Ugh Once again I need help please help!
2 answers:
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Answer:
Step-by-step explanation:
Given that:
The equation of the damped vibrating spring is y" + by' +2y = 0
(a) To convert this 2nd order equation to a system of two first-order equations;
let y₁ = y
y'₁ = y' = y₂
So;
y'₂ = y"₁ = -2y₁ -by₂
Thus; the system of the two first-order equation is:
y₁' = y₂
y₂' = -2y₁ - by₂
(b)
The eigenvalue of the system in terms of b is:
(c)
Suppose , then λ₂ < 0 and λ₁ < 0. Thus, the node is stable at equilibrium.
(d)
From λ² + λb + 2 = 0
If b = 3; we get
Now, the eigenvector relating to λ = -1 be:
Let v₂ = 1, v₁ = -1
Let Eigenvector relating to λ = -2 be:
Let m₂ = 1, m₁ = -1/2
∴
So as t → ∞
F(x) = 1/x is a special function. is called the multiplicative inverse or the reciprocal function. the specific shape of the graph is called a rectangular hyperbola.
negative values of x with produce negative values of f(x) and positive values will produce positive function values. so this hyperbola is symmetric with respect to the origin and all negative values are in Q3 and all positive values are in Q1.
as x gets farther and farther from 0 in either direction, you can see that 1/x will be an ever-increasingly smaller distance from the x-axis. indeed, the limit of this function is zero as x approaches either positive or negative infinity.
if x is between -1 and 1, then the value of f(x) increases until x = 0, the limit of the function as x approaches 0 being positive and negative infinity depending on which direction you are coming from.
the domain is all real numbers with the exception of 0
the range also happens to be all real numbers with the exception of zero. all that really means is that zero has no multiplicative inverse.
strictly speaking, this is an exponential function, as the function is f(x) = x^(-1).
negative values of x with produce negative values of f(x) and positive values will produce positive function values. so this hyperbola is symmetric with respect to the origin and all negative values are in Q3 and all positive values are in Q1.
as x gets farther and farther from 0 in either direction, you can see that 1/x will be an ever-increasingly smaller distance from the x-axis. indeed, the limit of this function is zero as x approaches either positive or negative infinity.
if x is between -1 and 1, then the value of f(x) increases until x = 0, the limit of the function as x approaches 0 being positive and negative infinity depending on which direction you are coming from.
the domain is all real numbers with the exception of 0
the range also happens to be all real numbers with the exception of zero. all that really means is that zero has no multiplicative inverse.
strictly speaking, this is an exponential function, as the function is f(x) = x^(-1).