Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term + ... 18, Student Edition Printed Access Card

8th Edition

ISBN: 9781305782198

Author: James Stewart

Publisher: Cengage Learning

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Differential Equations

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Chapter 14, Problem 2P

Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the Wood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point P(x, y) on the surface of seawater is approximated by

C(x, y) = e^{−(x2 + 2y2)/10x}

where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin.

(a) Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source.

(b) Suppose a shark is at the point (x₀, y₀) when it first detects the presence of blood in the water. Find an equation of the shark's path by setting up and solving a differential equation.

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

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Chapter 14, Problem 1P

Chapter 14, Problem 3P

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What is a solution to a differential equation? We said that a differential equation is an equation that describes the derivative, or derivatives, of a function that is unknown to us. By a solution to a differential equation, we mean simply a function that satisfies this description. 2. Here is a differential equation which describes an unknown position function s(t): ds dt 318 4t+1, ds (a) To check that s(t) = 2t2 + t is a solution to this differential equation, calculate you really do get 4t +1. and check that dt' (b) Is s(t) = 2t2 +++ 4 also a solution to this differential equation? (c) Is s(t)=2t2 + 3t also a solution to this differential equation? ds 1 dt (d) To find all possible solutions, start with the differential equation = 4t + 1, then move dt to the right side of the equation by multiplying, and then integrate both sides. What do you get? (e) Does this differential equation have a unique solution, or an infinite family of solutions?

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Chapter 14 Solutions

Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term + ... 18, Student Edition Printed Access Card

Ch. 14.1 - In Example 2 we considered the function W = f(T,...Ch. 14.1 - The temperature-humidity index I (or humidex, for...Ch. 14.1 - Prob. 3E Ch. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - A model for the surface area of a human body is...Ch. 14.1 - The wind-chill index W discussed in Example 2 has...Ch. 14.1 - The wave heights h in the open sea depend on the...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Let g(x, y) = cos(x + 2y). (a) Evaluate g(2, 1)....Ch. 14.1 - Let F(x,y)=1+4y2. (a) Evaluate F(3,1). (b) Find...

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Solution Summary: The author explains the level curves of a concentration function C(x,y)=e

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