Problem 86 - Precalculus

Question

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?

Step-by-Step Solution

Answer

The plot shows that \( y_2 \) approximates \( y_1 \) well near \( x=0 \), but diverges as \( x \) increases.

Step 1: Understand the Functions

First, identify the two functions we are dealing with. The function \( y_1 = e^{-x} \) is the exponential decay function, while \( y_2 = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{24} \) is the fourth-order Taylor series approximation of \( e^{-x} \) centered at \( x=0 \).

Step 2: Plot the Functions

Using a graphing tool or graph paper, plot \( y_1 = e^{-x} \), which will look like a curve that starts at \( (0,1) \) and approaches the x-axis as \( x \) increases. Plot \( y_2 \) on the same set of axes, noticing that it should also start at \( (0,1) \) due to the Taylor series approximation.

Step 3: Compare the Plots

Observe how the two plots correspond. \( y_2 \) closely follows \( y_1 \) near \( x=0 \), but begins to diverge as \( x \) moves further away from zero, particularly as \( x \) becomes larger.

Key Concepts

Exponential Decay

Exponential decay is a fundamental mathematical concept that describes processes which reduce or deplete over time in a fashion that is proportional to the current value. It is commonly seen in various disciplines such as physics, biology, and finance. The function \( y_1 = e^{-x} \) is a classic example of exponential decay. This function represents a rapidly decreasing curve that starts at 1 when \( x = 0 \), demonstrating how initially there is a substantial drop that becomes more gradual as \( x \) increases.

This behavior can be explained by looking at how the rate of change itself decreases. As the value of the exponent \( x \) increases, the whole expression \( e^{-x} \) becomes smaller, representing less change over time given a constant rate.

If you think of this in practical terms, imagine a substance that decreases in quantity by half every measured time interval. This essence of exponential behavior is captured in the mathematics by maintaining a consistent rate of decrease.

Function Plotting

Function plotting is a powerful visual tool that aids in understanding the behavior of mathematical functions. By plotting a function like \( y_1 = e^{-x} \) and comparing it to another function, such as its Taylor series approximation \( y_2 \), we can visually assess how two expressions relate to each other. Such plots allow us to easily see where the functions overlap or deviate from one another.

To plot a function, you generally follow these steps:

• Choose a range of values for \( x \).
• Calculate the corresponding \( y \) values for each \( x \).
• Represent these as points on a graph and connect these points to form a curve.

Graphs are especially useful in seeing how approximations like Taylor series perform, particularly in what range they are accurate and when they start to diverge from the actual function. This visual method can provide insight into both the accuracy and limitations of using polynomial approximations to represent complex behaviors.

Graphical Analysis

Graphical analysis helps deepen our understanding of mathematical concepts by interpreting graph behaviors and characteristics. When comparing the exponential decay function \( y_1 = e^{-x} \) with its Taylor series approximation \( y_2 = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{24} \), graphical analysis allows us to comprehend not just the numbers but the visual discrepancies and congruities between these functions.

The process involves:

• Identifying where the plots of \( y_1 \) and \( y_2 \) coincide or diverge.
• Evaluating how close the approximation is to the actual curve near \( x=0 \), where it's designed to be the most precise.
• Noticing the deviation that increases as \( x \) moves away from zero due to diminishing accuracy of the polynomial terms.

This analysis is crucial when using approximations in practice, as it informs decision-making on the validity of models over certain ranges and guides adjustments when greater accuracy is required. Thus, graphical analysis is a key component in leveraging the power of mathematics to model and solve real-world problems.