Problem 45 - Calculus Early Transcendentals

Question

Approximate mountains Suppose the elevation of Earth's surface over a \(16-\mathrm{mi}\) by \(16-\mathrm{mi}\) region is approximated by the function $$z=10 e^{-\left(x^{2}+y^{2}\right)}+5 e^{-\left((x+5)^{2}+(y-3)^{2}\right) / 10}+4 e^{-2\left((x-4)^{2}+(y+1)^{2}\right)}.$$ a. Graph the height function using the window \([-8,8] \times[-8,8] \times[0,15].\) b. Approximate the points \((x, y)\) where the peaks in the landscape appear. c. What are the approximate elevations of the peaks?

Step-by-Step Solution

Answer

Question: Graph the height function within the given window and find the approximate (x, y) coordinates and elevations of the peaks in the landscape. Solution: a) After graphing the given height function, we observe there are three distinct peaks. b) Approximate (x, y) coordinates of the peaks: 1. Peak 1: (x₁, y₁) ≈ (0, 0) 2. Peak 2: (x₂, y₂) ≈ (4, -1) 3. Peak 3: (x₃, y₃) ≈ (-5, 3) c) Approximate elevations of the peaks: 1. Peak 1: z₁ ≈ 10 2. Peak 2: z₂ ≈ 4 3. Peak 3: z₃ ≈ 5

Step 1: a. Graph the height function

To graph the height function within the given window \([-8,8] \times[-8,8] \times[0,15]\), use graphing software or a graphing calculator. Input the function: $$z=10 e^{-\left(x^{2}+y^{2}\right)}+5 e^{-\left((x+5)^{2}+(y-3)^{2}\right) / 10}+4 e^{-2\left((x-4)^{2}+(y+1)^{2}\right)}.$$ Ensure that the x, y, and z limits are set to the specified ranges. The graph will show the landscape represented by this function, allowing us to find the peaks visually or by exploring the surface plot.

Step 2: b. Approximate the points (x, y) where the peaks appear

By observing the graph, we see that there are three distinct peaks. Visually estimate or use the graphing software to find the \((x, y)\) coordinates for each peak: 1. Peak 1: \((x_1, y_1) \approx (0, 0)\) 2. Peak 2: \((x_2, y_2) \approx (4, -1)\) 3. Peak 3: \((x_3, y_3) \approx (-5, 3)\)

Step 3: c. Approximate the elevations of the peaks

To find the approximate elevations of the peaks, we plug the \((x, y)\) coordinates from step b into the height function: 1. Peak 1: \(z_1 \approx 10 e^{-\left({x_1}^{2}+{y_1}^{2}\right)}+5 e^{-\left(({x_1}+5)^{2}+({y_1}-3)^{2}\right) / 10}+4 e^{-2\left(({x_1}-4)^{2}+({y_1}+1)^{2}\right)} \approx 10\) 2. Peak 2: \(z_2 \approx 10 e^{-\left({x_2}^{2}+{y_2}^{2}\right)}+5 e^{-\left(({x_2}+5)^{2}+({y_2}-3)^{2}\right) / 10}+4 e^{-2\left(({x_2}-4)^{2}+({y_2}+1)^{2}\right)} \approx 4\) 3. Peak 3: \(z_3 \approx 10 e^{-\left({x_3}^{2}+{y_3}^{2}\right)}+5 e^{-\left(({x_3}+5)^{2}+({y_3}-3)^{2}\right) / 10}+4 e^{-2\left(({x_3}-4)^{2}+({y_3}+1)^{2}\right)} \approx 5\) So, the approximate elevations of the peaks are: 1. Peak 1: \(z_1 \approx 10\) 2. Peak 2: \(z_2 \approx 4\) 3. Peak 3: \(z_3 \approx 5\)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is elevated to a variable exponent. These functions have a distinct curve that either grows rapidly or decays, depending on the power. In the context of the problem, the function representing the height of the landscape is composed of several exponential terms. Each term adds to the complexity of the terrain, contributing to the peaks and valleys.

The first term, \(10 e^{-(x^2 + y^2)}\), describes a hump centered around the origin \((0, 0)\). The expression \(e^{-(x^2 + y^2)}\) diminishes as \(x\) and \(y\) move away from zero, resulting in a peak at the origin.

• Term 1: \(10 e^{-(x^2 + y^2)}\) creates a peak at \((0,0)\)
• Term 2: \(5 e^{-((x+5)^2 + (y-3)^2)/10}\) peaks near \((-5, 3)\)
• Term 3: \(4 e^{-2((x-4)^2 + (y+1)^2)}\) peaks near \((4, -1)\)

This characteristic behavior results from the sharp falling off of the exponential functions as the distance from the center increases.

Local Maximum

In mathematical terms, a local maximum is a point where a function reaches a peak value in a surrounding neighborhood. For the problem at hand, identifying the local maxima relates to finding the mountain peaks on the graph of the height function. These are the points where the elevation reaches a high relative to the surrounding landscape.

To approximate the local maxima, it's crucial to examine the graph or analyze the height function. At these peak points, any small movement around these coordinates would yield a lower height value. Thus, the peaks can be determined either by visual inspection of the graph or by calculations:

• Peak 1 is roughly at \((0, 0)\) with an elevation \(z \approx 10\).
• Peak 2 is around \((4, -1)\) with an elevation \(z \approx 4\).
• Peak 3 appears near \((-5, 3)\) with an elevation \(z \approx 5\).

Observing these points on a surface plot helps us understand where the function rises to significant heights, indicating the local maxima.

Surface Plot Analysis

Surface plot analysis allows us to visually interpret functions of two variables, by representing them as a three-dimensional surface. In this problem, the surface plot helps graphically display the elevation function over a region for better understanding of the landscape's topology.

The window specified, \([-8, 8] \times [-8, 8] \), includes all necessary parts to visualize the function's critical features. The elevation or \(z\)-axis ranges from 0 to 15, capturing the significant variations caused by the combined effects of multiple exponential terms.

• A 3D plot gives a clear perspective on where the surface rises and falls.
• Peaks show as ridges or summits in the plot.
• Surface plots help in comprehending spatial relationships within the function's values.

By analyzing the surface plot, one can estimate the coordinates and heights of the peaks, providing a comprehensive picture of the terrain modeled by the function.