Problem 380 For the following exercises, fin... [FREE SOLUTION]

Chapter 7: Problem 380

For the following exercises, find a function of the form $y=a b^{x}+c \sin \left(\frac{\pi}{2} x\right)$ that fits the given data. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} \\ \hline y & {6} & {29} & {96} & {379} \\ \hline\end{array} $$

Short Answer

Expert verified

The function is $ y = 6 \times 4^{x} + 5 \sin \left(\frac{\pi}{2} x\right) $.

Step by step solution

Understanding the Function Form

The function is of the form $ y = a b^{x} + c \sin \left(\frac{\pi}{2} x\right) $. We need to determine the values of $ a $, $ b $, and $ c $ that allow the function to fit the data points given. The sine component will oscillate between $-c$ and $c$ as $x$ goes through an integer value. Thus, $ c\sin\left(\frac{\pi}{2} x\right) $ will be either $ 0, c, 0, -c $ for $x = 0, 1, 2, 3$ respectively.

Establishing Equations for Each Point

Use the given points to create equations. For $ x = 0 $, $ y = a + 0 \times c = 6 $ gives $ a = 6 $. For $ x = 1 $, $ y = a b + c = 29 $ gives $ 6b + c = 29 $. For $ x = 2 $, $ y = a b^{2} + 0 = 96 $ gives $ 6b^{2} = 96 $. For $ x = 3 $, $ y = a b^{3} - c = 379 $ gives $ 6b^{3} - c = 379 $.

Solving for Constants a, b, and c

From the equation $ 6b^{2} = 96 $, solve for $ b $: $ b^{2} = 16 $ thus $ b = 4 $ (as b > 0 for an increasing sequence). Now substitute $ b = 4 $ into $ 6b + c = 29 $ and solve for $ c $: $ 6 \times 4 + c = 29 $, so $ c = 5 $. Now using $ 6b^{3} - c = 379 $ with $ b = 4 $, solve for consistency: $ 6 \times 64 - 5 = 379 $, the equation is consistent with found values, confirming $ c = 5 $.

Writing the Function Expression

With the values determined as $ a = 6 $, $ b = 4 $, and $ c = 5 $, the function can be expressed as: $ y = 6 \times 4^{x} + 5 \sin \left(\frac{\pi}{2} x \right) $. This function fits the given data table for $ x = 0, 1, 2, 3 $.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function

The exponential function is a mathematical function of the form $ y = a b^{x} $. It is often used to model situations where something grows or decays at a constant percentage rate. In our exercise, the exponential part helps us describe the rapidly increasing values of $ y $ as $ x $ increases.

To break it down further:

$ a $ is the initial value of the function when $ x = 0 $. In our function, $ a = 6 $, which means that the function starts at 6 when $ x = 0 $.

$ b $ is the base of the exponential component and reflects the rate of increase. In this case, $ b = 4 $, indicating the function grows multiplicatively by a factor of 4 as $ x $ increments. This is why the function values get so large quickly as $ x $ increases.

When dealing with data modeling, an exponential function is especially useful for modeling exponential growth scenarios, such as population growth or interest accumulation. The key characteristic is the constant proportionality that defines the exponential relationship.

Sine Function

The sine function in the given formula is represented by $ c \sin\left(\frac{\pi}{2} x\right) $. This component adds an oscillating pattern to the model due to the nature of the sine wave, which moves between -1 and 1.

Here's how it works:

The coefficient $ c $ determines the amplitude of the sine wave, or how far the wave moves up and down. In our case, $ c = 5 $, so the sine component varies between -5 and 5.

The argument of the sine function, $ \frac{\pi}{2} x $, dictates the period of oscillation. Each increase in $ x $ by 1 results in the sine function reaching 0, its maximum, 0 again, and then its minimum, cycling every 4 points.

The purpose of including the sine function in this data modeling exercise is to accommodate any cyclical pattern in the data that the exponential component alone cannot capture. By understanding how $ c $ and the sine function interact, you can effectively model periodic behaviors alongside exponential trends.

Data Modeling

Data modeling is the process of creating a mathematical representation of the relationships within a set of data. The aim is to capture patterns and trends that can explain past data and predict future behavior. In the given problem, we are tasked with creating a model that blends both exponential growth and periodic oscillation.

Some key steps in data modeling include:

Identify the type of functions involved: In this exercise, the model combines an exponential function and a sine function to fit the data.

Substitute data points into the function form to establish equations: Using the given data points, we create equations for each $ x $ value and solve for unknowns $ a $, $ b $, and $ c $.

Refine the model to fit the data: Once the constants are found, the function needs to be checked against the data to ensure accuracy.

By combining both exponential and sine components, the model can more accurately reflect complex real-world phenomena. This is crucial for making reliable predictions and the accurate interpretation of the underlying data.

91影视

For the following exercises, find a function of the form \(y=a b^{x}+c \sin \left(\frac{\pi}{2} x\right)\) that fits the given data. $$ \begin{array}{|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} \\ \hline y & {6} & {29} & {96} & {379} \\ \hline\end{array} $$

Short Answer

Step by step solution

Understanding the Function Form

Establishing Equations for Each Point

Solving for Constants a, b, and c

Writing the Function Expression

Key Concepts

Exponential Function

Sine Function

Data Modeling

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.