Please use python and show codes # Problem 4Consider the function\[ f(x, y) = \cos(0.2(x - 1)^2 + 0.5(y - 2)^2) e^{-x^2-y^2}. \]### Tasks1. **Plot the Function:** - Use Python's plotting capabilities, such as a surface plot or plotting a two-variable function. - Alternatively, use an online app like Desmos or Microsoft 3D Plotter. - Analyze the plot to determine if the function has a maximum or minimum.2. **Compute the Gradient:** - Determine the gradient of the function to understand its slope and direction.3. **System of Equations:** - Formulate the system of equations needed to find the maxima or minima of the function.4. **Solve the System:** - Attempt to solve the system of equations manually. - If unable, describe the methods and attempts made.### Visual RepresentationThe diagram or graph is not provided here, but you would typically expect a 3D surface plot where the x and y axes represent the variables, and the z-axis represents the value of the function \( f(x, y) \). Variations across the surface indicate the changes in function value, helping to identify peaks (maxima) or valleys (minima).

4. Problem 4: Consider the function f(x, y) = cos(0.2(x - 1)² +0.5(y-2)2)e-²-y² 1) Plot the function using either Python's plotting (this means you have to google "python surface plot" or "python plot two variable function") or an online app like Demos or Microsoft 3d plotter. Do you think the function has a maximum? A minimum? 2) Compute the gradient. 3) What is the system of equations for finding the max/min? 4) Are you able to solve the system of equations by hand? If not, explain what you tried.

4. Problem 4: Consider the function f(x, y) = cos(0.2(x - 1)² +0.5(y-2)2)e-²-y² 1) Plot the function using either Python's plotting (this means you have to google "python surface plot" or "python plot two variable function") or an online app like Demos or Microsoft 3d plotter. Do you think the function has a maximum? A minimum? 2) Compute the gradient. 3) What is the system of equations for finding the max/min? 4) Are you able to solve the system of equations by hand? If not, explain what you tried.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Please use python and show codes

# Problem 4

Consider the function

\[ f(x, y) = \cos(0.2(x - 1)^2 + 0.5(y - 2)^2) e^{-x^2-y^2}. \]

### Tasks

1. **Plot the Function:**
- Use Python's plotting capabilities, such as a surface plot or plotting a two-variable function.
- Alternatively, use an online app like Desmos or Microsoft 3D Plotter.
- Analyze the plot to determine if the function has a maximum or minimum.

2. **Compute the Gradient:**
- Determine the gradient of the function to understand its slope and direction.

3. **System of Equations:**
- Formulate the system of equations needed to find the maxima or minima of the function.

4. **Solve the System:**
- Attempt to solve the system of equations manually.
- If unable, describe the methods and attempts made.

### Visual Representation

The diagram or graph is not provided here, but you would typically expect a 3D surface plot where the x and y axes represent the variables, and the z-axis represents the value of the function \( f(x, y) \). Variations across the surface indicate the changes in function value, helping to identify peaks (maxima) or valleys (minima).

Expert Solution

Step 1

# This is the code required for python plotting

# Import libraries

from mpl_toolkits import mplot3d

import numpy as np

import matplotlib.pyplot as plt

x = np.outer(np.linspace(-3, 3, 32), np.ones(32))

y = x.copy().T # transpose

z = (x-1)**2+0.5*((y-1)**2)+x*y+2*x+3*y

fig = plt.figure(figsize =(14, 9))

ax = plt.axes(projection ='3d')

y_cmap = plt.get_cmap('hot')

surf = ax.plot_surface(x, y, z, cmap = my_cmap, edgecolor ='black')

fig.colorbar(surf, ax = ax, shrink = 0.5, aspect = 5)

ax.set_title('Surface plot for F(x,y)')

# show plot

plt.show()

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