The typical daily energy use of a household is described by a function f [-12, 12] → [0, ∞). The value of f(x) is the consumption rate at given time x. Both points x = -12 and x = 12 of the domain correspond to 3am so f(12) = f(-12). It is also known that 9am and 9pm (x = -6 and x = 6, respectively) are local maxima of the consumption rate. In this question we model the rate by a polynomial of degree 4 that is f(x) = a4x4 + A3x³ + a₂x² + α₁x + ao and assume that the total energy consumption during the day equals 24. (a) Write a linear system of equations describing the following properties: f(12) = f(-12), x = -6,6 are stationary points of f and the total energy consumption is 24. (b) Using Gaussian elimination, find all solutions of the linear system obtained in (a). (c) Find all functions f that satisfy all of the data given in the description of the question.
The typical daily energy use of a household is described by a function f [-12, 12] → [0, ∞). The value of f(x) is the consumption rate at given time x. Both points x = -12 and x = 12 of the domain correspond to 3am so f(12) = f(-12). It is also known that 9am and 9pm (x = -6 and x = 6, respectively) are local maxima of the consumption rate. In this question we model the rate by a polynomial of degree 4 that is f(x) = a4x4 + A3x³ + a₂x² + α₁x + ao and assume that the total energy consumption during the day equals 24. (a) Write a linear system of equations describing the following properties: f(12) = f(-12), x = -6,6 are stationary points of f and the total energy consumption is 24. (b) Using Gaussian elimination, find all solutions of the linear system obtained in (a). (c) Find all functions f that satisfy all of the data given in the description of the question.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![1. Energy consumption.
The typical daily energy use of a household is described by a function ƒ : [-12, 12] → [0, ∞).
The value of f(x) is the consumption rate at given time x. Both points x = −12 and x = 12
of the domain correspond to 3am so ƒ(12) = f(-12). It is also known that 9am and 9pm
(x = −6 and x = 6, respectively) are local maxima of the consumption rate. In this question
we model the rate by a polynomial of degree 4 that is ƒ(x) = axª + A3x³ + A2x² + a₁x + ªð
and assume that the total energy consumption during the day equals 24.
(a) Write a linear system of equations describing the following properties: ƒ(12) = f(−12),
x = −6, 6 are stationary points of f and the total energy consumption is 24.
(b) Using Gaussian elimination, find all solutions of the linear system obtained in (a).
(c) Find all functions f that satisfy all of the data given in the description of the question.
(d) Is it possible that the total energy consumed during the night hours from 1am to 5am
is smaller than 2 (that is the average over these four hours is less than half of the daily
average)?
If your answer is “yes” then you should give an example of such function f from (c),
otherwise justify why it is not possible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f9c13e2-5c54-4ec3-b4e3-dc49b76d4e30%2Fb12d7176-5936-4b4c-8862-ad0eb0d628b6%2Fabyxzrs_processed.png&w=3840&q=75)
Transcribed Image Text:1. Energy consumption.
The typical daily energy use of a household is described by a function ƒ : [-12, 12] → [0, ∞).
The value of f(x) is the consumption rate at given time x. Both points x = −12 and x = 12
of the domain correspond to 3am so ƒ(12) = f(-12). It is also known that 9am and 9pm
(x = −6 and x = 6, respectively) are local maxima of the consumption rate. In this question
we model the rate by a polynomial of degree 4 that is ƒ(x) = axª + A3x³ + A2x² + a₁x + ªð
and assume that the total energy consumption during the day equals 24.
(a) Write a linear system of equations describing the following properties: ƒ(12) = f(−12),
x = −6, 6 are stationary points of f and the total energy consumption is 24.
(b) Using Gaussian elimination, find all solutions of the linear system obtained in (a).
(c) Find all functions f that satisfy all of the data given in the description of the question.
(d) Is it possible that the total energy consumed during the night hours from 1am to 5am
is smaller than 2 (that is the average over these four hours is less than half of the daily
average)?
If your answer is “yes” then you should give an example of such function f from (c),
otherwise justify why it is not possible.
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