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Chapter 12: Problem 45

Approximate mountains Suppose the elevation of Earth's surface over a \(16-\mathrm{mi}\) by 16 -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?

Short Answer

Expert verified
Answer: The approximate locations of the peaks are: 1. Peak 1: (0, 0) 2. Peak 2: (-5, 3) 3. Peak 3: (4, -1) The approximate elevations of the peaks are: 1. Peak 1: 10 units 2. Peak 2: 5 units 3. Peak 3: 4 units

Step by step solution

01

Graph the height function

To graph the height function \(z=10e^{-\left(x^{2}+y^{2}\right)}+5e^{-\left((x+5)^{2}+(y-3)^{2}\right)/10}+4e^{-2\left((x-4)^{2}+(y+1)^{2}\right)}\) with the window \([-8,8] \times [-8,8] \times [0,15]\), you can use a graphing calculator or an online graphing tool such as Desmos or WolframAlpha. Plot the function using these tools setting the required range for x, y, and z axes as specified in the given window.
02

Approximate the points where the peaks appear

Once you have graphed the function, carefully examine the 3D plot. You will notice three distinct peaks in the landscape. These peaks correspond to the three terms in the height function. Approximate the (x, y) coordinates of these peaks by visually inspecting the plot. You can also use a tool like WolframAlpha to find the local maxima, which will provide you with the coordinates of these points with more precision. The approximate locations of the peaks are: 1. Peak 1: \((0, 0)\) 2. Peak 2: \((-5, 3)\) 3. Peak 3: \((4, -1)\)
03

Approximate the elevations of the peaks

To find the approximate elevations of the peaks, substitute the (x, y) coordinates of each peak into the height function z: 1. Peak 1: \(z(0, 0) = 10e^{-\left(0^{2}+0^{2}\right)}+5e^{-\left((0+5)^{2}+(0-3)^{2}\right)/10}+4e^{-2\left((0-4)^{2}+(0+1)^{2}\right)} \approx 10\) 2. Peak 2: \(z(-5, 3) = 10e^{-\left((-5)^{2}+3^{2}\right)}+5e^{-\left((-5+5)^{2}+(3-3)^{2}\right)/10}+4e^{-2\left((-5-4)^{2}+(3+1)^{2}\right)} \approx 5\) 3. Peak 3: \(z(4, -1) = 10e^{-\left(4^{2}+(-1)^{2}\right)}+5e^{-\left((4+5)^{2}+(-1-3)^{2}\right)/10}+4e^{-2\left((4-4)^{2}+(-1+1)^{2}\right)} \approx 4\) So, the approximate elevations of the peaks are: 1. Peak 1: 10 units 2. Peak 2: 5 units 3. Peak 3: 4 units

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Key Concepts

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

3D Surface Plot
Understanding the topography of a region is made significantly easier with a 3D surface plot. This type of visualization allows us to depict a function of two variables as a physical shape in a three-dimensional space. For instance, when students are tasked with graphing the height function of a mountainous region, creating a 3D surface plot provides a tangible representation of the terrain's elevations and depressions.

Using a 3D graph, the function is visually rendered over a grid of 'x' and 'y' values, showing how the 'z' value, which represents elevation in geography, changes accordingly. Graphing calculators or online tools can be utilized to convert complex multivariable functions into understandable 3D structures, promoting a comprehensive spatial comprehension that is crucial for fields like geography and environmental modeling.
Local Maxima
A local maxima refers to a point in a function's domain where its value is greater than the value at any other point in the immediate vicinity. When graphing multivariable functions, identifying local maxima is akin to pinpointing mountain peaks within a landscape. Each local maximum corresponds to a peak's summit, the highest point in that particular area.

To approximate the points where peaks appear, one would visually inspect the 3D surface plot or use mathematical tools to calculate precisely where the function reaches its local maxima. Understanding the concept of local maxima not only solidifies one's knowledge in calculus but also aids in practical surveying tasks such as identifying potential high points in a geographic terrain.
Function Visualization
The ability to create visual representations of mathematical concepts is integral in function visualization. This process allows for a tangible understanding of abstract mathematical theories, especially within multivariable calculus where functions are not easily graphed on a standard two-dimensional plane. Visualization includes the creation of 3D surface plots, contour maps, and vector fields, where the goal is to bridge the gap between numerical data and human intuition.

For instance, when a student explores the function defining a region's elevation, function visualization translates that into a viewable model. This supports the learning process, making it easier to grasp complex notions such as where a function increases or decreases, and the location of its peaks (local maxima) or valleys (local minima).
Calculus in Geography
The application of calculus in geography is an essential component for analyzing and understanding physical spaces on our planet. Calculus helps model and predict changes in landscapes, evaluate rates of erosion, and even in understanding weather patterns through the study of changes over space and time. Specifically, multivariable calculus, with its functions of several variables, plays a crucial role in representing 3D environments like mountain ranges.

Using calculus to graph a multivariable function of a geographical region, as in the given exercise, showcases the practical implementation of these mathematical principles. By identifying local maxima, we can determine the peaks of a mountainous landscape, and through 3D plots, we can visualize these undulating terrains, empowering learners and professionals alike to appreciate the intricacies of our earth's topography.

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Most popular questions from this chapter

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a).

Given positive numbers \(x_{1}, \ldots, x_{n},\) prove that the geometric mean \(\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n}\) is no greater than the arithmetic mean \(\left(x_{1}+\cdots+x_{n}\right) / n\) in the following cases. a. Find the maximum value of \(x y z,\) subject to \(x+y+z=k\) where \(k\) is a real number and \(x>0, y>0\), and \(z>0 .\) Use the result to prove that $$(x y z)^{1 / 3} \leq \frac{x+y+z}{3}.$$ b. Generalize part (a) and show that $$\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n} \leq \frac{x_{1}+\cdots+x_{n}}{n}.$$

Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=2 x^{2}+y^{2}+2 x-3 y ; R=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}$$

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum. (Source: Mathematics Magazine, May 1985, and Calculus and Analytical Geometry, 2nd ed., Philip Gillett, 1984)

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).
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