The curve starts at (1,1) and goes to the right, approaching the x-axis but never touching it. It also approaches the y-axis but never touches it. The curve is traced in the direction from (1,1) towards the positive x-axis as the parameter t increases.
To eliminate the parameter, we can solve for t in terms of x and substitute into the equation for y:
x = et --> t = ln(x)
y = e⁽⁻⁴ᵗ⁾ = e⁽⁻⁴⁾ln(x)) = x⁽⁻⁴⁾
So the Cartesian equation of the curve is y = x⁽⁻⁴⁾.
To sketch the curve, we can notice that as x increases, y decreases rapidly (since it is raised to the negative fourth power). The curve approaches the y-axis but never touches it. It also approaches the x-axis but is never quite horizontal. To indicate the direction in which the curve is traced as the parameter increases, we can use an arrow pointing to the right (since t = ln(x) increases as x increases).
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On an average, a metro train completes 4 round trips of 90 kilometres in a day. What is the average distance travelled by the metro?
On average, the metro train travels a distance of 90 kilometers in a single trip.
Since the metro train completes 4 round trips of 90 kilometres, the total distance traveled in a day would be 4290 = 720 kilometres (since a round trip is equivalent to two journeys of 90 kilometres).
To find the average distance traveled, we need to divide the total distance by the number of trips made. Since 4 round trips have been made, the number of trips made would be 4*2 = 8 (since each round trip is equivalent to 2 trips).
Therefore, the average distance traveled by the metro in a day would be 720/8 = 90 kilometres.
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Mrs. Smith has a bag containing colored counters, as shown below. Bag of Color Counters 2 If a student draws 1 counter out of the bag without looking, what is the probability that the counter will be orange?
If a student draws 1 counter out of the bag without looking, then the probability of drawing an orange counter is 0.25 or 25%.
What is probability?Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain to occur.
Probability is usually expressed as a fraction, decimal, or percentage. For example, if the probability of an event occurring is 0.5, this means there is a 50% chance that the event will occur.
The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if a fair six-sided die is rolled, the probability of rolling a 3 is 1/6, because there is one favorable outcome (rolling a 3) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
According to the given informationThe probability of drawing an orange counter out of the bag can be calculated by dividing the number of orange counters by the total number of counters in the bag.
The total number of counters in the bag is:
6 + 2 + 10 + 6 = 24
The number of orange counters is:
6
Therefore, the probability of drawing an orange counter is:
6/24 = 1/4 = 0.25
So the probability of drawing an orange counter is 0.25 or 25%.
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Find the inverse laplace transform of F(s)=(8s^2-4s+12)/s(s^2+4)
The inverse Laplace transform of F(s)=(8s^2-4s+12)/s(s^2+4) is 3 + 5cos(2t) - 2sin(2t).
Explanation:
To determine the inverse Laplace transform of(8s^2-4s+12)/s(s^2+4), follow these steps:
Step 1: To find the inverse Laplace transform of F(s)=(8s^2-4s+12)/s(s^2+4), use partial fraction decomposition:
F(s) = (A/s) + (Bs+C)/(s^2+4)
Step 2: Multiplying both sides by s(s^2+4) and equating coefficients, we get:
8s^2 - 4s + 12 = A(s^2+4) + (Bs+C)s
Simplifying, we get:
8s^2 - 4s + 12 = As^2 + 4A + B*s^2 + Cs
Equating coefficients, we get:
A + B = 8
C = -4
4A = 12
Solving for A, B, and C, we get:
A = 3
B = 5
C = -4
Therefore, we can write F(s) as:
F(s) = 3/s + (5s-4)/(s^2+4)
Step 3: Taking the inverse Laplace transform of each term separately, we get:
L^-1{3/s} = 3
L^-1{(5s-4)/(s^2+4)} = 5L^-1{s/(s^2+4)} - 2L^-1{1/(s^2+4)}
Using the table of Laplace transforms, we can find that:
L^-1{s/(s^2+4)} = cos(2t)
L^-1{1/(s^2+4)} = (1/2) sin(2t)
Therefore, the inverse Laplace transform of F(s) is:
L^-1{F(s)} = 3 + 5cos(2t) - 2sin(2t). where L^-1 denotes the inverse Laplace transform operator.
Therefore, the inverse Laplace transform of (8s^2-4s+12)/s(s^2+4) is 3 + 5cos(2t) - 2sin(2t).
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let f(x) = x4(x − 4)3. (a) find the critical numbers of the function f. (enter your answers from smallest to largest.)
The critical numbers of the function f(x) = [tex]x^{4} (x - 4)^{3}[/tex] are x = 0 and x = 4.
Find the critical numbers of the function f?To find the critical numbers of the function f(x) = [tex]x^{4}(x - 4)^{3}[/tex], we need to find the values of x at which the derivative of f(x) is equal to zero or undefined.
First, we will find the derivative of f(x) using the product rule:
f'(x) = [tex]4x^{3} (x - 4)^{3} + x^{4} 3(x - 4)^{2}(1)[/tex]
Simplifying this expression, we get:
f'(x) = [tex]4x^{3} (x - 4)^{2} (4 - x)[/tex]
Now, we can set f'(x) equal to zero and solve for x:
[tex]4x^{3} (x - 4)^{2} (4 - x)[/tex] = 0
From this equation, we can see that the critical numbers are x = 0, x = 4, and x = 4.
To check if x = 4 is a critical number, we need to find the limit of f'(x) as x approaches 4 from the left and from the right:
lim x→4- f'(x) = lim x→4- 4[tex]x^{3}[/tex][tex](x - 4)^{2}[/tex](4 - x) = 0
lim x→4+ f'(x) = lim x→4+ 4[tex]x^{3}[/tex][tex](x - 4)^{2}[/tex](4 - x) = 0
Since both limits are equal to zero, x = 4 is a critical number.
Therefore, the critical numbers of the function f(x) = [tex]x^{4} (x - 4)^{3}[/tex] are x = 0 and x = 4.
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complete the transformations below. then enter the final coordinates of the figure
The transformations of coordinates of A(2,2) is A'' = (-1, -1) , B(1, -2) is B'' = (0, 3) and C (4,-4) is C'' = (-3, 5).
The coordinates are given in the figure as,
A's coordinates are (2, 2) ;
B's coordinates are (1, -2) ;
and C's coordinates are (4, -4)
The coordinates are to be transformed as A", B" and C'' as,
First invert the number's sign and then to add up 1.
Therefore,
For A" :
At x- axis, +2 becomes -2 and the by adding 1 we get, -1
Similarly at y- axis, +2 becomes -2 and the by adding 1 we get, -1
Thus, A'' = (-1, -1)
For B" :
At x- axis, +1 becomes -1 and the by adding 1 we get, 0
Similarly at y- axis, -2 becomes +2 and the by adding 1 we get, +3
Thus, B'' = (0, 3)
For C" :
At x- axis, +4 becomes -4 and the by adding 1 we get, -3
Similarly at y- axis, -4 becomes +4 and the by adding 1 we get, +5
Thus, C'' = (-3, 5)
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Answer:A(-1,5) B(0,1) C(-3,-1)
Step-by-step explanation:I had the question
Find and calculate the value of c such that ∑ [infinity] n=0 e^nc = 3
The value of c is approximately -0.4055.
To find the value of c such that the sum ∑ (from n=0 to infinity) of e(nc) equals 3, we recognize this as a geometric series. For a geometric series to converge, the common ratio (r) must be between -1 and 1. In this case, r = ec.
The sum of an infinite geometric series is given by the formula S = a / (1 - r), where a is the first term and r is the common ratio.
In this problem, a = e(0c) = 1, and we want the sum S = 3. Plugging in the values:
3 = 1 / (1 - ec)
Now, solve for c:
1 - ec = 1/3
ec = 2/3
Take the natural logarithm (ln) of both sides:
ln(ec) = ln(2/3)
c = ln(2/3)
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Bus Company A claims that it is typically on time 95% of the time While Bus
Company B has a record of being on time 47 days out of the 50 days that it
operates. Which bus company seems to be doing b
Therefore, based on the information given, it seems that Bus Company A is doing slightly better in terms of on-time performance.
Which bus company seems to be doing better?To determine which bus company seems to be doing better, we need to compare their on-time performance.
Bus Company A claims that it is typically on time 95% of the time. This means that out of 100 trips, it expects to be on time for 95 of them.
On the other hand, Bus Company B has a record of being on time 47 days out of the 50 days that it operates. This means that its on-time performance is:
47/50 = 0.94 or 94%
Comparing the two percentages,
we can see that Bus Company A claims to have a higher on-time performance (95%) than Bus Company B's actual on-time performance (94%). Bus Company B's on-time performance is based on actual data from 50 days of operation.
Therefore, based on the information given, it seems that Bus Company A is doing slightly better in terms of on-time performance.
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Complete question:
Bus Company A claims that it is typically on time 95% of the time While Bus
Company B has a record of being on time 47 days out of the 50 days that it
operates. Which bus company seems to be doing better?
Segments HS and WB are equal in length. HS= (8x +15) and WB = (12-13). Which of the following is the value of x?
A) 3
B)4
C)6.5
D)7
Answer:lol it was 7
Step-by-step explanation:
a circle has an initial radius of 50 ft when the radius begins degreasing at the rate of 4 ft/min. what is the rate of change of area at the instant thate radius ois 20 ft?
The rate of change of the area at the instant when the radius is 20 ft is -160π square feet per minute.
How to find the area of a circle?The area of a circle is given by the formula A = π [tex]r^2[/tex], where r is the radius of the circle.
We are given that the radius of the circle is decreasing at a rate of 4 ft/min. This means that the rate of change of the radius with respect to time is -4 ft/min (negative because the radius is decreasing).
At the instant when the radius is 20 ft, we can calculate the rate of change of the area by taking the derivative of the area formula with respect to time:
dA/dt = d/dt (π[tex]r^2)[/tex]
Using the chain rule, we get:
dA/dt = 2πr (dr/dt)
Substituting r = 20 ft and dr/dt = -4 ft/min, we get:
dA/dt = 2π(20)(-4) = -160π
Therefore, the rate of change of the area at the instant when the radius is 20 ft is -160π square feet per minute.
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In a certain year, there were 80 days with measurable snowfall in Denver, and 63 days with measurable snowfall in Chicago. A meteorologist computes (80+1)/(365+2)=0.22,(63+1)/(365+2)=0.17,(80+1)/(365+2)=0.22,(63+1)/(365+2)=0.17, and proposes to compute a 95% confidence interval for the difference between the proportions of snowy days in the two cities as follows: 0.22−0.17±1.96(0.22)(0.78)367+(0.17)(0.83)3670.22−0.17±1.96367(0.22)(0.78)+367(0.17)(0.83)
Is this a valid confidence interval? Explain.
Yes, this is a valid confidence interval calculation for comparing the proportions of snowy days in Denver and Chicago.
The meteorologist is using a standard method for constructing a 95% confidence interval for the difference between two proportions. The formula applied is: (p1 - p2) ± z * sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)], where p1 and p2 are the proportions of snowy days in Denver and Chicago, respectively, n1 and n2 are the total number of days considered for each city, and z is the z-score corresponding to the desired level of confidence (1.96 for 95% confidence).In this case, p1 = 0.22, p2 = 0.17, n1 = n2 = 367 (365 days in a year plus 2 to account for the added 1 to both numerators). Plugging these values into the formula, the meteorologist computes the confidence interval as: 0.22 - 0.17 ± 1.96 * sqrt[(0.22 * 0.78 / 367) + (0.17 * 0.83 / 367)].This method assumes the samples are large enough and the proportions can be approximated by a normal distribution, which is reasonable given the sample sizes. The confidence interval provides an estimate of the range in which the true difference between the proportions of snowy days in the two cities lies, with 95% confidence.For more such question on confidence interval
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Syrinus and Natalia were filling their rectangular garden with dirt. The area of the garden is 30 feet. If the length of the garden is 6 feet, what is the width?
Answer:
The answer is 5ft
Step-by-step explanation:
A=L×B
A=L×W
30=6×W
30=6W
divide both sides by 6
W=5ft
The singular points of the differential equation y" + y'/x+y(x-2)/x-3=0 are Select the correct answer. a. 0 b. 0, 2, 3 c. 0, 3 d. 0, 2 e. none
The singular points of the differential equation are x=0 and x=3. the correct answer is (c) 0, 3.
The singular points of a differential equation are the points where the coefficients of y'', y' or y become infinite or undefined. In this case, the given differential equation is y" + y'/x + y(x-2)/(x-3) = 0.
To find the singular points, we need to check the coefficients of y'', y', and y for any infinite or undefined values.
- The coefficient of y'' is 1, which is finite for all values of x.
- The coefficient of y' is 1/x, which is infinite at x=0.
- The coefficient of y is (x-2)/(x-3), which is undefined at x=3.
Therefore, the singular points of the differential equation are x=0 and x=3. The correct answer is (c) 0, 3.
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Determine whether the nonhomogeneous system Ax = b is consistent, and if so, write the solution in the form x = xn + xp where xh is a solution of Ax = 0 and xp is a particular solution of Ax = b.
2x - 4y + 5z = 8
-7x + 14y + 4z = -28
3x - 6y + z = 12
The general solution of non-homogeneous system can be written as:
x = xh + xp = [2t + 1, t, -2s - 2] + [-1, -28, 1]
We can now write the augmented matrix of the system as:
[2 -4 5 8]
[-7 14 4 -28]
[3 -6 1 12]
We can use row reduction to determine whether the system is consistent and to find its solutions.
Performing the row reduction, we get:
[1 -2 0 2]
[0 0 1 -2]
[0 0 0 0]
From the last row of the row-reduced matrix, we can see that the system has a dependent variable, which means that there are infinitely many solutions. We can write the general solution as:
x = x1 = 2t + 1
y = y1 = t
z = z1 = -2s - 2
Here, t and s are arbitrary parameters.
To find a particular solution, we can use any method we like. One method is to use the method of undetermined coefficients. We can guess that xp is a linear combination of the columns of A, with unknown coefficients:
xp = k1[2 -7 3] + k2[-4 14 -6] + k3[5 4 1]
where k1, k2, and k3 are unknown coefficients.
We can substitute this into the system and solve for the coefficients. This gives:
k1 = -1
k2 = -2
k3 = 1
Therefore, a particular solution is:
xp = [-1 -28 1]
So the general solution can be written as:
x = xh + xp = [2t + 1, t, -2s - 2] + [-1, -28, 1]
where t and s are arbitrary parameters.
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Daisy and diesel play a game where diesels chance of winning is always 1/4. they play the game again and again until one of them wins 6 games. Find the probalbility that diesel will win her 6th game on the 18th game(that is, Diesel will win the 18th game, at which point she will have a total of 6 wins).
The probability that Diesel will win her 6th game on the 18th game is approximately 0.000568, or 0.0568%.
The probability that Diesel will win her 6th game on the 18th game is calculated using the binomial distribution formula.
Let's define the following terms:
- n = number of trials (games played) = 18
- k = number of successes (games won by Diesel) = 6
- p = probability of success (Diesel winning a game) = 1/4
- q = probability of failure (Daisy winning a game) = 1 - p = 3/4
The formula for the probability of exactly k successes in n trials is:
P(k) = (n choose k) * p^k * q^(n-k)
where (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials. It can be calculated as:
(n choose k) = n! / (k! * (n-k)!)
Plugging in the values, we get:
P(6) = (18 choose 6) * (1/4)^6 * (3/4)^12
= 18! / (6! * 12!) * (1/4)^6 * (3/4)^12
= 18564 * 0.00000244 * 0.0122
= 0.000568
Therefore, the probability that Diesel will win her 6th game on the 18th game is approximately 0.000568, or 0.0568%.
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suppose X ~N(8,5)show a normal bell curve for mu=8 and sigma=5, with x-axis scaling. add and label all the relevant features, including the percentile!
In this case, we can use a standard normal distribution table or a calculator to find that the value for P75 is approximately 12.1.
Here is a normal bell curve for a normal distribution with mean (μ) of 8 and standard deviation (σ) of 5:
The horizontal axis represents the range of possible values for the variable X, and the vertical axis represents the probability density of each value occurring.
The curve is symmetrical around the mean (μ=8), which is located at the peak of the curve. The standard deviation (σ=5) determines the width of the curve, with wider curves having larger standard deviations.
The shaded area under the curve represents the probability of a value falling within a certain range. For example, the area under the curve between X=3 and X=13 represents the probability of a value falling within 1 standard deviation of the mean, which is approximately 68%.
The percentile of a certain value can also be determined from the normal distribution. For example, the 75th percentile (represented as P75) is the value that separates the lowest 75% of values from the highest 25%. In this case, we can use a standard normal distribution table or a calculator to find that the value for P75 is approximately 12.1.
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What is the area? Round to the nearest tenth if necessary.
Answer:
Set your calculator to degree mode.
Draw a line from point O to a vertex of this octagon to form a right triangle.
tan(67.5°) = 17/x, so x = 17/tan(67.5°)
Area = (1/2)(34/tan(67.5°))(8)(17) = 957.7
[tex]\underset{ \textit{angle in degrees} }{\textit{area of a regular polygon}}\\\\ A=na^2\cdot \tan\left( \frac{180}{n} \right) ~~ \begin{cases} n=sides\\ a=apothem\\[-0.5em] \hrulefill\\ n=8\\ a=17 \end{cases}\implies A=(8)(17)^2\tan\left( \frac{180}{8} \right) \\\\\\ A=2312\tan(22.5^o)\implies A\approx 957.7[/tex]
Make sure your calculator is in Degree mode.
Write F2 + F4 + F6 + ... + F2n in summation notation. Then show that F2 + F4 + F6 + ... + F2n = F2n+1 - 1.
a) The summation notation of F₂ + F₄ + F₆ + ... + F₂ₙ is ∑ᵢ₌₁ᵗⁿ F₂ᵢ
b) Proved that F₂ + F₄ + F₆ + ... + F₂ₙ = F₂ₙ₊₁ - 1.
The Fibonacci sequence is defined as F₁ = 1, F₂ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3.
To express F₂ + F₄ + F₆ + ... + F₂ₙ in summation notation, we can observe that the terms are all even Fibonacci numbers. Thus, we can write:
∑ᵢ₌₁ᵗⁿ F₂ᵢ = ∑ᵢ₌₁ᵗⁿ₋₁ F₂ᵢ + F₂ₙ
where n is even.
We can then use the recurrence relation of the Fibonacci sequence to simplify this:
∑ᵢ₌₁ᵗⁿ F₂ᵢ = ∑ᵢ₌₁ᵗⁿ₋₁ F₂ᵢ + F₂ₙ
= (F₂₁ - 1) + F₂ₙ
= F₂ₙ₊₁ - 1
where we have used the fact that F₂₁ = F₁ = 1.
Therefore, we have shown that F₂ + F₄ + F₆ + ... + F₂ₙ = F₂ₙ₊₁ - 1.
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write three more equations for 1 2/3 that are all true and all different
One possible set of three equations that are all true and different for 1 2/3 is (5/3) + (1/3) = (8/3), (4/3) + (2/3) = (2), and (10/6) + (1/6) = (11/6).
Each of these equations represents a different way of expressing the same value of 1 2/3, which is equal to 5/3 or 1.6666... when expressed as a decimal.
The first equation shows that adding 1/3 to 5/3 results in a sum of 8/3, which is another way of expressing 1 2/3.The second equation shows that adding 2/3 to 4/3 results in a sum of 2, which is yet another way of expressing 1 2/3.Finally, the third equation shows that adding 1/6 to 10/6 results in a sum of 11/6, which is also equivalent to 1 2/3.Overall, these equations demonstrate the flexibility and versatility of mathematical expressions and show how different values can be represented in multiple ways through simple operations like addition and division.
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Think About the Process What is true about a figure and an image created by
a translation? The vertices of parallelogram GRAM are G(-9,-9), R(-8,-6),
A(-4,-6), and M(-5,-9). Graph GRAM and G'R'A'M', its image after a translation
10 units right and 2 units up.
What is true about a figure and an image created by a translation? Select all that apply.
A. Each point in the image has the same x-coordinate as the corresponding point
in the figure.
B. The figure and the image are the same shape.
C. The figure and the image are the same size.
D. Each point in the image moves the same distance and direction from the
figure.
Step-by-step explanation:
A. Each point in the image has the same x-coordinate as the corresponding point
in the figure.
D. Each point in the image moves the same distance and direction from the
figure.
These two statements are true about a figure and an image created by a translation. When a figure is translated, every point in the figure is moved the same distance and direction. This means that each point in the image has moved the same way as its corresponding point in the figure. Additionally, since a translation only involves moving a figure without changing its shape or size, the image and figure are the same shape and size, but just in different positions. As such, statement B and C are not true for figures and images created by translation.
To graph the image G'R'A'M', we need to add 10 to each x-coordinate and subtract 2 from each y-coordinate:
G': (-9+10, -9-2) = (1,-11)
R': (-8+10, -6-2) = (2,-8)
A': (-4+10, -6-2) = (6,-8)
M': (-5+10, -9-2) = (5,-11)
Graphing these points and connecting them gives us parallelogram G'R'A'M'.
A 2pi -periodic signal x(t) is specified over one period as x(t) = (1/A t 0 lessthanorequalto t < A 1 A lessthanorequalto t < pi 0 pi lessthanorequalto t < 2pi Sketch x(t) over two periods from t = 0 to 4pi. Show that the exponential Fourier series coefficients D_pi for this series are given by x(t) = {2 pi - A/4 pi n = 0 1/2 pi n (e^-j A n - 1/An) otherwise
The exponential Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$x(t)$[/tex] are:
[tex]$D_n n=\{(p i-A) / 2 p i$[/tex] for [tex]$n=0$[/tex]
[tex]\left\{(-1)^{\wedge} n /(n \mathrm{~A})\right.[/tex] for [tex]$n=+/-1,+/-2, \ldots$[/tex]
To sketch [tex]$x(t)$[/tex] over two periods from [tex]$t=0$[/tex] to [tex]$4 \mathrm{pi}$[/tex], we first need to plot one period of [tex]$x(t)$[/tex], which is given as:
[tex]$$\begin{aligned}& \mathrm{x}(\mathrm{t})=(1 / \mathrm{A}) \mathrm{t} 0 < =\mathrm{t} < \mathrm{A} \\& =\mathrm{A} \mathrm{A} < =\mathrm{t} < \mathrm{pi} \\& =0 \mathrm{pi} < =\mathrm{t} < 2 \mathrm{pi}\end{aligned}$$[/tex]
The plot of one period of [tex]x(t)[/tex] is shown below:
| /\
| / \
A | / \
| / \
| / \
|_____/ \_____
0 A pi 2pi
To sketch [tex]x(t)[/tex] over two periods, we need to repeat this pattern twice. Since [tex]x(t)[/tex] is a 2pi-periodic signal, we only need to sketch one period to represent the entire signal over any number of periods. Therefore, we can simply repeat the above plot twice to obtain the sketch of [tex]x(t)[/tex] over two periods from [tex]t = 0[/tex] to [tex]4pi[/tex], as shown below:
| /\ /\
| / \ / \
A | / \ / \
| / \ / \
|_____/ \__/ \_____
0 A pi 2pi 3pi
To find the exponential Fourier series coefficients [tex]D_n[/tex], we can use the formula:
[tex]$D_{\ldots} n=(1 / T) * \int[T] x(t) e^{\wedge}(-j n w 0 t) d t$[/tex]
where T is the period of [tex]$x(t)$[/tex], w0 is the fundamental angular frequency, and n is an integer. Since [tex]$x(t)$[/tex] is a 2pi-periodic signal, we have [tex]$T=2 p i$[/tex] and [tex]$\mathrm{wO}=2 \mathrm{pi} / \mathrm{T}=1$[/tex].
The Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$n=0,+/-1,+/-2, \ldots$[/tex] are given by:
[tex]$D_{\ldots} n=(1 / 2 p i) * \int[2 \mathrm{pi}] x(\mathrm{t}) \mathrm{e}^{\wedge}(-j n t) d t$[/tex]
For [tex]$\mathrm{n}=0$[/tex], we have:
[tex]{ D_0 }$[/tex][tex]=(1 / 2 p i)^* \int[2 \mathrm{pi}] \times(t) d t$[/tex]
[tex]=(1 / 2 \mathrm{pi}) *\left[(1 / \mathrm{A}) * \int[\mathrm{A}] \mathrm{t} d \mathrm{dt}+\mathrm{A}^* \int[\mathrm{pi}] \mathrm{dt}+0\right] \\& =(1 / 2 \mathrm{pi}) *\left[(1 / \mathrm{A}) *\left(\mathrm{~A}^{\wedge} 2 / 2\right)+\mathrm{A}(\mathrm{pi}-\mathrm{A})\right] \\[/tex]
[tex]& =(1 / 2 \mathrm{pi}) *[(\mathrm{~A} / 2)+\mathrm{A}(\mathrm{pi}-\mathrm{A})] \\& =(\mathrm{pi}-\mathrm{A} / 2 \mathrm{pi})\end{aligned}$$[/tex]
For [tex]$n=+/-1,+/-2, \ldots$[/tex], we have:
[tex]$$\begin{aligned}& D_n n=(1 / 2 p i)^* \int[2 p i] x(t) e^{\wedge}(-j n t) d t \\& =(1 / 2 p i)^*\left[(1 / A) * \int[A] t e^{\wedge}(-j n t) d t+A^* \int[\text { pi }] e^{\wedge}(-j n t) d t+0\right] \\& =(1 / 2 \text { pi })^*\left[(1 / A)^*\left((-1)^{\wedge} n-1\right)+A^*\left(1-(-1)^{\wedge} n\right) /(j n)\right] \\& =(-1)^{\wedge} n /(n A)\end{aligned}$$[/tex]
Therefore, the exponential Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$x(t)$[/tex] are:
[tex]$D_n n=\{(p i-A) / 2 p i$[/tex] for [tex]$n=0$[/tex]
[tex]$\left\{(-1)^{\wedge} n /(n \mathrm{~A})\right.$[/tex] for [tex]$n=+/-1,+/-2, \ldots$[/tex]
Using the formula for the inverse Fourier series, we can write the
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The exponential Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$x(t)$[/tex] are:
[tex]$D_n n=\{(p i-A) / 2 p i$[/tex] for [tex]$n=0$[/tex]
[tex]\left\{(-1)^{\wedge} n /(n \mathrm{~A})\right.[/tex] for [tex]$n=+/-1,+/-2, \ldots$[/tex]
To sketch [tex]$x(t)$[/tex] over two periods from [tex]$t=0$[/tex] to [tex]$4 \mathrm{pi}$[/tex], we first need to plot one period of [tex]$x(t)$[/tex], which is given as:
[tex]$$\begin{aligned}& \mathrm{x}(\mathrm{t})=(1 / \mathrm{A}) \mathrm{t} 0 < =\mathrm{t} < \mathrm{A} \\& =\mathrm{A} \mathrm{A} < =\mathrm{t} < \mathrm{pi} \\& =0 \mathrm{pi} < =\mathrm{t} < 2 \mathrm{pi}\end{aligned}$$[/tex]
The plot of one period of [tex]x(t)[/tex] is shown below:
| /\
| / \
A | / \
| / \
| / \
|_____/ \_____
0 A pi 2pi
To sketch [tex]x(t)[/tex] over two periods, we need to repeat this pattern twice. Since [tex]x(t)[/tex] is a 2pi-periodic signal, we only need to sketch one period to represent the entire signal over any number of periods. Therefore, we can simply repeat the above plot twice to obtain the sketch of [tex]x(t)[/tex] over two periods from [tex]t = 0[/tex] to [tex]4pi[/tex], as shown below:
| /\ /\
| / \ / \
A | / \ / \
| / \ / \
|_____/ \__/ \_____
0 A pi 2pi 3pi
To find the exponential Fourier series coefficients [tex]D_n[/tex], we can use the formula:
[tex]$D_{\ldots} n=(1 / T) * \int[T] x(t) e^{\wedge}(-j n w 0 t) d t$[/tex]
where T is the period of [tex]$x(t)$[/tex], w0 is the fundamental angular frequency, and n is an integer. Since [tex]$x(t)$[/tex] is a 2pi-periodic signal, we have [tex]$T=2 p i$[/tex] and [tex]$\mathrm{wO}=2 \mathrm{pi} / \mathrm{T}=1$[/tex].
The Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$n=0,+/-1,+/-2, \ldots$[/tex] are given by:
[tex]$D_{\ldots} n=(1 / 2 p i) * \int[2 \mathrm{pi}] x(\mathrm{t}) \mathrm{e}^{\wedge}(-j n t) d t$[/tex]
For [tex]$\mathrm{n}=0$[/tex], we have:
[tex]{ D_0 }$[/tex][tex]=(1 / 2 p i)^* \int[2 \mathrm{pi}] \times(t) d t$[/tex]
[tex]=(1 / 2 \mathrm{pi}) *\left[(1 / \mathrm{A}) * \int[\mathrm{A}] \mathrm{t} d \mathrm{dt}+\mathrm{A}^* \int[\mathrm{pi}] \mathrm{dt}+0\right] \\& =(1 / 2 \mathrm{pi}) *\left[(1 / \mathrm{A}) *\left(\mathrm{~A}^{\wedge} 2 / 2\right)+\mathrm{A}(\mathrm{pi}-\mathrm{A})\right] \\[/tex]
[tex]& =(1 / 2 \mathrm{pi}) *[(\mathrm{~A} / 2)+\mathrm{A}(\mathrm{pi}-\mathrm{A})] \\& =(\mathrm{pi}-\mathrm{A} / 2 \mathrm{pi})\end{aligned}$$[/tex]
For [tex]$n=+/-1,+/-2, \ldots$[/tex], we have:
[tex]$$\begin{aligned}& D_n n=(1 / 2 p i)^* \int[2 p i] x(t) e^{\wedge}(-j n t) d t \\& =(1 / 2 p i)^*\left[(1 / A) * \int[A] t e^{\wedge}(-j n t) d t+A^* \int[\text { pi }] e^{\wedge}(-j n t) d t+0\right] \\& =(1 / 2 \text { pi })^*\left[(1 / A)^*\left((-1)^{\wedge} n-1\right)+A^*\left(1-(-1)^{\wedge} n\right) /(j n)\right] \\& =(-1)^{\wedge} n /(n A)\end{aligned}$$[/tex]
Therefore, the exponential Fourier series coefficients [tex]$D_n n$[/tex] for [tex]$x(t)$[/tex] are:
[tex]$D_n n=\{(p i-A) / 2 p i$[/tex] for [tex]$n=0$[/tex]
[tex]$\left\{(-1)^{\wedge} n /(n \mathrm{~A})\right.$[/tex] for [tex]$n=+/-1,+/-2, \ldots$[/tex]
Using the formula for the inverse Fourier series, we can write the
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if a student stands 15m away directly west of a tree, what is the tree's bearing from the student?
Answer:
90 or 270
Step-by-step explanation:
To determine the bearing of the tree from the student, we need to use the reference direction of North. Typically, bearings are measured in degrees clockwise from North.
If the student stands 15 meters away directly west of the tree, then we can draw a line connecting the student and the tree, which would be a line going directly east to west.
Since we want to find the bearing of the tree from the student, we need to measure the angle between the line connecting the student and the tree and the North direction. Since the line between the student and the tree is going directly east to west, this angle will be either 90 degrees or 270 degrees, depending on which direction we consider as North.
If we consider North to be directly above the student, then the bearing of the tree from the student would be 90 degrees, since the line connecting the student and the tree is perpendicular to the North direction.
If we consider North to be directly below the student, then the bearing of the tree from the student would be 270 degrees, since the line connecting the student and the tree is perpendicular to the South direction (which is opposite to North).
Therefore, depending on the reference direction of North that we choose, the tree's bearing from the student will be either 90 degrees or 270 degrees.
The equation 5 factorial equals
Answer: 120
Step-by-step explanation:
The factorial function multiplies all numbers below that number going to 1. For example,
2! = 2 * 1 = 2
3! = 3 * 2 * 1 = 6
4! = 4 * 3 * 2 * 1 = 24
Thus, 5! would be 5 * 4 * 3 * 2 * 1 = 120.
find the derivative of the function ()=sin((2 −2))
The derivative of f(x) = sin(x²) is f'(x) = 2x × cos(x²).
The chain rule is a rule of calculus used to find the derivative of a composition of functions. It allows us to differentiate a function that is constructed by combining two or more functions, where one function is applied to the output of another function.
To find the derivative of the function f(x) = sin(x²), we need to apply the chain rule of differentiation, which states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) × h'(x).
Here, g(x) = sin(x) and h(x) = x². Therefore, g'(x) = cos(x) and h'(x) = 2x.
Applying the chain rule, we have
f'(x) = g'(h(x)) × h'(x) = cos(x²) × 2x
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Video Que Q.1 Pythagorean theorem 155 A flying squirrel lives in a nest that is 8 meters up in a tree, but wants to eat an acorn that is on the ground 2 meters away from the base of his tree. If the flying squirrel glides from his nest to the acorn, then scurries back to the base of the tree, and then climbs back up the tree to his nest, how far will the flying squirrel travel in total? If necessary, round to the nearest tenth
The distance that is being travelled by the flying squirrel in total would be = 18.2m.
How to calculate the distance covered by the flying squirrel?To calculate the distance covered by the squirrel, the Pythagorean formula should be used. That is;
C² = a² + b²
a = 8m
b = 2m
c²= 8² + 2²
= 64+4
c = √68
= 8.2m
The total distance travelled by the flying squirrel is to find the perimeter of the triangle covered by the squirrel.
perimeter = length+width+height
= 8+2+8.2 = 18.2m.
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The term error is used in two different ways in the context of a hypothesis test. First, there is the concept of standard error (i.e. average sampling error), and second, there is the concept of a Type I error.
a. What factor can a researcher control that will reduce the risk of a Type I error?
b. What factor can a researcher control that will reduce the standard error?
The following parts can be answered by the concept of hypothesis test.
a. To reduce the risk of a Type I error, a researcher can control the significance level or alpha level of their hypothesis test. By setting a lower alpha level (such as 0.01 instead of 0.05), the researcher is decreasing the likelihood of rejecting the null hypothesis when it is actually true.
b. To reduce the standard error, a researcher can increase the sample size of their study. As the sample size increases, the standard error decreases because there is less variability in the sample means. Additionally, ensuring that the sample is representative of the population can also help reduce standard error.
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Use the Integral Test to determine whether the series is convergent or divergen sigma^infinity_n=1 ne^(-9n) Evaluate the following integral^infinity-1 xe^-9x dx.
Since the integral _______ finite, the series is_______
To use the Integral Test to determine whether the series is convergent or divergent, we need to evaluate the integral ∫(xe^(-9x) dx) from 1 to ∞.
Let's evaluate the integral first:
∫(xe^(-9x) dx) from 1 to ∞
We use integration by parts for this, where:
u = x, dv = e^(-9x) dx
du = dx, v = -1/9 e^(-9x)
According to the integration by parts formula, ∫u dv = uv - ∫v du.
So, ∫(xe^(-9x) dx) = -1/9 x e^(-9x) - ∫(-1/9 e^(-9x) dx)
Now, we can integrate -1/9 e^(-9x) dx:
∫(-1/9 e^(-9x) dx) = (-1/9) * (-1/9) * e^(-9x) = 1/81 e^(-9x)
Thus, our integral becomes:
-1/9 x e^(-9x) - 1/81 e^(-9x)
Now we must evaluate the integral from 1 to ∞:
lim (x→∞) [ -1/9 x e^(-9x) - 1/81 e^(-9x) ] - [ -1/9 (1) e^(-9) - 1/81 e^(-9) ]
Since the exponential term e^(-9x) approaches 0 as x approaches ∞, the limit becomes:
0 - [ -1/9 e^(-9) - 1/81 e^(-9) ] = 1/9 e^(-9) + 1/81 e^(-9)
Since the integral is finite, the series is convergent.
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A blueprint for a cottage has a scale of 1:40 one room measures 3.4 m by 4.8 . calculate the dimensions of the room on the blueprint.
I need students to solve it, with operations
Answer: its 12.92
first i multiplide 3.4 by 4.8
Find the measurement of angle A and round the answer to the nearest tenth. :)
(Show work if you can plsss)
Answer:
40.82
Step-by-step explanation:
You need to use trig identities, which are sin(θ)=opposite length/hypotenuse length, cos(θ)=adjacent length/hypotenuse length, and tan(θ)=opposite length/adjacent length.
In your diagram, we see that the only available information is the length opposite of the angle x (19) and adjacent to angle x (22), so we will use the tan identity.
tan(x)=19/22
we need to solve for x, and so we need to get x alone. This can be done by using inverse tan: arctan or [tex]tan(x)^{-1}[/tex]. Note that we ARE NOT taking the equation to the exponent of -1, this is just notation for a trig identify.
arctan(x)tan(x)=x
x= arctan(19/22)
arctan(19/22)= 40.82
and so
x=40.82
Find the general solution of the differential equation. y (5) - 7y (4) + 13y" - 7y" +12y = 0. NOTE: Use C1, C2, C3, C4, and c5 for the arbitrary constants. C5 y(t) =
The general solution of the differential equation will be in the form: y(t) =[tex]C1 * e^(r1 * t) + C2 * e^(r2 * t) + C3 * e^(r3 * t) + C4 * e^(r4 * t) + C5 * e^(r5 * t),[/tex] where C1, C2, C3, C4, and C5 are arbitrary constants.
To find the general solution of the differential equation, we first need to find the characteristic equation by assuming a solution of the form y(t) = e^(rt). Plugging this into the differential equation, we get:
[tex]r^5 - 7r^4 + 13r^3 - 7r^2 + 12r = 0[/tex]
Factoring out an r term, we can simplify this to:
[tex]r(r^4 - 7r^3 + 13r^2 - 7r + 12) = 0[/tex]
We can solve for the roots of the polynomial using either factoring or the quadratic formula, but it turns out that there is only one real root, r = 1, with a multiplicity of 3, and two complex conjugate roots, r = 1 ± i. Therefore, the general solution is:
[tex]y(t) = C1 e^t + (C2 + C3 t + C4 t^2) e^(1+i)t + (C2 - C3 t + C4 t^2) e^(1-i)t + C5[/tex]
where C1, C2, C3, C4, and C5 are arbitrary constants to be determined by initial or boundary conditions. The last term, C5, represents the general solution to the homogeneous differential equation, since it contains no terms involving the roots of the characteristic equation.
To find the general solution of the given differential equation y(5) - 7y(4) + 13y'' - 7y' + 12y = 0, we first need to find the characteristic equation. The characteristic equation for this differential equation is:
[tex]r^5 - 7r^4 + 13r^3 - 7r^2 + 12r = 0.[/tex]
Now, we need to find the roots of this equation. Let's denote them as r1, r2, r3, r4, and r5.
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Write the summation in expanded form.k + 1 i(i!)i = 1 i(i!) + k(k!) + (k + 1)((k + 1)!)i(i!) + + (k + 1)((k + 1)!)1(1!) + 2(2!) + 3(3!) + + (i + 1)((i + 1)!)1(1!) + 2(2!) + 3(3!) + + (k + 1)((k + 1)!)1(1!) + 2(2!) + 3(3!) + + (k)(k!)
The given summation can be expanded as a series of terms, where each term is the product of two factors: one factor consists of the index variable, i or k+1, and the factorial i! or (k+1)!. The other factor consists of the sum of the first i or k terms of the corresponding factorial sequence, i.e., 1(1!), 2(2!), 3(3!), and so on.
Because i in the first term runs from 1 to k, the total is made up of the first k terms of the i! sequence multiplied by the appropriate value of i. Because k is the sole index variable in the second term, the total is composed of the first k terms of the k! sequence multiplied by each matching value of k.
The following terms have i ranging from k+1 to the summation's ultimate value, and the total is made up of the first i-1 terms of the i! sequence multiplied by each corresponding value of i. The first component, (k+1)!, accounts for the terms not included in the first two terms in the prior summations.
Overall, the summation represents a combination of factorials and their corresponding sum sequences, with the index variables determining the range of terms to include in each sum.
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