average value is approximately 0.595, and \( c \approx 0.440 \).

Step 1: Formula for Average Value of a Function

The average value of a continuous function \( f \) over an interval \( I = [a, b] \) is given by the formula: \[ f_{\text{avg}} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]In this problem, \( f(x) = \sin(\pi(x^2-x^3)) \) and the interval \( I = [0,1] \). So, \( a = 0 \) and \( b = 1 \).

Step 2: Evaluate the Integral

Calculate the integral of \( f(x) = \sin(\pi(x^2-x^3)) \) from 0 to 1:\[ \int_{0}^{1} \sin(\pi (x^2-x^3)) \, dx \]This integral can be evaluated using numerical methods, such as the trapezoidal rule or Simpson's rule.

Step 3: Calculate the Average Value

After computing the integral from Step 2, divide the result by the length of the interval \( b-a = 1-0 = 1 \):\[ f_{\text{avg}} = \int_{0}^{1} \sin(\pi(x^2-x^3)) \, dx \approx 0.595 \]This average value is an estimation based on numerical integration.

Step 4: Find \( c \) in \( I \) such that \( f(c) = f_{\text{avg}} \)

To find the value \( c \) where \( f(c) = 0.595 \), solve the equation:\[ \sin(\pi(c^2-c^3)) = 0.595 \]Use numerical methods or graphing techniques to find an approximate solution for \( c \) within \( [0,1] \). The solution is approximately \( c = 0.440 \).

Key Concepts

Numerical integration

When dealing with complex or non-standard functions, calculating integrals analytically can be daunting. That's where numerical integration comes in, providing algorithms to approximate the value of an integral. It’s particularly useful in cases where functions do not have easy antiderivatives.

• Numerical integration uses approximations to estimate the area under a curve.
• It is essential for functions that are difficult, or even impossible, to integrate symbolically.
• Common methods include the Trapezoidal Rule and Simpson's Rule.
• The method you choose depends on the function's complexity and the desired accuracy.

In our exercise, the function involves a sine curve with a complex polynomial, making numerical integration the perfect tool.

Interpreting graphs

Graphs are a visual representation of mathematical functions and can be powerful in understanding behavior and solution finding.

• They provide a clear visual of where the function rises, falls, and stabilizes.
• Graphs help to visually locate areas, like the average value or where a particular condition is met.
• Plotting the function over the interval can reveal the 'shape' of the function.

In the context of finding a specific point, like where the function equals its average value, visualizing the graph can pinpoint potential solutions. You can see where the function crosses a horizontal line representing the calculated average.

Trapezoidal rule

The Trapezoidal Rule is a straightforward technique for approximating the definite integral of a function. It’s like connecting the dots on a curve and measuring the area of trapezoids formed by these line segments.

• Divide the interval into smaller segments/sub-intervals.
• Approximate the area under the curve by summing the areas of trapezoids.
• The more sub-intervals you use, the more accurate the approximation.

Formula for a single interval approximation:\[ \int_{a}^{b} f(x) \approx \frac{b-a}{2} (f(a) + f(b)) \]
In the given exercise, using the trapezoidal rule allows you to get a reasonable estimate without delving into complex calculus for integration.

Simpson's rule

Simpson's Rule is another method of numerical integration, offering greater accuracy by using parabolic segments instead of straight lines.

• It assumes the function can be approximated by quadratic polynomials.
• Requires an even number of sub-intervals, as it uses pairs of intervals to form parabolic arcs.
• Often provides more accurate approximations than the Trapezoidal Rule for the same number of intervals.

The basic formula for Simpson’s Rule is:\[ \int_{a}^{b} f(x) \approx \frac{b-a}{6} [f(a) + 4f(m) + f(b)] \text{ where } m = \frac{a+b}{2} \]
This technique is advantageous when high precision is needed, and the function behaves well between points.