The general solution of the system of equations is given by: x(t) = c₁ + c₂t, y(t) = -5c₁ - 5c₂t. Where c₁ and c₂ are arbitrary constants.
Solving for General Solution of a SystemTo find the general solution of the system of equations:
X' = AX
where X = [x, y] and
A = [tex]\left[\begin{array}{ccc}5&1\\-4&1\end{array}\right][/tex]
we can proceed as follows:
Let's write the system of equations separately:
x' = 5x + y
y' = -4x + y
Taking the derivatives of x and y with respect to some variable (e.g., time), we obtain:
x'' = 5x' + y'
y'' = -4x' + y'
We can rewrite the system of equations in matrix form as:
X'' = AX'
Now, let's substitute X' with another variable, say V:
V = X'
We have:
X'' = AV
Therefore, we now have a new system of equations:
V = X'
X'' = AV
Substituting V back into the second equation, we get:
X'' = A(X')
This becomes:
X'' = AX'
This implies that X' is an eigenvector of A with eigenvalue 0.
Next, we need to find the eigenvectors of A. To do that, we solve the equation:
(A - 0I)V = 0
where I is the identity matrix and V is the eigenvector.
For A = [tex]\left[\begin{array}{ccc}5&1\\-4&1\end{array}\right][/tex] the matrix (A - 0I) becomes:
[tex]\left[\begin{array}{ccc}5&1\\-4&1\end{array}\right][/tex]V = [tex]\left[\begin{array}{ccc}5&1\\-4&1\end{array}\right][/tex][tex]\left[\begin{array}{ccc}v_{1} \\v_{2} \end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}0\\0\end{array}\right][/tex]
This gives us the following system of equations:
5v₁ + v₂ = 0
-4v₁ + v₂ = 0
We can solve this system of equations to find the eigenvectors:
5v₁ + v₂ = 0 --> v₂ = -5v₁
-4v₁ + v₂ = 0 --> v₂ = 4v₁
From these equations, we can choose a value for v₁ (e.g., 1) and calculate the corresponding v₂:
v₂ = -5(1) = -5
So, one eigenvector is v = [1, -5].
The general solution of the system of equations is given by:
X(t) = [tex]c_{1}e^{(\lambda_{1}t)v_{1}} + c_{2}e^{(\lambda_{2}t)v_{2}}[/tex]
where λ₁ and λ₂ are the eigenvalues and v₁ and v₂ are the corresponding eigenvectors.
In this case, since we have only one eigenvalue of 0 (due to X' being an eigenvector of A with eigenvalue 0), the general solution becomes:
X(t) = [tex]c_{1}e^{(0t)v_{1}} + c_{2}e^{(0t)v_{2}}[/tex]
Simplifying, we have:
X(t) = c₁v₁ + c₂tv₂
Substituting the values for v₁ and v₂, we get:
X(t) = c₁[1, -5] + c₂t[1, -5]
Expanding, we have:
x(t) = c₁ + c₂t
y(t) = -5c₁ - 5c₂t
where c₁ and c₂ are arbitrary constants.
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show that the vectors ⟨1,2,1⟩,⟨1,3,1⟩,⟨1,4,1⟩ do not span r3 by giving a vector not in their span
It is not possible to find a vector in R3 that cannot be written as a linear combination of ⟨1,2,1⟩,⟨1,3,1⟩, and ⟨1,4,1⟩.
It is required to show that the vectors ⟨1,2,1⟩,⟨1,3,1⟩,⟨1,4,1⟩ do not span R3 by providing a vector that is not in their span. Here is a long answer of 200 words:The given vectors are ⟨1,2,1⟩,⟨1,3,1⟩, and ⟨1,4,1⟩, and it is required to prove that they do not span R3.
The span of vectors is the set of all linear combinations of these vectors, which can be written as the following:Span {⟨1,2,1⟩, ⟨1,3,1⟩, ⟨1,4,1⟩} = {a ⟨1,2,1⟩ + b ⟨1,3,1⟩ + c ⟨1,4,1⟩ | a, b, c ∈ R}where R represents real numbers.To show that the given vectors do not span R3, we need to find a vector in R3 that cannot be written as a linear combination of ⟨1,2,1⟩,⟨1,3,1⟩, and ⟨1,4,1⟩.Suppose the vector ⟨1,0,0⟩, which is a three-dimensional vector, is not in the span of the given vectors.
Now, we need to prove it.Let the vector ⟨1,0,0⟩ be the linear combination of ⟨1,2,1⟩,⟨1,3,1⟩, and ⟨1,4,1⟩.⟨1,0,0⟩ = a⟨1,2,1⟩ + b⟨1,3,1⟩ + c⟨1,4,1⟩Taking dot products of the above equation with each of the given vectors, we get,⟨⟨1,0,0⟩, ⟨1,2,1⟩⟩ = a⟨⟨1,2,1⟩, ⟨1,2,1⟩⟩ + b⟨⟨1,3,1⟩, ⟨1,2,1⟩⟩ + c⟨⟨1,4,1⟩, ⟨1,2,1⟩⟩⟨⟨1,0,0⟩, ⟨1,2,1⟩⟩ = a(6) + b(8) + c(10)1 = 6a + 8b + 10c
Similarly,⟨⟨1,0,0⟩, ⟨1,3,1⟩⟩ = 7a + 9b + 11c⟨⟨1,0,0⟩, ⟨1,4,1⟩⟩ = 8a + 11b + 14cNow, we have three equations and three unknowns.
Solving these equations simultaneously, we geta = 1/2, b = -1/2, and c = 0
The vector ⟨1,0,0⟩ can be expressed as a linear combination of ⟨1,2,1⟩ and ⟨1,3,1⟩, which implies that it is not possible to find a vector in R3 that cannot be written as a linear combination of ⟨1,2,1⟩,⟨1,3,1⟩, and ⟨1,4,1⟩.
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The graph of a polynomial function is shown, State the interval(s) on which is increasing and the interval(s) on which is decreasing. (Enter your answers using interval notation)
increasing____
decreasing____
In the graph of a polynomial function shown below, it is required to determine the interval(s) on which it is increasing and the interval(s) on which it is decreasing. Polynomial Function Graph The solution can be found by determining the turning points of the polynomial function.
Turning points are points where the polynomial changes direction. This means that if we can determine the x-values of these turning points, we can identify the intervals of increasing and decreasing of the polynomial function.
The turning points of the polynomial function can be found by identifying the roots of its derivative. The roots of the derivative indicate the values of x where the function changes from increasing to decreasing or decreasing to increasing.
Thus, we differentiate the polynomial function to obtain its derivative.
f(x) = 2x³ - 3x² - 12x + 20
Differentiating both sides with respect to x gives;
f'(x) = 6x² - 6x - 12
Setting f'(x) equal to zero and solving for x yields: 6x² - 6x - 12 = 0
Factoring out 6 from the expression on the left gives;
6(x² - x - 2) = 0
Factorizing x² - x - 2 gives;
(x - 2)(x + 1) = 0
The roots of the equation are;`
[tex]x - 2 = 0 or x + 1 = 0[/tex]
Thus, the roots of the derivative are [tex]`x = 2` and `x = -1`[/tex]. Therefore, the polynomial function has two turning points at [tex]x = 2 and x = -1.[/tex]
The intervals of increasing and decreasing of the polynomial function can now be identified as shown below;*Interval of Decrease: [tex]`(-∞, -1) ∪ (2, ∞)[/tex]`*Interval of Increase:[tex]`(-1, 2)`[/tex]
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Counting Methods:
Question one: A pizza company advertises that it has 15
toppings from which to choose. Determine the number of two- topping
or three topping pizzas that company can make.
To determine the number of two-topping or three-topping pizzas that the company can make, we need to consider the combinations of toppings.
For two-topping pizzas:
The number of combinations of choosing 2 toppings from 15 is given by the formula:
C(15, 2) = 15! / (2! * (15-2)!)
= 15! / (2! * 13!)
= (15 * 14) / (2 * 1)
= 105
Therefore, the company can make 105 two-topping pizzas.
For three-topping pizzas:
The number of combinations of choosing 3 toppings from 15 is given by the formula:
C(15, 3) = 15! / (3! * (15-3)!)
= 15! / (3! * 12!)
= (15 * 14 * 13) / (3 * 2 * 1)
= 455
Therefore, the company can make 455 three-topping pizzas.
In total, the company can make 105 + 455 = 560 two-topping or three-topping pizzas.
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Assume you select seven bags from the total number of bags the farmers collected. What is the probability that three of them weigh between 86 and 91 lbs.
4.3.8 For the wheat yield distribution of exercise 4.3.5 find
A. the 65th percentile
B. the 35th percentile
Assuming that the seven bags are selected randomly, we can use the binomial probability distribution.
The binomial distribution is used in situations where there are only two possible outcomes of an experiment and the probabilities of success and failure remain constant throughout the experiment.
.Using the standard normal distribution table, we can find that the z-score corresponding to the 65th percentile is approximately 0.385. We can use the formula z = (x - μ) / σ to find the value of x corresponding to the z-score. Rearranging the formula, we get:x = zσ + μ= 0.385 * 80 + 1500≈ 1530.8Therefore, the 65th percentile is approximately 1530.8 lbs.B.
To find the 35th percentile, we can follow the same steps as above. Using the standard normal distribution table, we can find that the z-score corresponding to the 35th percentile is approximately -0.385. Using the formula, we get:x = zσ + μ= -0.385 * 80 + 1500≈ 1469.2Therefore, the 35th percentile is approximately 1469.2 lbs.
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A rental car company charges $40 plus 15 cents per each mile driven. Part1. Which of the following could be used to model the total cost of the rental where m represents the miles driven. OC=1.5m + 40 OC= 0.15m + 40 OC= 15m + 40 Part 2. The total cost of driving 225 miles is, 10 9 8 7 6 5 4 3 2 Member of People ILI 16-20 21-25 28-30 31-33 A frisbee-golf club recorded the ages of its members and used the results to construct this histogram. Find the number of members 30 years of age or younger
The total cost of driving 225 miles is $73.75. The given histogram is as follows: From the histogram, we can see that the number of members 30 years of age or younger is 12. Therefore, the correct answer is 12.
A rental car company charges $40 plus 15 cents per mile driven.
Part 1. Which of the following could be used to model the total cost of the rental where m represents the miles driven?OC=0.15m + 40
The given information tells us that a rental car company charges $40 plus 15 cents per mile driven. Here, m represents the miles driven.
Thus, the option that could be used to model the total cost of the rental where m represents the miles driven is:
OC = 0.15m + 40.
Part 2. The total cost of driving 225 miles isOC = 0.15m + 40 (given)
Now, we have to find the cost of driving 225 miles.
Thus, we have to put the value of m = 225 in the above equation.OC = 0.15m + 40OC = 0.15 × 225 + 40OC = 33.75 + 40OC = $73.75
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Let X be a continuous random variable with the probabilty density function; f(x) = kx 0
To determine the value of the constant k in the probability density function (PDF) f(x) = kx^2, we need to integrate the PDF over its entire range and set the result equal to 1, as the total area under the PDF must equal 1 for a valid probability distribution.
The given PDF is defined as:
f(x) = kx^2, 0 < x < 1
To find k, we integrate the PDF over its range:
∫[0,1] kx^2 dx = 1
Using the power rule for integration, we have:
k∫[0,1] x^2 dx = 1
Integrating x^2 with respect to x gives:
k * (x^3/3) | [0,1] = 1
Plugging in the limits of integration, we have:
k * (1^3/3 - 0^3/3) = 1
Simplifying, we get:
k/3 = 1
Therefore, k = 3.
Hence, the value of the constant k in the PDF f(x) = kx^2 is k = 3.
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Table 1 shows scores given to 4 sessions by a network intrusion detection system. The "True Label" column gives the ground truth (i.e., the type each session actually is). Sessions similar to the attack signature are expected to have higher scores while those dissimilar are expected to have lower scores. Draw an ROC curve for the scores in Table 1. Clearly show how you computed the ROC points. Assume "Attack" as the positive ('p') class.
Table 1. Intrusion detector's scores and corresponding "true" labels.
Session No. Score True Label
1
0.1
Normal
2
0.5
Attack
3
0.6
Attack
4
0.7
Normal
The ROC Curve can be used to evaluate the performance of the binary classifier that differentiates two classes.
The ROC Curve is generated by plotting the True Positive Rate (TPR) against the False Positive Rate (FPR) for a range of threshold settings.
The ROC Curve is a good way to visually evaluate the sensitivity and specificity of the binary classifier.
The ROC Curve is a graphical representation of the binary classifier's true-positive rate (TPR) versus its false-positive rate (FPR) for various classification thresholds.
The ROC Curve is often utilized to evaluate the sensitivity and specificity of binary classifiers. Since an ROC Curve can only be produced for binary classifiers, it is not appropriate for classifiers with more than two classes.
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Compute the following limit using L'Hospital's rule if appropriate. Use INF to denote oo and MINF to denote -oo.
lim x -> [infinity] (1 - 4/x)^x =
To compute the limit of the function (1 - 4/x)^x as x approaches infinity, we can apply L'Hôpital's rule.
Let's rewrite the function as:
f(x) = (1 - 4/x)^x
Taking the natural logarithm of both sides:
ln(f(x)) = ln[(1 - 4/x)^x]
Using the property ln(a^b) = b * ln(a):
ln(f(x)) = x * ln(1 - 4/x)
Now, we can find the limit of ln(f(x)) as x approaches infinity:
lim x -> infinity ln(f(x)) = lim x -> infinity x * ln(1 - 4/x)
This is an indeterminate form of infinity times zero. We can apply L'Hôpital's rule by taking the derivative of the numerator and denominator:
lim x -> infinity ln(f(x)) = lim x -> infinity [ln(1 - 4/x) - (x * (-4/x^2))] / (-4/x)
Simplifying the expression:
lim x -> infinity ln(f(x)) = lim x -> infinity [ln(1 - 4/x) + 4/x] / (-4/x)
As x approaches infinity, both ln(1 - 4/x) and 4/x approach 0:
lim x -> infinity ln(f(x)) = lim x -> infinity [0 + 0] / 0
This is an indeterminate form of 0/0. We can apply L'Hôpital's rule again by taking the derivative of the numerator and denominator:
lim x -> infinity ln(f(x)) = lim x -> infinity [(d/dx ln(1 - 4/x)) + (d/dx 4/x)] / (d/dx (-4/x))
Differentiating each term:
lim x -> infinity ln(f(x)) = lim x -> infinity [(-4/(x - 4)) * (-1/x^2) + (-4/x^2)] / (4/x^2)
Simplifying the expression:
lim x -> infinity ln(f(x)) = lim x -> infinity [4/(x - 4x) - 4] / (4/x^2)
As x approaches infinity, (x - 4x) becomes -3x:
lim x -> infinity ln(f(x)) = lim x -> infinity [4/(-3x) - 4] / (4/x^2)
Simplifying further:
lim x -> infinity ln(f(x)) = lim x -> infinity [-4/(3x) - 4] / (4/x^2)
Taking the limit as x approaches infinity, the terms with x in the denominator approach 0:
lim x -> infinity ln(f(x)) = [-4/(3 * infinity) - 4] / 0
Simplifying:
lim x -> infinity ln(f(x)) = (-4/INF - 4) / 0 = (-4/INF) / 0 = 0/0
Once again, we have an indeterminate form of 0/0. We can apply L'Hôpital's rule one more time:
lim x -> infinity ln(f(x)) = lim x -> infinity [(d/dx (-4/(3x))) + (d/dx -4)] / (d/dx 0).
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Solve the equation and in the answer sheet write down the sum of
the roots of the equation.
Solve the equation of the equation. 5x-2 x²+3x-1 3 4 = -1 and in the answer sheet write down the sum of the roots
The given equation is 5x - 2x² + 3x - 1/3 + 4 = -1 . The sum of the roots of the quadratic equation ax² + bx + c = 0. The sum of the roots of the equation is 4.
Step by step answer:
Step 1: Rearrange the equation5x - 2x² + 3x + 1/3 + 4 + 1 = 0 Multiplying the whole equation by 3, we get,15x - 6x² + 9x + 1 + 12 + 3 = 0
Step 2: Simplify the equation-6x² + 24x + 16 = 0 Dividing the whole equation by -2, we get,3x² - 12x - 8 = 0
Step 3: Find the roots of the quadratic equation
3x² - 12x - 8
= 0ax² + bx + c
= 0x
= [-b ± √(b² - 4ac)] / 2a
Here, a = 3,
b = -12,
c = -8x
= [12 ± √(12² - 4(3)(-8))] / 2(3)x
= [12 ± √216] / 6x
= [12 ± 6√6] / 6x
= 2 ± √6
Therefore, the roots of the quadratic equation are 2 + √6 and 2 - √6
Step 4: Find the sum of the roots The sum of the roots of the quadratic equation ax² + bx + c = 0 is given by the formula, Sum of roots = -b/a Here,
a = 3 and
b = -12
Sum of roots = -b/a= -(-12) / 3
= 4
Hence, the sum of the roots of the equation is 4.
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Let r be a primitive root of the odd prime p. Prove the following:
If p = 3 (mod4), then -r has order (p - 1)/2 modulo p.
Let r be a primitive root of the odd prime p.
Then, r has order (p - 1) modulo p.
This indicates that $r^{p-1} \equiv 1\pmod{p}$.
Therefore, $r^{(p-1)/2} \equiv -1\pmod{p}$.
Also, we can write that $(p-1)/2$ is an odd integer.
As p is 3 (mod 4), we can say that $(p-1)/2$ is an odd integer.
For example, when p = 7, (p-1)/2 = 3.
Let's consider $(-r)^{(p-1)/2} \equiv (-1)^{(p-1)/2} \cdot r^{(p-1)/2} \pmod{p}$;
as we know, $(p-1)/2$ is odd, we can say that $(-1)^{(p-1)/2} = -1$.
Therefore, $(-r)^{(p-1)/2} \equiv -1 \cdot r^{(p-1)/2} \equiv -1 \cdot (-1) = 1 \pmod{p}$.
This shows that the order of $(-r)^{(p-1)/2}$ modulo p is (p-1)/2.
As $(-r)^{(p-1)/2}$ has order (p-1)/2 modulo p, then -r has order (p-1)/2 modulo p.
This completes the proof.
The word "modulus" has not been used in the solution as it is a technical term in number theory and it was not necessary for this proof.
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Suppose x and y are positive real numbers. If x < y, then x^2 < y^2. Prove the statement using the method of direct proof.
Given that x and y are positive real numbers and x < y, we have to prove that x² < y² by direct proof. Method of direct proof Let P and Q are statements. To prove P → Q by the direct proof, we assume that P is true. Then we use only logic and the given information to prove that Q is true. It is also called a proof by deduction. Now, let's begin the proof. Assume that x < y, where x and y are positive real numbers. Squaring both sides, we get$x^2 < y^2$Therefore, it is proved that x² < y² by direct proof.
Hence, we have proved that if x < y, then x² < y² using the method of direct proof.
To prove the statement "If x < y, then x² < y²" using a direct proof, we will assume the premise that x < y and then show that x² < y².
Let's proceed with the direct proof:
Assumption: x < y
To prove: x² < y²
Proof:
Since x < y, we can multiply both sides of the inequality by x and y, respectively, without changing the inequality direction because both x and y are positive:
x * x < x * y (multiplying both sides by x)
y * x < y * y (multiplying both sides by y)
Simplifying the inequalities:
x² < xy
yx < y²
Since x < y, we know that xy < y² because multiplying a smaller number by y will result in a smaller product than multiplying y by itself.
Combining the two inequalities:
x² < xy < y²
Therefore, x² < y²
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\Use the chain rule to find the partial derivatives w = xy + yz + zx, x = rcose, y = rsine, z = r0,- , when r = 2,0 = = aw aw ar' de Q3(c). A rectangular box without a lid to be made from 12m² of cardboard. Find the maximum volume of such a box.
To find the maximum volume of a rectangular box made from 12m² of cardboard, we need to maximize the volume function subject to the constraint that the surface area is equal to 12m².
Let's denote the length, width, and height of the box as x, y, and z, respectively. The volume of the box is given by V = xyz. According to the given information, the surface area of the box is 12m², which gives us the constraint equation 2xy + 2xz + 2yz = 12. To find the maximum volume, we can use the method of Lagrange multipliers. We define the Lagrangian function L(x, y, z, λ) as the volume function V minus the constraint equation multiplied by a Lagrange multiplier λ:
L(x, y, z, λ) = xyz - λ(2xy + 2xz + 2yz - 12)
Next, we need to find the partial derivatives of L with respect to x, y, z, and λ, and set them equal to zero to find the critical points.
∂L/∂x = yz - 2λy - 2λz = 0
∂L/∂y = xz - 2λx - 2λz = 0
∂L/∂z = xy - 2λx - 2λy = 0
∂L/∂λ = 2xy + 2xz + 2yz - 12 = 0
Solving this system of equations will give us the critical points. From there, we can determine which point(s) correspond to the maximum volume. Once we find the critical points, we substitute their values into the volume function V = xyz to calculate the corresponding volumes. The largest volume among these points will be the maximum volume of the box. By comparing the volumes obtained at the critical points, we can determine the maximum volume of the rectangular box that can be made from 12m² of cardboard.
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Solve the system. Give answers as (x, y, z)
6x-3y-5z= -21
12x+3y-4z= 12
-24x + 3y + 1z = -9
Therefore, the solution of the system is (x, y, z) = (-5/3, -10.067, -2.8).
(x, y, z) = (-5/3, -10.067, -2.8).
The given system of linear equations is 6x - 3y - 5z = -21, 12x + 3y - 4z = 12 and -24x + 3y + z = -9.
To solve the system, we'll use elimination method to find the values of x, y, and z:1.
Multiply the first equation by 2:6x - 3y - 5z = -2112x - 6y - 10z = -42
Adding both equations will eliminate y and z:18x = -30x = -30/18x = -5/32.
Substituting the value of x in the first and third equation will eliminate y:-24(-5/3) + 3y + z = -9-40 + 3y + z = -9
→ 3y + z = 31 ... (i)6(-5/3) - 3y - 5z = -21-10 + 3y + 5z = 21
→ 3y + 5z = 31 ... (ii)From (i) and (ii), we have:
3y + z = 31 ... (i)
3y + 5z = 31 ... (ii)
Multiplying (i) by -5 and adding to (ii) will eliminate
y:3y + z = 31 ... (i)-15y - 5z = -155z = -14z = 14/-5z = -2.8
Substituting z = -2.8 and x = -5/3 in the second equation will give y:-24(-5/3) + 3y - 2.8 = -9 40 + 3y - 2.8 = -9 3y = -30.2y = -10.067
Therefore, the solution of the system is (x, y, z) = (-5/3, -10.067, -2.8).
(x, y, z) = (-5/3, -10.067, -2.8).
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A rectangular pond has a width of 50m and a length of 400m. The area of the pond covered by an alga is denoted by A (in mm²) and is measured at time t (in weeks) after a biologist begins to observe the growth. The rate at which A is changing can be modelled as be modelled as being proportional to √Ā. Initially the algae cover an area of 900m² and three weeks later this has increased to 1296m². How many days after the initial observation will it take for the algae to cover more than 10% of the pond's surface?
To determine the number of days it will take for the algae to cover more than 10% of the pond's surface, we need to find the relationship between the area covered by the algae and time.
The rate of change of the area is proportional to the square root of the area. By setting up a differential equation and solving it, we can find the time required for the algae to exceed 10% of the pond's surface area.
Let A(t) represent the area covered by the algae at time t. According to the problem, the rate of change of A is proportional to √A. This can be expressed as dA/dt = k√A, where k is the constant of proportionality.
We know that initially, A(0) = 900 m², and after three weeks, A(3) = 1296 m².
To find the value of k, we can substitute the given values into the differential equation:
dA/dt = k√A
√A dA = k dt
Integrating both sides, we have:
(2/3)[tex]A^(3/2)[/tex] = kt + C
Using the initial condition A(0) = 900, we can solve for C:
(2/3)[tex](900)^(3/2)[/tex] = k(0) + C
C = (2/3)[tex](900)^(3/2)[/tex]
Now we can solve for the time when the algae covers more than 10% of the pond's surface area, which is 0.10 * (50m * 400m) = 2000 m²:
(2/3)[tex]A^(3/2)[/tex] = kt + (2/3)[tex](900)^(3/2)[/tex]
Solving for t, we find the number of days it will take for the algae to exceed 10% of the pond's surface area.
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n 3n2 + n. 2. For every integer n > 1, prove that Σ(6i – 2) 1=1
Answer:
Here the answer
Step-by-step explanation:
Hope you get it
For the curve y = 3x², find the slope of the tangent line at the point (3, 7). O a. 14 b. 18 O c. 13 O d. 6
The slope of the tangent line at the point (3, 7) for the curve y = 3x² is 18.
To find the slope of the tangent line at a given point on a curve, we need to take the derivative of the curve equation with respect to x. The derivative represents the rate of change of the curve at any given point.
For the equation y = 3x², we can take the derivative using the power rule of differentiation. The power rule states that if we have a term of the form a[tex]x^n[/tex], the derivative will be na[tex]x^{(n-1)}[/tex]. Applying this rule, the derivative of 3x² becomes:
dy/dx = d/dx (3x²)
= 2 * 3[tex]x^{(2-1)[/tex]
= 6x
Now we have the derivative, which represents the slope of the curve at any point. To find the slope at the point (3, 7), we substitute x = 3 into the derivative:
dy/dx = 6(3)
= 18
Therefore, the slope of the tangent line at the point (3, 7) is 18.
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For each probability and percentile problem, draw the picture. A random number generator picks a number from 1 to 8 in a uniform manner. Part (a) Give the distribution of X.
Part (b) Part (c) Enter an exact number as an integer, fraction, or decimal. f(x) = ____, where ____
Part (d) Enter an exact number as an integer, fraction, or decimal. μ = ___
Part (e) Round your answer to two decimal places. σ = ____
Part (f) Enter an exact number as an integer, fraction, or decimal. P(3.75 < x < 7.25) = ____
Part (g) Round your answer to two decimal places. P(x > 4.33) =____ Part (h) Enter an exact number as an integer, fraction, or decimal. P(x > 5 | x > 3) =____ Part (i) Find the 90th percentile. (Round your answer to one decimal place.)
To answer the given probability and percentile problems, let's go through each part step by step.
(a) The distribution of X is a discrete uniform distribution with values ranging from 1 to 8, inclusive.
(b) The probability mass function (PMF) is given by:
f(x) = 1/8 for x = {1, 2, 3, 4, 5, 6, 7, 8}; 0 otherwise
(c) The PMF is:
f(x) = 1/8, where x = {1, 2, 3, 4, 5, 6, 7, 8}
(d) The mean (μ) is the average of the values in the distribution, which in this case is:
μ = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8
= 4.5
(e)The standard deviation (σ) is a measure of the dispersion of the values in the distribution. For a discrete uniform distribution, it can be calculated using the formula:
σ = [tex]\sqrt{{((n^2 - 1) / 12)\\} }[/tex], where n is the number of values in the distribution.
In this case, n = 8, so:
σ =[tex]\sqrt{ ((8^2 - 1) / 12)\\}[/tex]
= [tex]\sqrt{(63 / 12)}[/tex]
≈ 2.29
(f) To find the probability of a specific range, we need to calculate the cumulative probability for the lower and upper bounds and subtract them.
P(3.75 < x < 7.25) = P(x < 7.25) - P(x < 3.75)
Since the distribution is discrete, we round the bounds to the nearest whole number:
P(x < 7.25) = P(x ≤ 7)
= 7/8
P(x < 3.75) = P(x ≤ 3)
= 3/8
P(3.75 < x < 7.25) = (7/8) - (3/8)
= 4/8
= 1/2
= 0.5
(g) To find the probability of x being greater than a specific value, we need to calculate the cumulative probability for that value and subtract it from 1.
P( > 4.33) = 1 - P(x ≤ 4)
= 1 - 4/8
= 1 - 1/2
= 1/2
= 0.5
(h) To find the conditional probability of x being greater than 5 given that x is greater than 3, we calculate:
P(x > 5 | x > 3) = P(x > 5 and x > 3) / P(x > 3)
Since the condition "x > 3" is already satisfied, we only need to consider the probability of x being greater than 5:
P(x > 5 | x > 3) = P(x > 5)
= 1 - P(x ≤ 5)
= 1 - 5/8
= 3/8
= 0.375
(i) The percentile represents the value below which a given percentage of observations falls.
To find the 90th percentile, we need to determine the value x such that 90% of the observations fall below it.
For a discrete uniform distribution, each value has an equal probability, so the 90th percentile corresponds to the value at the 90th percentile rank.
Since the distribution has 8 values, the 90th percentile rank is:
90th percentile rank = (90/100) * 8
= 7.2
Since the values are discrete, we round up to the nearest whole number:
90th percentile ≈ 8
Therefore, the 90th percentile is 8 (rounded to one decimal place).
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Question (1): (20 points) The input to a weakly symmetric channel is a two-symbol alphabet Ex = {A, B}. The output of the channel is a three-symbol alphabet Ey = { C, D, E} according to the following: If the input is A, the output is either C or D or E with probabilities (1/3, 1/6, 1/2), respectively. If the input is B, the output is either C or D or E with probabilities (1/3, 1/2, 1/6), respectively. Find the channel transition matrix Q. (5 points) (10 points) Compute the channel capacity if the input symbols are equiprobable. Compute log() - H(column of Q) and comment on its value. (5 points)
The channel transition matrix Q for the given weakly symmetric channel can be calculated as follows:
The input alphabet Ex = {A, B} has 2 symbols, and the output alphabet Ey = {C, D, E} has 3 symbols. The probabilities of the output symbols given the input symbols are provided.
To construct the channel transition matrix Q, we assign the probabilities to each entry in the matrix. The rows of the matrix represent the input symbols, and the columns represent the output symbols.
Using the given probabilities, we have:
Q =
| 1/3 1/6 1/2 |
| 1/3 1/2 1/6 |
The channel capacity can be computed using the formula:
C = max[ΣΣ p(x) p(y|x) log2(p(y|x) / p(y))]
In this case, since the input symbols are equiprobable, p(A) = p(B) = 1/2. We can calculate the conditional probabilities p(y|x) and the marginal probabilities p(y) using the channel transition matrix Q.
The column probabilities of Q represent the marginal probabilities p(y). Therefore:
p(C) = 1/3 + 1/3 = 2/3
p(D) = 1/6 + 1/2 = 2/3
p(E) = 1/2 + 1/6 = 2/3
Substituting these values into the channel capacity formula and calculating the values for each output symbol, we obtain:
C = (1/2 * 2/3 * log2(2/3 / 2/3)) + (1/2 * 2/3 * log2(2/3 / 2/3)) + (1/2 * 2/3 * log2(2/3 / 2/3)) = 0
The value log2(1) = 0 indicates that the output symbols do not provide any additional information beyond what is already known from the input symbols.
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problem 1: let's calculate the average density of the red supergiant star betelgeuse. betelgeuse has 16 times the mass of our sun and a radius of 500 million km. (the sun has a mass of 2 × 1030 kg.)
The average density of the red supergiant star Betelgeuse is 1.45 × 10⁻¹¹ kg/m³.
To calculate the average density of the red supergiant star Betelgeuse,
we need to use the formula for average density, which is:
Average density = Mass/VolumeHere,
Betelgeuse has 16 times the mass of our sun.
Therefore, its mass (M) is given by:
M = 16 × (2 × 10²³) kg
M = 32 × 10²³ kg
M = 3.2 × 10²⁴ kg
Betelgeuse has a radius (r) of 500 million km.
We need to convert it to meters:r = 500 million
km = 500 × 10⁹ m
The volume (V) of Betelgeuse can be calculated as:
V = 4/3 × π × r³V = 4/3 × π × (500 × 10⁹)³
V = 4/3 × π × 1.315 × 10³⁵V = 2.205 × 10³⁵ m³
Therefore, the average density (ρ) of Betelgeuse can be calculated as:
ρ = M/Vρ = (3.2 × 10²⁴) / (2.205 × 10³⁵)
ρ = 1.45 × 10⁻¹¹ kg/m³
Thus, the average density of the red supergiant star Betelgeuse is 1.45 × 10⁻¹¹ kg/m³.
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All vectors and subspaces are in R". Check the true statements below: A. If W is a subspace of R" and if v is in both W and W, then v must be the zero vector. B. In the Orthogonal Decomposition Theorem, each term y=y.u1/u1.u1 u1 +.... + y.up/up.up up is itself an orthogonal projection of y onto a subspace of W.
C. If y = 21 + 22, where 2₁ is in a subspace W and z2 is in W, then 2₁ must be the orthogonal projection of Y onto W. D. The best approximation to y by elements of a subspace W is given by the vector y – projw(y). E. If an n x p matrix U has orthonormal columns, then UUT x = x for all x in R".
A. The statement given is true.
This is because if v is in both W and W, then it must be the zero vector.
B. The statement given is also true. In the Orthogonal Decomposition Theorem, each term
y=y.u1/u1.u1 u1 +.... + y.up/up.up up is itself an orthogonal projection of y onto a subspace of W. C.
The best approximation to y by elements of a subspace W is given by the vector y – projw(y).E. If an n x p matrix U has orthonormal columns, then UUT x = x for all x in R".The summary of the answers are:A is true.B is true.C is false.D is true.E is true.
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4. (20) In two jars (jar-1, jar-2) containing black and white balls, the probability of drawing a white ball from jar-1 is equal to drawing a black ball from jar-2. The balls are drawn according to the following rules: • The balls are drawn without replacement (i.e. the ball drawn is put back to the jar). • If a black ball is drawn, the next ball is drawn from the other jar. Else the next ball is drawn from the same jar. If an is the probability of having nth draw from jar-1 (a) (10) Prove that an+1 equals drawing a black ball from jar-2 (b) (10) If the first ball is drawn from jar-1, what is the probability of drawing 1000th ball from jar-1?
(a) an+1 = probability of drawing a black ball from jar-2 (b) The probability of drawing the 1000th ball from jar-1, given that the first ball was drawn from jar-1, is the same as the probability of drawing a white ball from jar-1.
How to calculate probabilities in ball-drawing scenario?(a) To prove that an+1 equals drawing a black ball from jar-2, we can analyze the different possibilities for the nth draw:
1. If the nth draw is from jar-1 and a white ball is drawn, then an+1 will be equal to an (drawing from jar-1 again).
2. If the nth draw is from jar-1 and a black ball is drawn, then an+1 will be equal to the probability of drawing a black ball from jar-2 (since the next draw will be from jar-2).
3. If the nth draw is from jar-2 and a white ball is drawn, then an+1 will be equal to the probability of drawing a white ball from jar-1 (since the next draw will be from jar-1).
4. If the nth draw is from jar-2 and a black ball is drawn, then an+1 will be equal to an (drawing from jar-2 again).
Based on these possibilities, it can be concluded that an+1 equals drawing a black ball from jar-2.
(b) If the first ball is drawn from jar-1, the probability of drawing the 1000th ball from jar-1 can be calculated as the product of probabilities for each draw. Since the balls are drawn with replacement (put back after each draw), the probability of drawing a ball from jar-1 remains the same for each draw. Therefore, the probability of drawing the 1000th ball from jar-1 is the same as the probability of drawing the first ball from jar-1, which is given as the probability of drawing a white ball from jar-1.
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For the matrix A shown below, x = (0, 1,-1) is an eigenvector corresponding to a second order eigenvalue X. Use x to find X. Hence determine a vector of the form y = (1, a, b) such that x and y form an orthogonal basis for the subspace spanned by the eigenvectors coresponding to eigenvalue X. 1 2 2 A = 1 2 -1 -1 1 4 Enter your answers as follows: If any of your answers are integers, you must enter them without a decimal point, e.g. 10 • If any of your answers are negative, enter a leading minus sign with no space between the minus sign and the number. You must not enter a plus sign for positive numbers. If any of your answers are not integers, then you must enter them with at most two decimal places, e.g. 12.5 or 12.34, rounding anything greater or equal to 0.005 upwards. Do not enter trailing zeroes after the decimal point, e.g. for 1/2 enter 0.5 not 0.50. These rules are because blackboard does an exact string match on your answers, and you will lose marks for not following the rules. Your answers: a: b:
For the dot product to be zero, a must be equal to b. So, we can choose a = b , a vector y of the form (1, a, a) will form an orthogonal basis with x.
To find the eigenvalue corresponding to the eigenvector x = (0, 1, -1), we need to solve the equation Ax = Xx, where A is the given matrix. Substituting the values, we have:
A * (0, 1, -1) = X * (0, 1, -1)
Simplifying, we get:
(2, -1, 1) = X * (0, 1, -1)
From the equation, we can see that the second component of the vector on the left side is -1, while the second component of the vector on the right side is X. Therefore, we can conclude that X = -1.
To find a vector y = (1, a, b) that forms an orthogonal basis with x, we need y to be orthogonal to x. This means their dot product should be zero. The dot product of x and y is given by:
x · y = 0 * 1 + 1 * a + (-1) * b = a - b
For the dot product to be zero, a must be equal to b. So, we can choose a = b. a vector y of the form (1, a, a) will form an orthogonal basis with x.
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find the radius of convergence, r, of the series.[infinity](−9)nnnxnn = 1
The radius of convergence, r, of the series is 1/9.
To obtain the radius of convergence, we can use the ratio test.
The ratio test states that if we have a power series of the form ∑(aₙxⁿ), then the radius of convergence, r, is given by:
r = lim┬(n→∞)|aₙ/aₙ₊₁|
In this case, we have the series ∑((-9)ⁿⁿ/n!)xⁿ.
Let's apply the ratio test to find the radius of convergence.
We start by evaluating the ratio:
|aₙ/aₙ₊₁| = |((-9)ⁿⁿ/n!)xⁿ / ((-9)ⁿ⁺¹⁺¹/(n+1)!)xⁿ⁺¹|
= |-9ⁿ⁺¹⁺¹xⁿ / (-9)ⁿⁿ⁺¹ xⁿ⁺¹(n+1)/n!|
Simplifying the expression:
|aₙ/aₙ₊₁| = |(-9)(n+1)/(n+1)|
= 9
Taking the limit as n approaches infinity:
lim┬(n→∞)|aₙ/aₙ₊₁| = 9
Since the limit is a finite positive number (9), the radius of convergence is given by:
r = 1 / lim┬(n→∞)|aₙ/aₙ₊₁| = 1/9
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Write the system of linear equations in the form Ax = b and solve this matrix equation for x. -2x1 3x2 -11 6x1 + X2 H -39 CHCE =
The given system of linear equations is as follows:-2x1 + 3x2 = -11 (Equation 1)6x1 + x2 = -39 (Equation 2)To write the above system of linear equations in the form Ax = b.
we can represent it as given below:
A = [ -2 3 ; 6 1 ]
x = [ x1 ; x2 ]
b = [ -11 ; -39 ]
Therefore, Ax = b becomes [ -2 3 ; 6 1 ] [ x1 ; x2 ] = [ -11 ; -39 ]Now, to solve this matrix equation, we need to find the inverse of matrix A. Let A^-1 be the inverse of matrix A, then we can write x = A^-1 b
So, first we find the determinant of matrix A using the formula: Determinant of
A = (ad - bc)
where, a = -2, b = 3, c = 6 and d = 1.So, Determinant of A = (-2)(1) - (3)(6) = -20
As the determinant is not equal to zero, the inverse of matrix A exists. Now, we find the inverse of matrix A using the formula: A^-1 = (1/Determinant of A) [ d -b ; -c a ]where, a = -2, b = 3, c = 6 and d = 1.So, A^-1 = (1/-20) [ 1 -3 ; -6 -2 ]= [ -1/20 3/20 ; 3/10 1/10 ]
Now, we can find the solution to the given system of linear equations as follows:
x = A^-1 b= [ -1/20 3/20 ; 3/10 1/10 ] [ -11 ; -39 ]
= [ 2 ; -5 ]
Therefore, the solution to the given system of linear equations isx1 = 2 and x2 = -5.
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show working out clearly
A. Given the function f(x) = x(3x - x²). Determine: i. The critical value/s; ii. The nature of the critical point/s. (4 marks) (6 marks)
The function f(x) = x(3x - x²) can be written as f(x) = 3x² - x³, and we will find its critical value/s and the nature of the critical point/s.i).
To find the critical value/s, we need to find the derivative of the function: `f'(x) = 6x - 3x²`. Now we need to solve for x to get the critical values:`f'(x) = 0`Solving for x, we get:`6x - 3x² = 0`Factorizing, we get:`3x(2 - x) = 0`So the critical values are x = 0 and x = 2.ii) To find the nature of the critical points, we can use the second derivative test. We know that `f''(x) = 6 - 6x`.Substituting x = 0, we get:`f''(0) = 6 - 0 = 6`Since `f''(0) > 0`, the function has a local minimum at x = 0.Substituting x = 2, we get:`f''(2) = 6 - 12 = -6`Since `f''(2) < 0`, the function has a local maximum at x = 2.Therefore, the critical values are x = 0 and x = 2, and the nature of the critical points is a local minimum at x = 0 and a local maximum at x = 2.
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How does the formula for determining degrees of freedom in
chi-square differ from the formula in t-tests and ANOVA?
For one-way ANOVA, the degrees of freedom are calculated using the formula:df = k - 1where k is the number of groups being compared. For two-way ANOVA, the degrees of freedom are calculated using the formula:df = (a-1)(b-1)where a is the number of levels in factor A and b is the number of levels in factor B.
The formula for determining degrees of freedom in chi-square is different from the formula in t-tests and ANOVA in several ways. Chi-square tests are used to examine the relationship between categorical variables, while t-tests and ANOVA are used to compare means between two or more groups. The degrees of freedom in a chi-square test depend on the number of categories being compared, while in t-tests and ANOVA, the degrees of freedom depend on the number of groups being compared.
In chi-square, the degrees of freedom are calculated using the formula:df = (r-1)(c-1)where r is the number of rows and c is the number of columns in the contingency table. T-tests and ANOVA, on the other hand, have different formulas for calculating degrees of freedom depending on the type of test being conducted. For a two-sample t-test, the degrees of freedom are calculated using the formula:df = n1 + n2 - 2where n1 and n2 are the sample sizes for each group.
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let , be vectors in given by a) find a vector with the following properties: for any linear transformation which satisfies we must have . enter the vector in the form
If the result is zero, then we need to choose another vector and repeat the process. Therefore, we choose any non-zero vector and apply T to it.
Given, vectors , are given as:
We need to find a vector such that for any linear transformation T satisfying we must have , i.e.,
Here, is the null space of the linear transformation T.
Let us first find the basis for the null space of T.
Let be the matrix representing the linear transformation T with respect to the standard basis.
Since the columns of A represent the images of the standard basis vectors under T, the null space of A is precisely the space of all linear combinations of the vectors that map to zero.
Therefore, we can find a basis for the null space of A by computing the reduced row echelon form of A and looking for the special solutions of the corresponding homogeneous system.
Now, we need to find a vector which is not in the null space of T.
This can be done by taking any non-zero vector and applying T to it. If the result is non-zero, then we have found our vector.
If the result is zero, then we need to choose another vector and repeat the process.
Therefore, we choose any non-zero vector and apply T to it.
Let . Then,
Since this is non-zero, we have found our vector. Therefore, we can take as our vector.
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Use the information below to find the probability that a flight arrives on time given that it departed on time.
The probability that an airplane flight departs on time is 0.890
The probability that a flight arrives on time is 0.87
The probability that a flight departs and arrives on time is 0.83
The probability that a flight arrives on time given that it departed on time is.......
Therefore, the probability that a flight arrives on time given that it departed on time is approximately 0.932.
To find the probability that a flight arrives on time given that it departed on time, we can use the formula for conditional probability:
P(Arrival on time | Departure on time) = P(Arrival on time and Departure on time) / P(Departure on time)
From the given information, we have:
P(Arrival on time and Departure on time) = 0.83
P(Departure on time) = 0.890
Plugging these values into the formula, we get:
P(Arrival on time | Departure on time) = 0.83 / 0.890 ≈ 0.932
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Solve the following eigenvalue problem AX = 2X, 1-1 1 A= 1 1 1 1 1 1
The eigenvalues and eigenvectors of matrix $A$ are,λ = 0, with eigenvector $X_1 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$λ = 3, with eigenvectors $X_2 = \begin{bmatrix}1\\1\\1\end{bmatrix}$ and $X_3 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$.
The given eigenvalue problem is, $AX=2X$,
where $A=\begin{bmatrix}1 & -1 & 1\\1 & 1 & 1\\1 & 1 & 1\end{bmatrix}$First, we need to find the eigenvalues of matrix $A$.
The characteristic equation of matrix $A$ is given by,|A-λI| = 0Where, λ is the eigenvalue and I is the identity matrix of order 3.
Substituting A, we get,$\begin{vmatrix}1-λ & -1 & 1\\1 & 1-λ & 1\\1 & 1 & 1-λ\end{vmatrix}=0$Expanding the above determinant,
we get,$\begin{aligned}&(1-λ)\begin{vmatrix}1-λ & 1\\1 & 1-λ\end{vmatrix}-\begin{vmatrix}-1 & 1\\1 & 1-λ\end{vmatrix}+\begin{vmatrix}-1 & 1-λ\\1 & 1\end{vmatrix}\\&=(1-λ)[(1-λ)^2-1]-[(-1)(1-λ)-(1)(1)]+[-1(1-λ)-1(1)]\\&=(λ-3)λ^2=0\end{aligned}$Hence, the eigenvalues of matrix $A$ are λ = 0, λ = 3.
Now, we need to find the eigenvectors corresponding to the eigenvalues of matrix $A$.For λ = 0,$(A-0I)X=0$Therefore, $\begin{bmatrix}1 & -1 & 1\\1 & 1 & 1\\1 & 1 & 1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$
On solving, we get the eigenvector as,$X_1 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$For λ = 3,$(A-3I)X=0$Therefore, $\begin{bmatrix}-2 & -1 & 1\\1 & -2 & 1\\1 & 1 & -2\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$On solving,
we get the eigenvectors as,$X_2 = \begin{bmatrix}1\\1\\1\end{bmatrix}$ and $X_3 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$Therefore, the eigenvalues and eigenvectors of matrix $A$ are,λ = 0,
with eigenvector $X_1 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$λ = 3, with eigenvectors $X_2 = \begin{bmatrix}1\\1\\1\end{bmatrix}$ and $X_3 = \begin{bmatrix}-1\\1\\0\end{bmatrix}$.
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find a power series representation for the function. (give your power series representation centered at x = 0.) f(x)=1/(3 x)
The power series representation for the function is [tex]f(x) = \sum\limits^{\infty}_{0} {(-\frac x3)^n}[/tex]
How to find the power series for the functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = 1/(3 + x)
Rewrite the function as
[tex]f(x) = \frac{1}{3(1 + \frac x3)}[/tex]
Expand
[tex]f(x) = \frac{1}{3(1 - - \frac x3)}[/tex]
So, we have
[tex]f(x) = \frac{1}{3} * \frac{1}{(1 - (-\frac x3)}[/tex]
The power series centered at x = 0 can be calculated using
[tex]f(x) = \sum\limits^{\infty}_{0} {r^n}[/tex]
In this case
r = -x/3 i.e. the expression in bracket
So, we have
[tex]f(x) = \sum\limits^{\infty}_{0} {(-\frac x3)^n}[/tex]
Hence, the power series for the function is [tex]f(x) = \sum\limits^{\infty}_{0} {(-\frac x3)^n}[/tex]
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Question
Find a power series representation for the function. (give your power series representation centered at x = 0
f(x) = 1/(3 + x)