Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find \(a\) and \(b\) so that \(\sin (a x+b)\) closely approximates the data $$\begin{array}{|r|r|r|r|r|r|}\hline \mathrm{x} & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{y} & 0.97 & 0.70 & 0.26 & -0.26 & -0.5 \\\\\hline\end{array}$$ Observe that for \(y=\sin (a x+b), \arcsin y=a x+b\). Therefore, fit \(a x+b\) to the number pairs, \((x, \arcsin y)\) using linear least squares.

Step-by-Step Solution

Answer

The values are \(a \approx -0.292\) and \(b \approx 1.293\).

Step 1: Calculate arcsin of y-values

First, calculate the arcsine of each y-value since we have the equation \( y = \sin(a x + b) \). Compute \( \arcsin(y) \) for each given data point: \( y = 0.97, 0.70, 0.26, -0.26, -0.50 \). These will be our target values for fitting.For \( y = 0.97 \), \( \arcsin(0.97) \approx 1.33 \).For \( y = 0.70 \), \( \arcsin(0.70) \approx 0.775 \).For \( y = 0.26 \), \( \arcsin(0.26) \approx 0.264 \).For \( y = -0.26 \), \( \arcsin(-0.26) \approx -0.264 \).For \( y = -0.50 \), \( \arcsin(-0.50) \approx -0.524 \).

Step 2: Set up the linear equation system

We have the equation \( \arcsin(y) = a x + b \). Using the point pairs \((x, \arcsin(y))\), we can create a system of equations for linear regression. The points are: - \((0, 1.33)\)- \((1, 0.775)\)- \((2, 0.264)\)- \((3, -0.264)\)- \((4, -0.524)\)

Step 3: Apply Least Squares Method

Set up equations based on the points and the form \( y = a x + b \) to minimize \( \sum (\arcsin(y_i) - (a x_i + b))^2 \). Use normal equations to solve for \( a \) and \( b \):\[\begin{align*}5b + 10a &= \sum \arcsin(y_i) \10b + 30a &= \sum x_i \cdot \arcsin(y_i) \\end{align*} \]Calculate \( \sum \arcsin(y_i) \approx 1.581 \) and \( \sum x_i \cdot \arcsin(y_i) \approx 2.903 \).

Step 4: Solve Linear Equations

Using the normal equations from Step 3:\[\begin{align*}5b + 10a &= 1.581 \10b + 30a &= 2.903 \\end{align*}\]Solve this to find \(a\) and \(b\). The solution gives us:\[ a \approx -0.292, \quad b \approx 1.293 \]

Step 5: Verify the Fit

Substitute \( a \approx -0.292 \) and \( b \approx 1.293 \) back into the equation \( y = \sin(a x + b) \) and check how closely it fits the original \( y \) values by calculating \( \sin \) outputs for each \( x \). If the outputs closely match the provided \( y \) values, the approximation is satisfactory. If not, re-evaluate calculations.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics for describing periodic phenomena. They include sine, cosine, tangent, and their inverses like arcsine. In this exercise, we focus on the sine function and its inverse, arcsine. The sine function, often denoted as \( \sin(x) \), maps angles to coordinates on the unit circle. It is periodic, repeating every \(2\pi\) radians. The arcsine function, \( \arcsin(x) \), is the inverse of sine, mapping a value from -1 to 1 back to an angle between \(-\pi/2\) and \(\pi/2\).
Understanding these functions is crucial when dealing with oscillatory data, common in fields like physics and engineering.
In our problem, the data is approximated using \( y = \sin(a x + b) \), indicating a relationship that can be explored through the sine function's properties.
Trigonometric functions also aid in modeling natural phenomena such as sound waves and seasonal patterns, making them indispensable tools in scientific modeling and data fitting.

Data Fitting

Data fitting involves finding a mathematical function that closely approximates a set of data points. In our context, we want to fit the function \( y = \sin(a x + b) \) to provided data points. This requires adjusting the parameters \(a\) and \(b\) to minimize the differences between the actual data values and those predicted by the model.
Data fitting is important in many applications, from simple trends in business analytics to complex simulations in meteorology.

• Linear data fitting, like in our task, approximates data using a straight line or simple transformations, and is often easier to interpret.
• We use the least squares method to reduce the sum of the squared differences between the observed values and our model's predicted values.

The goal is a model that accurately represents the data's underlying trends, allowing for predictions and insights about the data's behavior.

Normal Equations

Normal equations are a critical part of the least squares method, used to find the optimal parameters of a linear fit. They stem from calculus, finding the derivative of the squared differences and setting it to zero to minimize error. For a simple linear regression, we derive two equations involving sums of the given data.
In our example, the linear equation is \( \arcsin(y) = a x + b \). From here, we set up:

• \( \sum (\arcsin(y_i) - (a x_i + b))^2 \)

and derive to get the normal equations:
\[\begin{align*}5b + 10a &= \sum \arcsin(y_i) \ 10b + 30a &= \sum x_i \cdot \arcsin(y_i) \end{align*}\]
Solving these equations gives the parameters \(a\) and \(b\), which minimize the error. These key equations play an essential role in efficiently determining the line of best fit in data fitting tasks, especially when dealing with large data sets.