the slope of the curve at a = 2 is √3 / 3.
To find the slope of the curve at the given values of a, we will use the definition of the derivative:
f'(a) = lim(h→0) [f(a + h) - f(a)] / h
Let's calculate the derivative at each given value of a:
(a) a = 5:
Using the definition of the derivative, we have:
f'(5) = lim(h→0) [f(5 + h) - f(5)] / h
Substituting the function f(x) = √(2x - 1), we get:
f'(5) = lim(h→0) [√(2(5 + h) - 1) - √(2(5) - 1)] / h
Simplifying inside the square roots:
f'(5) = lim(h→0) [√(10 + 2h - 1) - √9] / h
f'(5) = lim(h→0) [√(2h + 9) - 3] / h
Now, we can proceed to evaluate the limit. Let's simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator:
f'(5) = lim(h→0) [(√(2h + 9) - 3) * (√(2h + 9) + 3)] / (h * (√(2h + 9) + 3))
Expanding the numerator:
f'(5) = lim(h→0) [(2h + 9) - 9] / (h * (√(2h + 9) + 3))
f'(5) = lim(h→0) [2h / (h * (√(2h + 9) + 3))]
Canceling out the common factor of h:
f'(5) = lim(h→0) [2 / (√(2h + 9) + 3)]
Now, we can evaluate the limit as h approaches 0:
f'(5) = 2 / (√(2(0) + 9) + 3)
f'(5) = 2 / (√9 + 3)
f'(5) = 2 / (3 + 3)
f'(5) = 2 / 6
f'(5) = 1/3
Therefore, the slope of the curve at a = 5 is 1/3.
(b) a = 2:
Using the definition of the derivative, we have:
f'(2) = lim(h→0) [f(2 + h) - f(2)] / h
Substituting the function f(x) = √(2x - 1), we get:
f'(2) = lim(h→0) [√(2(2 + h) - 1) - √(2(2) - 1)] / h
Simplifying inside the square roots:
f'(2) = lim(h→0) [√(4 + 2h - 1) - √3] / h
f'(2) = lim(h→0) [√(2h + 3) - √3] / h
We can proceed to evaluate the limit. Let's simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator:
f'(2) = lim(h→0) [(√(2h + 3) - √3) * (√(2h + 3) + √3)] / (h * (√(2
h + 3) + √3))
Expanding the numerator:
f'(2) = lim(h→0) [(2h + 3) - 3] / (h * (√(2h + 3) + √3))
f'(2) = lim(h→0) [2h / (h * (√(2h + 3) + √3))]
Canceling out the common factor of h:
f'(2) = lim(h→0) [2 / (√(2h + 3) + √3)]
Now, we can evaluate the limit as h approaches 0:
f'(2) = 2 / (√(2(0) + 3) + √3)
f'(2) = 2 / (√3 + √3)
f'(2) = 2 / (2√3)
f'(2) = 1 / √3
To rationalize the denominator, we multiply both the numerator and denominator by √3:
f'(2) = (1 / √3) * (√3 / √3)
f'(2) = √3 / 3
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Is the function given by f(x) = 2x+5, for x 52, 4x-1, for x>2, Choose the correct answer below. continuous at x=2? Why or why not? OA. The given function is not continuous at x = 2 because lim f(x) does not exist. x-2 B. The given function is not continuous at x=2 because f(2) does not exist. C. The given function is continuous at x = 2 because the limit is 6. D. The given function is continuous at x = 2 because lim f(x) does not exist. X-2
The answer is B. The given function is not continuous at x=2 because f(2) does not exist.
The given function is not continuous at x = 2 because f(2) does not exist. f(x) = { 2x + 5 , x ≤ 2 ; 4x - 1, x > 2 }There are different types of discontinuity.
The function is said to be discontinuous if there exists a point in its domain that does not have a corresponding limit, and that point can either be isolated or non-isolated (removable, jump or infinite discontinuity).
As the value of x approaches 2 from the left, the function f(x) approaches 2(2) + 5 = 9.
As x approaches 2 from the right, the function f(x) approaches 4(2) - 1 = 7.
Therefore, the left and right-hand limits of the function f(x) as x approaches 2 exist.
However, there is no point f(2) in the domain of the function. Since f(x) does not exist at x = 2, there is a discontinuity at x = 2, which is a non-isolated type of discontinuity, specifically, a jump discontinuity. Hence, the answer is B.The given function is not continuous at x=2 because f(2) does not exist.
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Find parametric equations for the line that is tangent to the given curve at the given parameter value. r(t)=(5sint)i+(t 4
−2cost)j+(e 4t
)k,t=0 What is the standard parameterization for the tangent line? x=
y=
z=
(Type expressions using t as the variable.)
The standard parameterization for the tangent line is given by:x = 5t + 5y = 0t + 0z = 4t + 0
Given curve is r(t) = (5sin(t))i + (t4 - 2cos(t))j + (e4t)k and the given parameter value is t = 0.
We need to find the parametric equations for the line that is tangent to the given curve at t = 0 and the standard parameterization for the tangent line.
We know that the tangent to a curve at a point is given by the first derivative of the curve at that point.
Therefore, the parametric equation for the line tangent to the given curve at t = 0 is given by:
r'(t) = 5cos(t)i + 4t³j + 4e⁴tk
Now, at t = 0, we have:r'(0) = 5cos(0)i + 4(0)³j + 4e⁴(0)k = 5i + 0j + 4k = <5, 0, 4>
Therefore, the parametric equations for the line tangent to the given curve at t = 0 is given by
:x = 5t + 5y = 0t + 0z = 4t + 0
The standard parameterization for the tangent line is given by:x = 5t + 5y = 0t + 0z = 4t + 0
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For approximately what values of x can you replace sin(x) by x−(x3/6) with an error of magnitude no greater than 4∗10−3 ?
We can replace sin(x) by [tex]x - (x³/6)[/tex] with an error of magnitude no greater than [tex]`4*10^(-3)`[/tex] for `x` in the range of [tex]`(-1.0268, 1.0268)`.[/tex]
We need to approximate sin(x) by [tex]x - (x³/6)[/tex] with an error of magnitude no greater than [tex]4∗10−3.[/tex]
Therefore, we have to use the Taylor series of sin(x) as given below;`
[tex]sin(x) = x - x³/3! + x^5/5! - x^7/7! + ...`[/tex]
And we have to find the range of values of x for which `sin(x)` can be replaced with `x - x³/6` with an error of magnitude no greater than
[tex]`4*10^(-3)`i.e. `|sin(x) - (x - x³/6)| ≤ 4*10^(-3)`[/tex]
We know that the error of a Taylor series approximation can be bounded by the next term in the series, thus;
[tex]`|(x⁵/5!) - (x⁷/7!) + ...| ≤ 4*10^(-3)`[/tex]
Here, we can assume that the error is dominated by the first neglected term.
Thus; [tex]`|x⁵/5!| ≤ 4*10^(-3)`[/tex]
or
[tex]`|x⁵| ≤ 4*(10^(-3))*(5!)`[/tex]
or
[tex]`|x| ≤ 1.0268`[/tex]
Therefore, we can replace sin(x) by [tex]x - (x³/6)[/tex] with an error of magnitude no greater than [tex]`4*10^(-3)`[/tex] for `x` in the range of [tex]`(-1.0268, 1.0268)`.[/tex]
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solve the polynomial inequality and graph the solution set on the number line. Express the solution set in interval notation (x−9)(x+2)>0 Use the inequality in the form f(x)>0, to write the intervals determined by the boundary points as they appear from ieft to right on a number line. Solve the inequality. What is the solution set? Select the correct choice below and, if necessary. fill in the answer box fo complete your choice A. The solution is (Type yout answer in inteval notation. Simplify your answer) B. The solution set is the empty set.
The solution set in interval notation is (-∞, -2) ∪ (9, ∞).
To solve the polynomial inequality (x-9)(x+2) > 0, we can follow these steps:
Find the critical points by setting the expression inside the inequality to zero: x - 9 = 0 and x + 2 = 0. Solving these equations gives x = 9 and x = -2.
Test the intervals created by the critical points. We have three intervals: (-∞, -2), (-2, 9), and (9, ∞).
Choose a test point within each interval and evaluate the expression (x-9)(x+2) to determine its sign.
For x < -2, we can choose x = -3: (-3-9)(-3+2) = (-12)(-1) = 12, which is greater than zero (+).
For -2 < x < 9, we can choose x = 0: (0-9)(0+2) = (-9)(2) = -18, which is less than zero (-).
For x > 9, we can choose x = 10: (10-9)(10+2) = (1)(12) = 12, which is greater than zero (+).
Determine the intervals where the expression (x-9)(x+2) is greater than zero. The solution set consists of the intervals (-∞, -2) and (9, ∞).
Therefore, the solution set in interval notation is (-∞, -2) ∪ (9, ∞).
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What is the x-coordinate of point B? Write a decimal coordinate.
On a coordinate plane, point B is 1.5 units to the left and 3.5 units up.
To find the x-coordinate of point B, we need to consider the information given. Since point B is 1.5 units to the left of the origin, its x-coordinate will be negative.
The x-coordinate of point B can be calculated by subtracting 1.5 units from the x-coordinate of the origin (0). Therefore, the x-coordinate of point B is -1.5.
Hence, the decimal coordinate for point B is (-1.5, y), where y is the y-coordinate of point B.
Answer:
1) The x-coordinate of point B is -1.5.
2) Example of a decimal coordinate is: 115°28.315'W, 32°52.189'N
Step-by-step explanation:
Based on the information you have provided, point B being 1.5 units on the left indicates it falls on the negative x-axis, bearing in mind that the horizontal plane is the x-axis, where anything to the right of it is positive and to the left is negative. This is how we arrive at the -1.5 value.
An example of how to illustrate a decimal coordinate is given above. Note that it is a random example, as no specific figures have been given in your question.
The total distance flown from N Y ’ s J F K airport to Moscow, Russia is 5015 miles. A typical flight goes from NY to Toronto, Canada, to Helsinki, Finland, to Moscow. The distance from NY to Toronto is 347 miles. The distance from Toronto to Helsinki is 3552 miles more than the distance from Helsinki to Moscow. What is the distance between Toronto and Helsinki, and what is the distance between Helsinki and Moscow?
Please show your work!
Answer:
The distance between Toronto and Helsinki is 4110 miles, and the distance between Helsinki and Moscow is 558 miles.
Step-by-step explanation:
Let's assign variables to the unknown distances:
Distance from Toronto to Helsinki = x
Distance from Helsinki to Moscow = x - 3552
According to the given information, the total distance flown from NY to Moscow is 5015 miles, and the distance from NY to Toronto is 347 miles. Using these values, we can set up the equation:
347 + x + (x + x - 3552) = 5015
Simplifying the equation:
347 + 2x - 3552 = 5015
Combining like terms:
2x - 3205 = 5015
Adding 3205 to both sides:
2x = 8220
Dividing both sides by 2:
x = 4110
Therefore, the distance between Toronto and Helsinki is 4110 miles, and the distance between Helsinki and Moscow is 4110 - 3552 = 558 miles.
Write a recursive formula for the geometric sequence. an={32,61,241,961,…} a1= an=
\(r = 4\), so the recursive formula for the geometric sequence is \(a_n = 4 \cdot a_{n-1}\) where \(a_1 = 32\) is the initial term of the sequence.
To find the recursive formula for the geometric sequence \(a_n = \{32, 61, 241, 961, \ldots\}\), we need to identify the common ratio \(r\) between consecutive terms.
To find \(r\), we can divide any term by its previous term. Let's take the second and first terms:
\(\frac{a_2}{a_1} = \frac{61}{32}\)
Similarly, let's take the third and second terms:
\(\frac{a_3}{a_2} = \frac{241}{61}\)
And finally, the fourth and third terms:
\(\frac{a_4}{a_3} = \frac{961}{241}\)
From these ratios, we can observe that the common ratio \(r\) is consistent and equal to 4.
Now, to write the recursive formula, we can express each term \(a_n\) in terms of the previous term \(a_{n-1}\) using the common ratio:
\(a_n = r \cdot a_{n-1}\)
In this case, \(r = 4\), so the recursive formula for the geometric sequence is:
\(a_n = 4 \cdot a_{n-1}\)
where \(a_1 = 32\) is the initial term of the sequence.
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Write the given system as a set of scalar equations. Let x' = col (x₁ '(t), ×₂'(t)). 1 *[40] X + e - 1 6 X' = x₁ ' (t) = X₂' (t) = t 5
Therefore, the set of scalar equations for the given system is:
x₁ ' (t) = 4x₁ + 6(e - 1)
x₂' (t) = 6x₂
To write the given system as a set of scalar equations, we can expand the matrix equation into two separate equations by multiplying the matrix and column vector:
1 * 4x₁ + (e - 1) * 6 = x₁ ' (t)
6 * x₂ = x₂' (t)
Simplifying further, we have:
4x₁ + 6(e - 1) = x₁ ' (t)
6x₂ = x₂' (t)
These equations represent the scalar equations for the given system. The first equation describes the derivative of the variable x₁ with respect to t, which is equal to 4x₁ plus 6 times the quantity (e - 1). The second equation describes the derivative of the variable x₂ with respect to t, which is equal to 6 times x₂.
Therefore, the set of scalar equations for the given system is:
x₁ ' (t) = 4x₁ + 6(e - 1)
x₂' (t) = 6x₂
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velocity time graph question :)
The distance travelled by the train at a velocity greater than 30 m/s is 3,300 m.
What is the distance traveled by the train?
The distance traveled by the train for a velocity greater than 30 m/s is calculated by applying the following formula for velocity time graph.
The total distance traveled by the train is calculated from the area of the triangle;
A = ¹/₂ x base x height
A = ¹/₂ x (120 - 0)s x (60 - 0 ) m/s
A = 3600 m
The distance traveled by the train below 30 m/s is calculated as;
A(30) = ¹/₂ x (20 - 0 ) s x (30 - 0 ) m/s
A(30) = 300 m
The distance travelled by the train at a velocity greater than 30 m/s is calculated as
= 3,600 m - 300 m
= 3,300 m
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The chance that a PNG Provincial Police Commander believes the death penalty "significantly reduces the number of murders" is 1 in 4. If a random sample of 8 police commanders is selected: (a) determine the probability distribution function of X (X: number of police commanders). (b) find the expected number of commanders and the standard deviation of commanders. (c) find the probability that exactly 5 commanders believe that the death penalty significantly reduces the number of murders. (d) find the probability that at most 3 commanders believe that the death penalty significantly reduces the number of murders.
The probability distribution function is P(X = k) = (8 choose k) × [tex](1/4)^k[/tex] × [tex](3/4)^(8-k)[/tex], for k = 0, 1, 2, 3,4,5,6,7, 8.
b. The expected number of commanders is 2 while the standard deviation of the commander is 1
c. The probability that exactly 5 commanders believe that the death penalty significantly reduces the number of murders. 0.0916
d. The probability that at most 3 commanders believe that the death penalty significantly reduces the number of murders is 0.6046
How to determine probability
The probability distribution function follows a binomial distribution with parameters n = 8 and p = 1/4.
Thus,
P(X = k) = (8 choose k) * (1/4)^k * (3/4)^(8-k),
for k = 0, 1, 2, ..., 8.
The expected number of commanders who believe the death penalty significantly reduces the number of murders is:
E(X) = n * p = 8 * 1/4 = 2.
where
E(X) is the expected number
The standard deviation of commanders who believe the death penalty significantly reduces the number of murders is
SD(X) = √(n * p * (1 - p)) = √(8 * 1/4 * 3/4) = 1.
The probability that exactly 5 commanders believe that the death penalty significantly reduces the number of murders is:
P(X = 5) = (8 choose 5) * ([tex]1/4)^5 * (3/4)^3[/tex] = 0.0916 (rounded to four decimal places).
The probability that at most 3 commanders believe that the death penalty significantly reduces the number of murders is:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
=[tex]8 choose 0) * (1/4)^0 * (3/4)^8 + (8 choose 1) * (1/4)^1 * (3/4)^7+ (8 choose 2) * (1/4)^2 * (3/4)^6 + (8 choose 3) * (1/4)^3 * (3/4)^5[/tex]
= 0.6046
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Using the Chinese Remainder Theorem, find all the solutions of the linear system 2x≡1(mod3),3x≡2(mod4),4x≡2(mod5)
All the solutions of the linear system are: x ≡ 216 (mod 60)
To solve the following linear system of congruences using the Chinese Remainder Theorem:
2x ≡ 1 (mod 3),3x ≡ 2 (mod 4),4x ≡ 2 (mod 5)
we need to break down the system into individual congruences using the Chinese Remainder Theorem.
The given congruences are:
2x ≡ 1 (mod 3) ...(i)
3x ≡ 2 (mod 4) ...(ii)
4x ≡ 2 (mod 5) ...(iii)
The Chinese Remainder Theorem states that for a system of m linear congruences, each given in the form:
x ≡ a1 (mod m1), x ≡ a2 (mod m2),...x ≡ am (mod mm)
where the mi are pairwise relatively prime, the system has a unique solution (mod M), where M = m1m2...mm.
So, now we need to solve each of the given congruences and find the values of x.
Let's do this one by one:
2x ≡ 1 (mod 3)
=> x ≡ 2 (mod 3) ....(1)
3x ≡ 2 (mod 4)
=> x ≡ 2 (mod 4) ....(2)
4x ≡ 2 (mod 5)
=> 2x ≡ 1 (mod 5) [dividing by 2 both sides]
x ≡ 3 (mod 5) ....(3)
Now, applying the Chinese Remainder Theorem on (1), (2), and (3) above:
x ≡ a1M1y1 + a2M2y2 + a3M3y3(mod M)
where M = m1m2m3, M1 = m/m1, M2 = m/m2, and M3 = m/m3
Now, we have:
M1 = (3 x 4) / 3 = 4
M2 = (3 x 5) / 4 = 15/4, so we will multiply throughout by 4 to get M2 = 15
M3 = (4 x 3) / 5 = 12/5, so we will multiply throughout by 5 to get M3 = 12
So, M = m1m2m3 = 3 x 4 x 5 = 60
Applying the Euclidean Algorithm, we get:
60 = 15 x 4 + 0
Therefore, y1 = 4.
So, x = 2 x 4 x 15 + 2 x 15 x 2 + 3 x 12 = 120 + 60 + 36 = 216
Thus, all the solutions of the linear system are: x ≡ 216 (mod 60)
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Categorize the type of sampling used in the situation below: To estimate the mean number of pets in households in a small region, you assign each household a number (i.e. 1 through 600). You then select every 8th household for inspection or surveying.
A. Random
B. Cluster
C. Systematic
D. Convenience
The correct answer is C Systematic, In systematic sampling, the population is ordered, and a fixed interval is used to select samples
In systematic sampling, the population is ordered, and a fixed interval is used to select samples. In this case, the households are assigned numbers, and every 8th household is selected for inspection or surveying.
This follows a systematic pattern of selection based on a predetermined interval. Therefore, the correct categorization is systematic sampling.
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A coin is tossed 57 times and 30 heads are observed. Would we infer that this is a fair coin? Use a 92% level confidence interval to base your inference. The sample statistic for the proportion of heads is: (3 decimals) The standard error in this estimate is: (3 decimals) The correct z∗ value for a 92% level confidence interval is: (3 decimals) The lower limit of the confidence interval is: (3 decimals) The upper limit of the confidence interval is: (3 decimals) Based on this confidence interval, it is that the coin is fair. How would a 99% confidence interval compare to the 92% you constructed? The 99% CI would be narrower. The 99% CI would be wider. They would have the same center. There is no way to tell how they would compare. They would have different centers.
Comparing a 99% confidence interval to the 92% interval, the 99% confidence interval would be wider. This is because a higher confidence level requires a larger interval to capture the true parameter value with greater certainty.
To determine whether the coin is fair, we can construct a confidence interval for the proportion of heads based on the observed data.
The sample proportion of heads is calculated by dividing the number of heads observed (30) by the total number of tosses (57):
Sample proportion (p-hat) = 30/57 ≈ 0.526 (rounded to 3 decimal places)
To calculate the standard error, we use the formula:
Standard error = sqrt((p-hat * (1 - p-hat)) / n)
where p-hat is the sample proportion and n is the sample size. Substituting the values:
Standard error = sqrt((0.526 * (1 - 0.526)) / 57) ≈ 0.065 (rounded to 3 decimal places)
To find the z*-value for a 92% confidence interval, we need to find the critical value corresponding to a 4% significance level (100% - 92% = 8% divided by 2 = 4%).
Using a standard normal distribution table, we find that the z*-value for a 4% significance level is approximately 1.751 (rounded to 3 decimal places).
Now we can construct the confidence interval using the formula:
Confidence interval = p-hat ± (z* * standard error)
Confidence interval = 0.526 ± (1.751 * 0.065) ≈ 0.526 ± 0.114 (rounded to 3 decimal places)
The lower limit of the confidence interval is 0.526 - 0.114 ≈ 0.412, and the upper limit is 0.526 + 0.114 ≈ 0.640.
Based on this confidence interval, we can say with 92% confidence that the true proportion of heads for the coin falls between 0.412 and 0.640.
Comparing a 99% confidence interval to the 92% interval, the 99% confidence interval would be wider. This is because a higher confidence level requires a larger interval to capture the true parameter value with greater certainty.
The center of the interval may or may not be the same, but the width of the interval would be greater for a 99% confidence level compared to a 92% confidence level.
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Find the equation for the tangent to the graph of y at y=sin : 00 * (₁²) is y= (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed) The equation of the line tangent to the graph of y at
The equation of the tangent at y = sin(0°) is y = x or x – y = 0.
The given equation is y = sin(x°), we have to find the equation of tangent line at y = sin(0°).
The equation of tangent is of the form y – y1 = m(x – x1), where (x1, y1) is the point of tangency, and m is the slope of the tangent.
The given equation is y = sin(x°).Differentiating both sides with respect to x, we get,dy/dx = cos(x°) …………….(1)
Now, the equation of tangent is of the form y – y1 = m(x – x1)At y = sin(0°), we have x = 0°
Also, substituting x = 0° in (1), we get,dy/dx = cos(0°) = 1
Therefore, slope of the tangent, m = dy/dx| x=0° = 1
Substituting m = 1 and (x1, y1) = (0°, sin(0°)) in the equation of tangent, we get,y – sin(0°) = 1(x – 0°) => y – 0 = x => y = x …………….(2)
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Five years ago, an alumnus of a university donated $74,316.4 to establish a permanent endowment for scholarships. The first scholarships were awarded 1 year after the contribution. If the amount awarded each year, that is, the interest on the endowment, is $4,752.99, the rate of return earned on the fund is closest to:
Given, An alumnus of a university donated $74,316.4 to establish a permanent endowment for scholarships.
The first scholarships were awarded 1 year after the contribution.
The amount awarded each year, that is, the interest on the endowment is $4,752.99.
[tex]To find the rate of return earned on the fund, we will use the formula for simple interest that is,I = P × r × twhere I = interest, P = principal, r = rate of interest, and t = time.[/tex]
Let's substitute the given values into the formula,[tex]I = 74316.4 × r × 1The interest on the fund is $4,752.99.[/tex]
Therefore,74316.4 × r × 1 = 4752.99
Simplifying the above expression by dividing both sides by 74316.4 × 1, we get = 0.064 or 6.4% (rounded to one decimal place)Therefore, the rate of return earned on the fund is closest to 6.4%.
Thus, the correct option is (D) 6.4%.
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"**Missing part B for both**
A function \( f(x) \) is of the form \[ f(x)=a+\tan (b x) \] where \( a \) and bare constants, and \[ -5.5
where[tex]$n$[/tex] is any integer except where are constants. Thus, the function [tex]$f(x) = a + \tan(bx)$[/tex] becomes:
[tex]$$f(x) = 3 + \tan(n \pi x)$$[/tex]where n is any integer except 0.
From the given information, we have[tex]$f(0) = a + \tan (0) = 3$[/tex].
Therefore, [tex]$a=3$[/tex].Now, we are given that [tex]$f(2) = 5$[/tex], which implies that [tex]$a + \tan(2b) = 5$.[/tex]
Thus,[tex]$\tan(2b) = 5 - a = 5 - 3 = 2$[/tex].
Using the identity,[tex]$\tan(2\theta) = \frac{2 \tan \theta}{1- \tan^2 \theta}$,[/tex]
we can write:[tex]n$$\frac{2 \tan b}{1 - \tan^2 b} = 2$$[/tex]Cross-multiplying and rearranging,
we get:[tex]$$\tan^2 b = 0$$[/tex]
Therefore[tex], $\tan b = 0$ or $\tan b$[/tex] is undefined.
But since[tex]$-5.5 < bx < 5.5$[/tex], we must have [tex]$\tan(bx) \neq \pm \infty$.[/tex]
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Introduction to Chemical Engineering Thermodynamics (7th Edition) Chapter 13. solution 48
Earlier you sended my solution to wrong problem. Please send me solution for 13 chapter, solution 48.
C3H8(g) -> C3H6(g) + H2(g) (I) C3H8(g) -> C2H4(g) + CH4(g) (II)
In the given chemical reaction, the reaction (I) shows the conversion of propane (C3H8) into propene (C3H6) and hydrogen gas (H2), while the reaction (II) shows the conversion of propane (C3H8) into ethene (C2H4) and methane (CH4).
In reaction (I), one molecule of propane (C3H8) is converted into one molecule of propene (C3H6) and one molecule of hydrogen gas (H2). The reaction can be represented as:
C3H8(g) -> C3H6(g) + H2(g)
In reaction (II), one molecule of propane (C3H8) is converted into one molecule of ethene (C2H4) and one molecule of methane (CH4). The reaction can be represented as:
C3H8(g) -> C2H4(g) + CH4(g)
These reactions involve the breaking and formation of chemical bonds. In reaction (I), a carbon-carbon bond in propane is broken, resulting in the formation of a double bond in propene. In addition, a hydrogen atom is removed from propane, leading to the formation of hydrogen gas. In reaction (II), a carbon-carbon bond in propane is broken, resulting in the formation of a double bond in ethene. A carbon-hydrogen bond is also broken, leading to the formation of methane.
Overall, these reactions demonstrate the conversion of propane into different products, propene and hydrogen gas in reaction (I), and ethene and methane in reaction (II).
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Let y = f(x), where f(x) = 9x^3/2 + x^1/2 Find the differential of the function. dy =
Let y = f(x), where f(x) = 9x3/2 + x1/2. Find the differential of the function. dy = ?The given function is:f(x) = 9x3/2 + x1/2The differential of the function is given by:dy/dx = df/dx ........ (1)
The first step is to differentiate f(x) with respect to x. Then we have: f(x) = 9x3/2 + x1/2 Differentiating the above equation with respect to x, we get:df/dx = d/dx [9x3/2 + x1/2]df/dx = d/dx [9x3/2] + d/dx [x1/2]df/dx = 9 × d/dx [x3/2] + 1/2 × d/dx [x]df/dx = 9 × (3/2) × x(3/2)-1 + 1/2 × 1x(1/2)
-1df/dx = (27/2) x(1/2) + 1/2 x(-1/2)df/dx = (27/2) √x + 1/(2√x) Substitute df/dx = dy/dx in equation (1).dy/dx = df/dxdy/dx = (27/2) √x + 1/(2√x)Therefore, the differential of the function is dy = (27/2) √x + 1/(2√x) which can be simplified as follows:dy = 27x1/2/2 + x-1/2/2 or(dy)/(dx) = 27/2√x + 1/2x1/2
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Find the solution of the following polynomial inequality.
Express your answer in interval notation.
x(x+2)2(x−5)≤0
We are given a polynomial inequality as: x(x+2)2(x−5)≤0In order to find the solution to the given polynomial inequality, we need to follow the following steps:
Step 1: Find the critical points by solving the polynomial equation obtained by equating the given polynomial inequality to 0x(x+2)2(x−5) = 0Therefore, the critical points are x = 0, x = -2 and x = 5
Step 2: Plot the critical points on the number line as shown below:
Step 3: Test each of the intervals on the number line using the test values to find whether the polynomial inequality is positive or negative in that interval
Test 1: Let x = -3 which is in the interval (-∞, -2)Now, x(x+2)2(x−5) = (-3)(-1)2(-8) = 24
Since the test value of x(-3) is positive, therefore, the polynomial inequality is positive in the interval (-∞, -2)
Test 2: Let x = -1 which is in the interval (-2, 0)Now, x(x+2)2(x−5) = (-1)(1)2(-6) = 6
Since the test value of x(-1) is positive, therefore, the polynomial inequality is positive in the interval (-2, 0)
Test 3: Let x = 1 which is in the interval (0, 5)Now, x(x+2)2(x−5) = (1)(3)2(-4) = -36
Since the test value of x(1) is negative, therefore, the polynomial inequality is negative in the interval (0, 5)
Test 4: Let x = 6 which is in the interval (5, ∞)Now, x(x+2)2(x−5) = (6)(8)2(1) = 96
Since the test value of x(6) is positive, therefore, the polynomial inequality is positive in the interval (5, ∞)
Step 4: Thus, the solution to the given polynomial inequality in interval notation is:(-∞, -2] U [0, 5]
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Let A= ⎣
⎡
2
−1
2
−1
−3
−2
3
−2
−3
⎦
⎤
a) [10pts] Find the cofactors of a 12
,a 21
, and a 33
. b) [10pts] Evaluate the determinant of (adjA) using expansion along the second row.
For the matrix A the cofactor of a_12 = 3, a_21 = -12, and a_33 = -7 and the determinant of adj(A) using expansion along the second row is 122.
a) To determine the cofactors of the matrix:
A = [2 -1 2]
[-1 -3 -2]
[3 -2 -3]
The cofactor of an element a_ij is obtained by C_ij = (-1)^(i+j) * M_ij, where M_ij is the determinant of the matrix obtained by removing the i-th row and j-th column from matrix A.
Cofactor of a_12:
C_12 = (-1)^(1+2) * M_12
Removing the 1st row and 2nd column from A, we obtain:
M_12 = [-1 -2]
[3 -3]
Now, we can calculate the determinant of M_12:
M_12 = (-1) * (-3) - (-2) * 3 = -3
Thus, C_12 = (-1)^(1+2) * (-3) = 3.
Cofactor of a_21:
C_21 = (-1)^(2+1) * M_21
Removing the 2nd row and 1st column from A, we have:
M_21 = [2 2]
[3 -3]
Now, we calculate the determinant of M_21:
M_21 = 2 * (-3) - 2 * 3 = -12
Hence, C_21 = (-1)^(2+1) * (-12) = -12.
Cofactor of a_33:
C_33 = (-1)^(3+3) * M_33
Removing the 3rd row and 3rd column from A, we obtain:
M_33 = [2 -1]
[-1 -3]
Calculating the determinant of M_33:
M_33 = 2 * (-3) - (-1) * (-1) = -7
Therefore, C_33 = (-1)^(3+3) * (-7) = -7.
b) To evaluate the determinant of adj(A) using expansion along the second row:
adj(A) represents the adjugate matrix of A, which is obtained by taking the transpose of the matrix of cofactors of A.
The cofactor matrix of A is:
C = [C_11 C_12 C_13]
[C_21 C_22 C_23]
[C_31 C_32 C_33]
Taking the transpose of C, we get:
adj(A) = [C_11 C_21 C_31]
[C_12 C_22 C_32]
[C_13 C_23 C_33]
Now, we evaluate the determinant of adj(A) by expanding along the second row:
det(adj(A)) = C_12 * adj(A)_12 + C_22 * adj(A)_22 + C_32 * adj(A)_32
Since we are expanding along the second row, adj(A)_12, adj(A)_22, and adj(A)_32 are the elements of the second row of adj(A).
adj(A)_12 = C_21
adj(A)_22 = C_22
adj(A)_32 = C_23
Substituting these values, we have:
det(adj(A)) = C_12 * C_21 + C_22 * C_22 + C_32 * C_23
Plugging in the calculated values of the cofactors:
det(adj(A)) = 3 * (-12) + (-12) * (-12) + (-7) * (-2)
∴ det(adj(A)) = 122
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FIND
Find \( c \) if \( a=2.65 \mathrm{mi}, b=3.17 \mathrm{mi} \) and \( \angle C=41 \) degrees. Enter \( c \) rounded to 3 decimal places. \( c= \) mi; Assume \( \angle A \) is opposite side \( a, \angle
The value of T falls within the range of 0 to 1.
To test the hypothesis H0: p = 0.572 versus H1: p > 0.572, where p is the population proportion, we can use a one-sample proportion test.
Given:
n = 564 (sample size)
x = 340 (number of observed "successes")
First, we calculate the sample proportion:
ˆp = x/n = 340/564 ≈ 0.6028
Next, we compute the test statistic z-score:
[tex]z = (ˆp - p) / sqrt(p*(1-p)/n)[/tex]
Here, p represents the null hypothesis value, which is 0.572.
z = (0.6028 - 0.572) / sqrt(0.572*(1-0.572)/564)
z ≈ 1.503
To test the null hypothesis at the 91 percent level of significance, we compare the z-score to the critical value corresponding to a 91% confidence level.
The critical value can be obtained from a standard normal distribution table or using statistical software. For a one-sided test at the 91% confidence level, the critical value is approximately 1.695.
Since the calculated z-score (1.503) is less than the critical value (1.695), we do not reject the null hypothesis H0.
Now let's calculate Q1, Q2, and Q3 using the given formulas:
Q1 = ˆp ≈ 0.6028
Q2 = z ≈ 1.503
Since we do not reject H0, Q3 = 0
Now, we can calculate Q using the given formula:
Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|)
Q = ln(3 + |0.6028| + 2|1.503| + 3|0|)
Q = ln(3 + 0.6028 + 2*1.503)
Q ≈ ln(3 + 0.6028 + 3.006)
Q ≈ ln(6.6088)
Q ≈ 1.885
Finally, we calculate T using the formula T = 5sin^2(100Q):
T = [tex]5sin^2(1001.885)[/tex]
T ≈ [tex]5sin^2(188.5)[/tex]
Since[tex]sin^2(188.5[/tex]) is greater than 0, the minimum value for T is 0. Therefore, we have:
0 ≤ T < 1.
Therefore, the answer is (A) 0 ≤ T < 1.
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Explain The reaction. Hydrogen tan + Oxyzenjug → Water on Gails be written as fellow Hydrogen + Oxysen → Water;2H_2(t) + 10_2(k) →; Remnmber to include the phases and the coefficient. For example. In the above reaction, if you simply type: O_2 is it is wrong. The correct answer is 1O_2 is Leave a space between cocifficent, formula, and phase. espacelformula(spicelphase Write stie balancred formula equation for: Sodium bicarbonate is acetic acid → sodium acetate ( sa) + carbon dioxide + dihydrogen monaxidey
The balanced equation for the reaction between sodium bicarbonate and acetic acid to form sodium acetate, carbon dioxide, and water is as follows:
2 NaHCO3(s) + CH3COOH(aq) → 2 CH3COONa(aq) + CO2(g) + H2O(l)
Let's break down the equation step by step:
1. Begin by identifying the reactants and products:
Reactants: Sodium bicarbonate (NaHCO3) and acetic acid (CH3COOH)
Products: Sodium acetate (CH3COONa), carbon dioxide (CO2), and water (H2O)
2. Write the unbalanced equation:
NaHCO3 + CH3COOH → CH3COONa + CO2 + H2O
3. Balance the equation by adjusting the coefficients:
2 NaHCO3 + 2 CH3COOH → 2 CH3COONa + CO2 + H2O
This step ensures that the number of atoms on each side of the equation is equal.
4. Finally, indicate the phases of the substances involved:
2 NaHCO3(s) + 2 CH3COOH(aq) → 2 CH3COONa(aq) + CO2(g) + H2O(l)
(s) represents a solid, (aq) represents an aqueous solution, and (g) represents a gas.
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Use the given conditions to find the exact values of sin(2u), cos(2u), and tan(20) using the double-angle formulas. sin(u) = -4/5, 3/2
Using the given conditions, the exact values are: The value of sin(2u) = -24/25, The value of cos(2u) = 7/25, The value of tan(20) = 7/24
To find the exact values of sin(2u), cos(2u), and tan(20), we can utilize the double-angle formulas. Let's start with sin(2u):
sin(2u) = 2sin(u)cos(u)
Given sin(u) = -4/5, we can use the Pythagorean identity to find cos(u):
cos(u) = √(1 - sin²(u))
cos(u) = √(1 - (-4/5)²)
cos(u) = √(1 - 16/25)
cos(u) = √(9/25)
cos(u) = 3/5
Now we can substitute the values of sin(u) and cos(u) into the double-angle formula for sin(2u):
sin(2u) = 2(-4/5)(3/5)
sin(2u) = -24/25
Moving on to cos(2u), we can use the double-angle formula:
cos(2u) = cos²(u) - sin²(u)
Using the values of sin(u) and cos(u) we found earlier:
cos(2u) = (3/5)² - (-4/5)²
cos(2u) = 9/25 - 16/25
cos(2u) = -7/25
Finally, let's calculate tan(20) using the formula:
tan(2u) = sin(2u) / cos(2u)
Substituting the values we found for sin(2u) and cos(2u):
tan(20) = (-24/25) / (-7/25)
tan(20) = 24/7
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A bee is flying with a velocity of 28 feet per second at an angle of 7° above the horizontal. Find the vertical and horizontal components of the velocity (in ft/s). (Round your answers to one decimal
Given that a bee is flying with a velocity of 28 feet per second at an angle of 7° above the horizontal. We need to find the vertical and horizontal components of the velocity (in ft/s).
Horizontal component of velocity = v cos θ = 28 cos 7° ≈ 27.41 ft/sVertical component of velocity = v sin θ = 28 sin 7° ≈ 2.22 ft/s. Therefore, the horizontal component of velocity is 27.41 ft/s and the vertical component of velocity is 2.22 ft/s.
Therefore, the horizontal component of velocity is 27.41 ft/s and the vertical component of velocity is 2.22 ft/s. Given that a bee is flying with a velocity of 28 feet per second at an angle of 7° above the horizontal. We need to find the vertical and horizontal components of the velocity (in ft/s).
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research statistic and citation for bmi
The researchers analyzed data from more than two million individuals across multiple countries and found that both low and high BMI levels were associated with increased mortality risks.
Body Mass Index (BMI) is a commonly used statistical measure to assess an individual's body composition and determine if they are underweight, normal weight, overweight, or obese. BMI is calculated by dividing a person's weight (in kilograms) by the square of their height (in meters).
Here is a citation for a relevant research article on BMI:
Title: "Body Mass Index and Mortality: A Systematic Review and Meta-Analysis of Observational Studies"
Authors: Katherine M. Flegal, Barry I. Graubard, David F. Williamson, and Mitchell H. Gail
Journal: JAMA (Journal of the American Medical Association)
Year: 2005
Volume: 293
Issue: 15
Pages: 1861-1867
DOI: 10.1001/jama.293.15.1861
This article provides a comprehensive review and meta-analysis of multiple observational studies to examine the association between BMI and mortality. The researchers analyzed data from more than two million individuals across multiple countries and found that both low and high BMI levels were associated with increased mortality risks. The study concluded that maintaining a BMI within the normal range (18.5-24.9) was associated with the lowest mortality risk.
Citing this research article can provide valuable information about the relationship between BMI and mortality rates, which helps to understand the implications of BMI on health outcomes.
Please note that there is a vast amount of research available on BMI, and depending on your specific area of interest or focus, there may be other relevant articles that address different aspects or populations related to BMI.
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need help all information is in the picture. thanks!
The correct option is the last one, the linear equation is : -15 = 8x - 3y
How to find the linear equation?Remember that a general linear equation is written as:
y = ax + b
Where a is the slope and b is the y-intercept.
Two lines are parallel if the lines have the same slope and different y-intercept, then if our line is parallel to y = (8/3)x + 1, we can write our line as:
y = (8/3)x + b
To find the value of b, we use the fact that our line passes through (-3 , -3), then:
-3 = (8/3)*-3 + b
-3 = -8 + b
-3 + 8 = b
5 = b
The line is:
y = (8/3)*x + 5
Now rewrite this in standard form:
y = (8/3)*x + 5
-5 = (8/3)*x - y
3*-5 = 3*(8/3)*x - 3y
-15 = 8x - 3y
The correct option is the last one.
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Suppose that a multiple regression model contains two predictors, X1 and X2. You decide to remove X2 from the model. After removing X2 what happens to the estimate of the coefficient of X1?
Question 21 options:
a.The estimate for the coefficient of X1 does not change.
b.The estimate for the coefficient of X1 usually increases.
c.The estimate for the coefficient of X1 always decreases.
d.None of the above answers are correct.
The correct answer is a) The estimate for the coefficient of X1 does not change.
When you remove X2 from the multiple regression model, it means that you are estimating the relationship between the response variable and X1 while holding all other predictors constant. Removing X2 does not directly affect the estimate of the coefficient of X1 because the coefficient represents the change in the response variable associated with a one-unit change in X1, while holding all other predictors constant.
Removing a predictor from the model does not alter the relationship between the remaining predictor and the response variable. Therefore, the estimate for the coefficient of X1 remains the same after removing X2. However, it is important to note that the standard error, t-value, and significance of the coefficient may change as a result of removing X2.
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No solution No Credit. Problem Solving. (25 points) 1. Find the laplace transform of sin(t)sin(2t)sin(3t), using festf(t)dt. 2. Find the inverse laplace transform of (sª - 4s³ + 8s² - 5s + 14]/[(s+2)(s²+16) (s²+4s+4)]. 3. Find the simplified z transform of k²cos(k*a). 4. Find the inverse z transform of F(z) = (8z - z³)/(4-z)³.
The answer to the given problem solving is:Laplace Transform of sin(t)sin(2t)sin(3t):
Let f(t) = sin(t)sin(2t)sin(3t).
Taking Laplace Transform of f(t), we get:L{f(t)} = L{sin(t)sin(2t)sin(3t)}=> L{sin(t)} * L{sin(2t)} * L{sin(3t)}=> [1/(s²+1)] * [2/(s²+4)] * [3/(s²+9)]=> 6s/[(s²+1)(s²+4)(s²+9)]
6s/[(s²+1)(s²+4)(s²+9)] is the Laplace transform of sin(t)sin(2t)sin(3t).
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Please explain Henry's and Raoult's law and consequently vapor-liquid
Henry's law states that the concentration of a gas in a liquid is directly proportional to its partial pressure in the gas phase, while Raoult's law states that the partial pressure of a component in a solution is directly proportional to its mole fraction in the liquid phase.
Henry's law applies to the solubility of gases in liquids. It states that at a constant temperature, the concentration of a gas dissolved in a liquid is directly proportional to the partial pressure of the gas in the gas phase. Mathematically, it can be represented as C = kH * P, where C is the concentration of the gas, kH is the Henry's law constant, and P is the partial pressure of the gas.
Raoult's law, on the other hand, describes the behavior of ideal solutions. It states that the partial pressure of a component in a solution is directly proportional to its mole fraction in the liquid phase. Raoult's law assumes ideal mixing between the components and no interactions between them. Mathematically, it can be expressed as P = P° * x, where P is the partial pressure of the component in the solution, P° is the vapor pressure of the pure component, and x is the mole fraction of the component in the liquid phase.
Both Henry's law and Raoult's law are important in understanding vapor-liquid equilibrium. In ideal solutions, the vapor phase and the liquid phase reach equilibrium when the partial pressures of the components in the gas phase follow Raoult's law, and the concentrations of dissolved gases in the liquid phase follow Henry's law. These laws provide a foundation for understanding the behavior of solutions and predicting the vapor pressures of components in mixtures.
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4. Find the Fourier series of the function \( f(x)=4+5 x,-\pi \leq x \leq \pi . \) (30 pts.)
The Fourier series of f(x) = 4 + 5x on the interval -π ≤ x ≤ π is given by f(x) = 8/2 + Σ [(8/n)cos(nx) + (5/(n²)) × sin(nx)]
To find the Fourier series of the function f(x) = 4 + 5x on the interval -π ≤ x ≤ π,
Determine the coefficients of the Fourier series.
The Fourier series representation of f(x) is ,
f(x) = a₀/2 + Σ [aₙcos(nx) + bₙsin(nx)]
where a₀, aₙ, and bₙ are the Fourier coefficients.
To find the coefficients, calculate the following integrals,
a₀ = (1/π) × ∫[f(x)] dx, from -π to π
aₙ = (1/π) × ∫[f(x)cos(nx)] dx, from -π to π
bₙ = (1/π) × ∫[f(x)sin(nx)] dx, from -π to π
Let's start by calculating the coefficients,
a₀ = (1/π) × ∫[(4 + 5x)] dx, from -π to π
Integrating 4 with respect to x gives
a₀ = (1/π) × [4x] from -π to π
= (1/π) × [4π - (-4π)]
= (1/π) × [8π]
= 8
Next, let's calculate aₙ,
aₙ = (1/π) × ∫[(4 + 5x) × cos(nx)] dx, from -π to π
Integrating (4 + 5x) × cos(nx) with respect to x,
aₙ = (1/π) × [(4/n)sin(nx) + (5/(n²)) × cos(nx)] from -π to π
= (1/π) × [(4/n)sin(nπ) + (5/(n²)) × cos(nπ) - (4/n)sin(-nπ) - (5/(n²)) × cos(-nπ)]
Since sin(-nπ) = 0 and cos(-nπ) = cos(nπ), we have,
aₙ = (1/π) × [(4/n)sin(nπ) + (5/(n²)) × cos(nπ) - (4/n)sin(nπ) - (5/(n²)) × cos(nπ)]
= 0
Finally, let's calculate bₙ,
bₙ = (1/π) × ∫[(4 + 5x) × sin(nx)] dx, from -π to π
Integrating (4 + 5x) × sin(nx) with respect to x
bₙ = (1/π) × [-(4/n)cos(nx) + (5/(n²)) × sin(nx)] from -π to π
= (1/π) × [-(4/n)cos(nπ) + (5/(n²)) × sin(nπ) - (-(4/n)cos(-nπ) + (5/(n²)) × sin(-nπ))]
Since cos(-nπ) = cos(nπ) and sin(-nπ) = 0, we have,
bₙ = (1/π) × [-(4/n)cos(nπ) + (5/(n²)) × sin(nπ) - (-(4/n)cos(nπ))]
= (1/π) × [(8/n)cos(nπ) + (5/(n²)) × sin(nπ)]
The summation includes all values of n excluding n = 0.
Therefore, the required Fourier series of f(x) on the given interval is equal to f(x) = 8/2 + Σ [(8/n)cos(nx) + (5/(n²)) × sin(nx)]
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The above question is incomplete , the complete question is:
Find the Fourier series of the function
f(x) = 4 + 5x , -π ≤ x ≤ π