Problem 85 Plot \(y_{1}=e^{x}\) and \(y_{2}... [FREE SOLUTION]

Chapter 3: Problem 85

Plot \(y_{1}=e^{x}\) and \(y_{2}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing screen. What do you notice?

Short Answer

Expert verified

The polynomial \( y_2 \) approximates \( y_1 = e^x \) well near \( x = 0 \) but deviates as \( x \) moves further from zero.

Step by step solution

Understanding the Functions

In this problem, we need to plot two functions, \( y_1 = e^x \) and \( y_2 = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \). The first function, \( y_1 \), represents the exponential function. The second function, \( y_2 \), is a polynomial approximation of \( e^x \) using the first five terms of its Taylor series expansion at \( x = 0 \).

Plotting the Functions

To compare both functions, we plot them on the same set of axes. This involves choosing a range of \( x \) values, typically around -2 to 2, as both functions will show significant changes over this interval. Using graphing software or a graphing calculator, input both functions to visualize their behavior.

Analyzing the Plot

Upon plotting, you will notice that \( y_2 \), the polynomial, closely follows \( y_1 \), the exponential function, near \( x = 0 \), but deviates as \( x \) moves further away from zero. This reflects the accuracy of the Taylor series approximation: it is most precise near the expansion point \( x = 0 \).

Drawing Conclusions

The primary observation is that the polynomial \( y_2 \) closely resembles \( y_1 = e^x \) near \( x = 0 \) but deviates for larger \( |x| \) values. This illustrates the concept of Taylor series approximation, which is particularly effective near its expansion point.

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Key Concepts

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

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is expressed in the form \( y = a^x \). One of the most common exponential functions is \( y = e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. Exponential functions have distinct properties:

The rate of growth (or decay) is proportional to their value, meaning they increase rapidly or decrease slowly.
They have a constant multiplicative rate of change, making them useful for modeling phenomena like population growth, radioactive decay, and interest compounding.
For \( e^x \), the graph is always increasing and never crosses the horizontal axis.

Understanding the behavior of exponential functions is crucial when learning about polynomial approximations and how they represent curves like \( e^x \) near specific points.

Polynomial Approximation

Polynomial approximation involves representing complex functions using simpler polynomial expressions. A common technique for this is the Taylor series, which expands functions into infinite sums of polynomials. The idea is to approximate a function like \( e^x \) using a finite number of terms.Here, \( y_2 = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} \) is a polynomial approximation of \( e^x \), derived from the Taylor series at \( x = 0 \). Each term in this series represents a degree of the polynomial:

The zeroth term, \( 1 \), ensures the approximation starts at the function's value at \( x = 0 \).
The first term, \( x \), accounts for the initial slope.
Higher degree terms like \( \frac{x^2}{2} \), \( \frac{x^3}{6} \), and \( \frac{x^4}{24} \) fine-tune the curve to mirror the original function more closely near \( x = 0 \).

This polynomial provides a close approximation for \( e^x \) around \( x = 0 \), but its accuracy diminishes as \( x \) moves away from zero.

Graphing Functions

Graphing functions is essential for visually understanding the behavior and relationship between mathematical expressions. When graphing both \( y_1 = e^x \) and its polynomial approximation, \( y_2 \), key insights emerge:

The plot for \( y_1 = e^x \) is smooth and continuously increases, maintaining its exponential nature.
The polynomial approximation \( y_2 \) closely matches \( y_1 \) near \( x = 0 \), demonstrating how the Taylor series provides an accurate local representation.
As \( x \) diverges from zero, \( y_2 \) begins to stray from the true exponential path, showing where the approximation becomes less reliable.

By using graphing tools, students can see these functions side by side, which reinforces the concept that polynomial approximations are best used for estimating function values near a specific point. Understanding graphing helps illustrate how close an approximation can get to the actual function's graph in a given interval.

91影视

Plot \(y_{1}=e^{x}\) and \(y_{2}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing screen. What do you notice?

Short Answer

Step by step solution

Understanding the Functions

Plotting the Functions

Analyzing the Plot

Drawing Conclusions

Key Concepts

Exponential Function

Polynomial Approximation

Graphing Functions

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