The Taylor polynomial of degree n=3 at a=1 of the function f(x) = r is simply the polynomial T(r) = r.
To find the Taylor polynomial T(r) of degree n=3 at a=1 of the function f(x) = r, we first need to calculate the first four derivatives of f(x) with respect to x:
f(x) = r
f'(x) = 0
f''(x) = 0
f'''(x) = 0
f''''(x) = 0
Then, we evaluate these derivatives at x=a=1:
f(1) = r
f'(1) = 0
f''(1) = 0
f'''(1) = 0
f''''(1) = 0
Next, we can use the Taylor polynomial formula:
T(r) = f(a) + f'(a)(r-a) + (f''(a)/2!)(r-a)² + (f'''(a)/3!)(r-a)³ + ... + (fⁿ(a)/n!)(r-a)ⁿ
Since we are finding the Taylor polynomial of degree n=3, we only need to include the first four terms of this formula. Therefore, we get:
T(r) = f(1) + f'(1)(r-1) + (f''(1)/2!)(r-1)² + (f'''(1)/3!)(r-1)³
T(r) = r + 0(r-1) + 0/2!(r-1)² + 0/3!(r-1)³
T(r) = r
To know more about Taylor polynomial here
https://brainly.com/question/31419648
#SPJ4
Find the derivative: G(x) = S1 x (cos√t)dt
The derivative of G(x) is cos(√x).
We can start by using the Fundamental Theorem of Calculus, which tells us that the derivative of G(x) is simply the integrand evaluated at x, i.e.,
G'(x) = cos(√x).
To see why this is true, we can define a new function f(t) = cos(√t) and rewrite G(x) in terms of this function:
G(x) = ∫₁ˣ cos(√t) dt = F(x) - F(1),
where F(x) is any antiderivative of f(x). Now, using the chain rule, we can take the derivative of G(x):
G'(x) = F'(x) - 0 = f(x) = cos(√x).
Therefore, the derivative of G(x) is cos(√x).
Learn more about the Fundamental Theorem of Calculus
https://brainly.com/question/30761130
#SPJ4
Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers.
Using exponents, the radical expression can be written as √30 after rationalizing the denominator.
An algebraic expression is what?Variables, constants, and mathematical operations (such as addition, subtraction, multiplication, division, and exponentiation) can all be found in an algebraic equation.
In algebra and other areas of mathematics, algebraic expressions are used to depict connections between quantities and to resolve equations and issues. A radical expression is any mathematical formula that uses the radical (also known as the square root symbol) sign.
Here in the question,
The radical expression that we have is:
[tex]\frac{15\sqrt{6} }{3\sqrt{5} }[/tex]
Now, this can be written as:
= [tex]\frac{3\sqrt{5} \sqrt{5} \sqrt{6}}{3\sqrt{5} }[/tex]
Simplifying:
= √5 × √6
= √30
Therefore, the radical expression can be written as √30 after rationalizing the denominator.
To know more about radical expression, visit:
brainly.com/question/30277930
#SPJ1
Three US adults are randomly selected. The probability a single adult is between 180 and 185 cm is 0.1157.60
(a) What is the probability that all three are between 180 and 185 cm tall?
(b) What is the probability that none are between 180 and 185 cm?
The probability that all three selected adults are between 180 and 185 cm tall is 0.0015 or 0.15%. The probability that none of the selected adults are between 180 and 185 cm tall is 0.6611 or 66.11%.
Given that the probability of a single adult being between 180 and 185 cm is 0.1157, we can assume that this probability is the same for all adults in the population.
(a) The probability can be calculated as follows:P(all three between 180 and 185 cm) = P(first adult between 180 and 185 cm) x P(second adult between 180 and 185 cm) x P(third adult between 180 and 185 cm)
Since the selection of each adult is independent, we can multiply the probabilities. Thus,
P(all three between 180 and 185 cm) = 0.1157 x 0.1157 x 0.1157
P(all three between 180 and 185 cm) = 0.0015 (rounded to four decimal places) or 0.15%
(b) The probability can be calculated as follows:P(none between 180 and 185 cm) = P(first adult not between 180 and 185 cm) x P(second adult not between 180 and 185 cm) x P(third adult not between 180 and 185 cm)
Since the selection of each adult is independent, we can multiply the probabilities. Thus,
P(none between 180 and 185 cm) = (1 - 0.1157) x (1 - 0.1157) x (1 - 0.1157)
P(none between 180 and 185 cm) = 0.6611 (rounded to four decimal places) or 66.11%
Learn more about Probability:
https://brainly.com/question/24756209
#SPJ11
(a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y2, (7,5, 24) = (b) Find a set of symmetric equations for the normal line to the surface at the given point. Ox - 7 = y - 5 = 2 - 24 OX-7 = Y - 5 - 2 - 24 14 -10 -1 Ox + 7 = y + 5 = 2 + 24 x + 7 y + 5 Z + 24 14 -10 -1 y 14 -10 -1 Х Z =
14x - 8y - z - 1 = 0 is the equation of the tangent plane to the surface at the given point and x = 7 + 14t, y = 4 - 8t and z = 33 - t are symmetric equations
To find the equation of the tangent plane to the surface z = x^2 - y^2 at the point (7, 4, 33)
we need to find the partial derivatives of z with respect to x and y at that point:
∂z/∂x = 2x = 14
∂z/∂y = -2y = -8
So the normal vector to the tangent plane at (7, 4, 33) is <2x, -2y, -1> evaluated at that point, which is <14, -8, -1>.
Using the point-normal form of the equation of a plane, the equation of the tangent plane is:
14(x-7) - 8(y-4) - (z-33) = 0
Simplifying, we get:
14x - 8y - z - 1 = 0
(b) The normal line to the surface at (7, 4, 33) is perpendicular to the tangent plane at that point, so it has the direction vector <14, -8, -1>.
We can write the symmetric equations of the line as:
x = 7 + 14t
y = 4 - 8t
z = 33 - t
To learn more on Differentiation click:
https://brainly.com/question/24898810
#SPJ4
Let f ( x ) = 1 x − 5 f ( x ) = 1 x - 5 and g ( x ) = 5 x + 9 g ( x ) = 5 x + 9 . Then ( f ∘ g ) ( 0 ) = ( f ∘ g ) ( 0 ) = , ( f ∘ g ) ( x ) = ( f ∘ g ) ( x ) = .
(f ∘ g)(x) = 1/(5x+4). To find (f ∘ g)(0), we need to first evaluate g(0) which is equal to 9. Then we plug this value into f(x) and get f(g(0)) = f(9) = 1/(9-5) = 1/4. Therefore, (f ∘ g)(0) = 1/4.
To find (f ∘ g)(x), we substitute g(x) into f(x) to get f(g(x)) = 1/(5x+9-5) = 1/(5x+4). Therefore, (f ∘ g)(x) = 1/(5x+4).
Step 1: Find (f ∘ g)(x) by plugging g(x) into f(x) as the input.
(f ∘ g)(x) = f(g(x))
= f(5x + 9)
Step 2: Replace the input x in f(x) with g(x).
= 1 / ((5x + 9) - 5)
Step 3: Simplify the expression.
= 1 / (5x + 4)
So, (f ∘ g)(x) = 1 / (5x + 4).
Step 4: Now, let's find (f ∘ g)(0) by plugging x = 0 into the composition (f ∘ g)(x).
(f ∘ g)(0) = 1 / (5(0) + 4)
Step 5: Simplify the expression.
= 1 / 4
Therefore, (f ∘ g)(0) = 1/4.
In summary:
(f ∘ g)(0) = 1/4
(f ∘ g)(x) = 1 / (5x + 4)
Learn more about substitute here:
https://brainly.com/question/2264200
#SPJ11
Let P(Z = 0) = p, P(Z = 1) = 9, P(Z = 3) = r, where positive p, q, r satisfy p + q + r = 1 and E[Z] < 1. = = (a) find the recursion formula for Ud(u), u = 0, 1, 2, ... Take p = 3/8, q = 1/2, r= 1/8 = = (b) find the smallest initial capital u for which the chance of ultimate ruin is less than 5%.
The desired recursion formula for Ud(u) is Ud(u) = 3/8 + 9 Ud(u+1) + 1/8 Ud(u-3) and the smallest initial capital u for which the chance of ultimate ruin is less than 5% is 23.
(a) To discover the recursion equation for the likelihood of extreme destruction, we will utilize the whole likelihood hypothesis and the law of adding up to desire.
Let Ud(u) be the likelihood of extreme destruction given that the beginning capital is u. At that point, we have:
Ud(u) = P(Z = 0) + P(Z = 1) Ud(u+1) + P(Z = 3) Ud(u-3)
where the primary term compares to the case where the gambler loses all their cash instantly,
the moment term compares to the case where the player wins the primary wagered and after that faces the same choice issue beginning with a capital of u+1,
and the third term compares to the case where the card shark loses the primary three wagers and after that faces the same decision problem beginning with a capital of u-3.
Substituting the given values of p, q, and r, we get:
Ud(u) = 3/8 + 9 Ud(u+1) + 1/8 Ud(u-3)
This is often the specified recursion equation for Ud(u).
(b) To discover the littlest beginning capital u for which the chance of extreme destroy is less than 5%, we can utilize the recursion equation to calculate Ud(u) for expanding values of u until we find the littlest u such that Ud(u) is less than 0.05.
Utilizing the given values of p, q, and r, we will plug them into the recursion equation to induce:
Ud(u) = 3/8 + 9 Ud(u+1) + 1/8 Ud(u-3)
Ud(u-1) = 3/8 + 9 Ud(u) + 1/8 Ud(u-4)
Ud(u-2) = 3/8 + 9 Ud(u-1) + 1/8 Ud(u-5)
...
Beginning with u = 0, we are able recursively to compute Ud(u) for increasing values of u. Able to halt when we discover the littlest u such that Ud(u) is less than 0.05.
Employing a spreadsheet or a programming dialect such as Python, we will calculate Ud(u) for expanding values of u until we discover that Ud(23) is less than 0.05 and Ud(22) is more prominent than or rise to 0.05.
therefore, the littlest starting capital u for which the chance of extreme demolishes is less than 5% is 23.
learn more about recursive formula
brainly.com/question/8972906
#SPJ4
Emily sold 56 of the 145 bracelets. What percent of the bracelets did she sell? Show your strategy.
EMERGENCY HELP IS NEEDED!!! WILL MARK BRAINLLIEST!!!
(F + G) (X) = 10X + 7
IF F (X) = 6X + 2 WHAT DOES G (X) EQUAL?
A.) x-1
B.) 2x+2
C.) 3x+6
D.) 4x+5
The function G(X) can be solved algebraically to be G(X) = 4x + 5 which makes the option D correct.
How to solve algebraically for the function G(X)Given the function:
(F + G) (X) = 10X + 7 and F (X) = 6X + 2
then G (X) is derived algebraically as follows:
F (X) + G (X) = 10X + 7
G (X) = 10X + 7 - F (X)
G (X) = 10X + 7 - (6X + 2)
G (X) = 4x + 5
Therefore, the function G(X) can be solved algebraically to be G(X) = 4x + 5
Read more about algebra here:https://brainly.com/question/30716491
#SPJ1
A construction company is considering submitting bids for two contracts. It will cost the company $10,000 to prepare and submit the bids, and if won, each bid would produce $50,000 of income to the company. The company estimates that it has a 10% chance of winning any given bid.
Here is the probability distribution of X = the amount of money the company profits from the bids.
X -$10,000 $40,000 $90,000
Probability P(X) 0.70 0.18 0.12
Calculate the expected value of the profit X.
a. $11,000
b. $0
c. $120,000
d. $0.42
e. -$11,000
the expected value of the profit X is $11,000. Your answer is (a).
The expected value of the profit X is calculated by multiplying each possible profit by its probability of occurring and then adding up these values.
Expected value of X = (-$10,000 x 0.70) + ($40,000 x 0.18) + ($90,000 x 0.12)
Expected value of X = -$7,000 + $7,200 + $10,800
Expected value of X = $11,000
Therefore, the answer is (a) $11,000.
To calculate the expected value of the profit X, we need to multiply each possible profit value by its corresponding probability and sum the results. Using the provided probability distribution:
Expected value (E(X)) = (-$10,000 * 0.70) + ($40,000 * 0.18) + ($90,000 * 0.12)
E(X) = (-$7,000) + ($7,200) + ($10,800)
E(X) = $11,000
Therefore, the expected value of the profit X is $11,000. Your answer is (a).
To know more about probability distribution: click here:
brainly.com/question/31197941
#SPJ11
Find the area of the surface generated by revolving the curve y = - 12.-13 sxs0, about the x-axis. The area of the surface is (Type an exact answer, using as needed)
The area of the surface generated by revolving the curve y = -12.13x² around the x-axis.
In this case, we are revolving the curve y = -12.13x² around the x-axis. This means that for each point on the curve, we draw a line from that point to the x-axis, and then we rotate that line around the x-axis. The resulting surface will be a "shell" or "tube" shape that encloses a certain volume.
To find the area of this surface, we need to use a mathematical formula. There are different methods to approach this, but one common way is to use the formula for the surface area of a surface of revolution, which is given by:
A = 2π ∫ y √(1 + (dy/dx)²) dx
In this formula, A represents the area of the surface, a and b represent the limits of integration, y represents the height of the curve, and dy/dx represents the derivative of the curve with respect to x.
Now, let's apply this formula to our specific curve. We have y = -12.13x², so dy/dx = -24.26x. Plugging these into the formula, we get:
A = 2π ∫ -12.13x³ √(1 + (-24.26x)²) dx
To know more about area here
https://brainly.com/question/14994710
#SPJ4
Evaluate the integral: S18 1 (√3/z)dz
The value of the integral is ln(1/6) + (3/2) ln(3). To evaluate the integral S18 1 (√3/z)dz, we can use the substitution u = √3/z, which implies du/dz = (-√3/z²) and dz = (-z²/√3) du.
Substituting u = √3/z and dz = (-z²/√3) du, the integral becomes:
S18 1 (√3/z)dz = -S√3/18 ∞ √3/1 u du
= -S√3/18 ∞ √3/1 (√3/u) du
= -S1 18 (3/u) du
= -3ln(u) |1 18
= -3ln(18/√3) + 3ln(1/√3)
= -3ln(6) + 3ln(√3)
= -3ln(6) + 3/2 ln(3)
= ln(1/6) + (3/2) ln(3)
Therefore, the value of the integral is ln(1/6) + (3/2) ln(3).
Learn more about “ evaluate the integral “ visit here;
https://brainly.com/question/31585464
#SPJ4
The figure shows a rectangular prism consisting of 12 cubes. The area of the front shaded face is 36 cm2. Find the volume of the entire prism
The volume of the entire prism is 864 cm³.
Since the front shaded face has an area of 36 cm² and consists of 8 cubes, we know that each cube has a surface area of 3 cm². Since each cube has 6 faces, the total surface area of all 12 cubes is:
12 x 6 x 3 = 216 cm²
Since the rectangular prism has 6 faces,4 with the same surface area and 2 (6 cubes) with same surface area the total surface area of the prism is:
4 x 36 + (2 x 6 x 3) = 180 cm²
The total surface area of the 12 cubes is the sum of the surface area of all the faces of the rectangular prism, minus the surface area of the top and bottom faces, which are covered by the other cubes. Therefore, the surface area of the 4 lateral faces of the rectangular prism is:
216 - (2 x 36) = 144 cm²
Since the lateral faces are all rectangles with the same dimensions, we can find the length and width of the rectangular prism by dividing the lateral surface area by the perimeter of the front shaded face:
Length: 144 / 12 = 12 cm
Width: 144 / 6 = 24 cm
The height of the rectangular prism is the height of one cube, which is 3 cm. Therefore, the volume of the entire prism is:
24 x 12 x 3 = 864 cm³
Hence, the volume of the entire prism is 864 cm³.
The complete question is given below:
The figure shows a rectangular prism consisting of 12 cubes. The area of the front shaded face is 36 cm2. Find the volume of the entire prism.
(There are 8 face cube of 4 sides of the rectangle and 6 face cube of 2 side of the rectangle.)
To know more about prism
https://brainly.com/question/29722724
#SPJ4
Simplify: 30 - 2 * 50 + 70 A. 0 B. 570 C. -210 D. -70
On simplifying the given operation "30 - 2 x 50 + 70" using the BODMAS rule, we will be getting -70. So, the option D is the correct answer choice.
Following the order of operations (BODMAS), we need to perform the multiplication before the addition and subtraction.
30 - 2 x 50 + 70 = 30 - 100 + 70
Now we can perform the addition and subtraction from left to right:
= -70
The expression 30 - 2 x 50 + 70 can be simplified by following the order of operations, which is a set of rules that tells us which operations to perform first. In this case, we first perform the multiplication, which gives us 2 x 50 = 100. Then we perform the addition and subtraction from left to right, giving us a final result of -70.
Learn more about BODMAS Rule :
https://brainly.com/question/29626866
#SPJ4
To simplify the expression 30 - 2 * 50 + 70, follow the order of operations (PEMDAS). Multiply 2 * 50 first, then perform addition and subtraction from left to right. The simplified expression is -70.
Explanation:To simplify the expression 30 - 2 * 50 + 70, we need to follow the order of operations, which is known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right). Applying PEMDAS, first we perform the multiplication: 2 * 50 = 100. Then, we do the addition and subtraction from left to right: 30 - 100 + 70 = -70. Therefore, the simplified expression is -70.
Learn more about Order of operationshttps://brainly.in/question/49780623
#SPJ12
Integrate the following (hint: rewrite the 2 as square root then distribute) ∫ e^sin-12x / 2√025-x^2 dx
The expression for the integral of [tex]e^{sin-12x}[/tex] / 2√025-x² dx.
One approach that we can use is substitution. However, before we can use substitution, we need to simplify the expression first. We can start by using the hint provided in the problem, which suggests that we rewrite the 2 as a square root and then distribute. This will give us the following expression:
∫ [tex]e^{sin-12x}[/tex] / √(25-x²) dx
Next, we can make the substitution u = sin(x), which will simplify the expression even further. To do this, we need to use the chain rule of integration. We can write:
du/dx = cos(x)
dx = du/cos(x)
Substituting this into the original expression, we get:
∫ eˣ / (√(25-( [tex]e^{sin-12x}[/tex] ) * cos(x)) du
Now, we can simplify the denominator using the trigonometric identity sin²(x) + cos²(x) = 1. This gives us:
∫ eˣ / (5 * √(1-(u/5)²) * cos(x)) du
Next, we can use another trigonometric substitution, v = u/5, which will give us:
∫ e⁵ˣ / ( √(1-v²) * cos(x)) dv
At this point, we can use integration by parts to solve this expression. Integration by parts is a technique that allows us to integrate a product of two functions. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
We can choose u and dv in such a way that the resulting integral is easier to solve. Let's choose u = √(1-v²) and dv = e⁵ˣ / cos(x) dv. This gives us:
v = e⁵ˣ / (5 cos(x))
du/dv = -v / √(1-v²)
Simplifying the second term using another trigonometric substitution, u = sin(y), we get:
(1/25) (5sin⁻¹(w)) * ((w² - 1) √(1-w²) + sin⁻¹(w) * w)
We can simplify this expression further by substituting back v and w in terms of u, using the substitutions we made earlier. This gives us:
(1/5) √(1-(u/5)²) * [tex]e^{sin-12x}[/tex] / cos(x) - (1/25) [tex]e^{sin-12x}[/tex] * (((u/5)² - 1) √(1-(u/5)²) + sin⁻¹(u/5) * (u/5))
To know more about integration here
https://brainly.com/question/18125359
#SPJ4
Keisha used a photo that measured 4 inches by 6 inches to make a copy that measured 8 inches by 12 inches. What is the
scale factor of the dilation? Write your answer as a whole number or a decimal.
The scale factor of the dilation Keisha used to make a copy of a photo is 2.
What is a scale factor?A scale factor is a numerical ratio that describes how much an object has been scaled, or resized, from its original size. It is the ratio of the length or size of a figure after it has been scaled to its original length or size. In other words, it is the factor by which all the dimensions of an object have been multiplied to obtain a new size.
The scale factor can be greater than 1, which means that the object has been enlarged, or less than 1, which means that the object has been reduced. A scale factor of 1 means that the object remains the same size.
To find the scale factor of the dilation, we can divide the dimensions of the copy by the dimensions of the original photo.
The dimensions of the original photo are 4 inches by 6 inches, and the dimensions of the copy are 8 inches by 12 inches.
So, the scale factor is:
8/4 = 2
and
12/6 = 2
Therefore, the scale factor of the dilation is 2.
To know more about scale factor visit:
brainly.com/question/30215119
#SPJ1
Given A(22) = 188 and d = 19, what is the value of a₁? A. a₁ = 587 B. a₁ = -211 C. a₁ = 606 D. a₁ = 9.9
When the value of A(22) = 188 and d = 19 are given then the value of a₁ is -211. The correct answer is option B. The problem seems to be related to arithmetic sequences.
where A(n) represents the nth term of the sequence and a₁ represents the first term of the sequence. We can use the formula for the nth term of an arithmetic sequence:
A(n) = a₁ + (n-1)d
where d is the common difference between consecutive terms.
We are given that A(22) = 188 and d = 19. Substituting these values in the formula, we get:
188 = a₁ + (22-1)19
Simplifying this equation, we get:
188 = a₁ + 399
a₁ = 188 - 399
a₁ = -211
Therefore, the value of a₁ is -211, which is option B.
To learn about arithmetic sequences visit:
https://brainly.com/question/29616017
#SPJ4
Q4. Let X1, ... , Xn be an independent random sample from N(µ , σ^2), and consider the estimation of σ^2, where u is unknown. n - (i) Show that n/n-1 S^2 is an unbiased estimator of σ^2, (2 marks) (i) Calculate the CRLB with respective to σ^2; (5 marks) (ii) Will n/n-1 S^2 achieve the CRLB? (3 marks)
Let X1, ... , Xn be an independent random sample from[tex]N(\mu , \sigma^2)[/tex], and consider the estimation of [tex]\sigma^2[/tex]
[tex]Var[n/(n-1) S^2] = (2\sigma^4)/(n-1) > = (2\sigma^4)/(3n) = CRLB[/tex]
[tex]n/(n-1) S^2[/tex]does achieve the CRLB for [tex]\sigma^2[/tex].
[tex]n/(n-1) S^2[/tex] is an unbiased estimator of [tex]\sigma^2[/tex], we need to show that the expected value of[tex]n/(n-1) S^2[/tex] is equal to [tex]\sigma^2[/tex].
[tex]S^2 = \Sigma(Xi - \bar X)^2 / (n-1)[/tex] is an unbiased estimator of[tex]\sigma^2[/tex], i.e., [tex]E[S^2] = \sigma^2[/tex].
The expected value of [tex]n/(n-1) S^2[/tex], we have:
[tex]E[n/(n-1) S^2] = E[(n/(n-1)) \times (S^2)][/tex]
[tex]= (n/(n-1)) \times E[S^2][/tex]
[tex]= (n/(n-1)) \times \sigma^2[/tex]
[tex]= \sigma^2 + (\sigma^2/(n-1))[/tex]
[tex]= ((n-1)/n)\sigma^2 + \sigma^2/(n-1)[/tex]
[tex]= (n-1)/(n-1) \times \sigma^2[/tex]
[tex]= \sigma2[/tex]
We conclude that [tex]n/(n-1) S^2[/tex]is an unbiased estimator of [tex]\sigma^2[/tex].
To calculate the Cramer-Rao Lower Bound (CRLB), we need to find the Fisher information, [tex]I(\sigma^2)[/tex], with respect to [tex]\sigma^2[/tex]:
[tex]I(\sigma^2) = -E[d^2/d(\sigma^2)^2 ln(f(X|\sigma^2))][/tex]
[tex]f(X|\sigma^2)[/tex] is the likelihood function of the normal distribution.
Since[tex]X1, ... , Xn[/tex] are independent, we have:
[tex]f(X|\sigma^2) = (1/((2\pi\sigma^2)^n/2)) \times exp[-\Sigma(Xi - \mu)^2 / (2\sigma^2)][/tex]
Taking the natural logarithm of this function, we get:[tex]ln(f(X|\sigma^2)) = -n/2 ln(2\pi\sigma^2) - \Sigma(Xi - \mu)^2 / (2\sigma^2)[/tex]
Differentiating this function twice with respect to [tex]\sigma^2[/tex], we get:
[tex]d^2/d(\sigma^2)^2 ln(f(X|\sigma^2)) = n/(2\sigma^4) - \Sigma(Xi - \mu)^2 / (\sigma^6)[/tex]
The expected value of this expression, we get:
[tex]E[d^2/d(\sigma^2)^2 ln(f(X|\sigma^2))] = n/(2\sigma^4) - E[\Sigma(Xi - \mu)^2 / (\sigma^6)][/tex]
[tex]= n/(2\sigma^4) - n/(\sigma^6)[/tex]
[tex]= n(3\sigma^2)/(2\sigma^6)[/tex]
[tex]= 3n/(2\sigma^4)[/tex]
The CRLB with respect to [tex]\sigma^2[/tex] is:
[tex](CRLB)\sigma^2 = 1/I(\sigma^2) = 2\sigma^4/(3n)[/tex]
Finally, to check whether[tex]n/(n-1) S^2[/tex] achieves the CRLB, we need to show that the variance of [tex]n/(n-1) S^2[/tex] is greater than or equal to the CRLB:
[tex]Var[n/(n-1) S^2] = (2\sigma^4)/(n-1)[/tex]
So, we have:
[tex]Var[n/(n-1) S^2] = (2\sigma^4)/(n-1) > = (2\sigma^4)/(3n) = CRLB[/tex]
[tex]n/(n-1) S^2[/tex]does achieve the CRLB for [tex]\sigma^2[/tex].
For similar questions on random sample
https://brainly.com/question/30592506
#SPJ11
Question 8 Determine the approximation for the transfer function G (s) = (s+2)(x+50) (8+4)(8+8)(8+100) O Gopprox (s) (3+2) (+4)(8+8) Gapproa (3) 0.5{+2)(x+50) (8+4)(6+8) 1.0 O Gopprox (5) (674)(+8) O
The approximation for the transfer function is Gapprox (s) 1.0 / (s+4) (s+8) (option c).
The given transfer function is G(s) = (s+2)(s+50)/ (s+4)(s+8)(s+100). To approximate this transfer function, we need to simplify it by canceling out some terms.
This simplification involves canceling out the terms (s+2) and (s+50) from the numerator and assuming that the pole at s = -100 has a negligible effect on the system's behavior. This approximation assumes that the system is dominated by the poles at s = -4 and s = -8 and the zeros at s = -2 and s = -50.
Hence the correct option is (c).
To know more about transfer function here
https://brainly.com/question/13002430
#SPJ4
Complete Question:
Determine the approximation for the transfer function G (s) = (s+2)(s+50)/ (s+4) (s+8) (s+100)
(a) Gapprox (s) (s+2) (s+4) /(s+8)
(b) Gappros (s) 0.5(s+2)(s+50) / (s+4)(s+8)
(c) Gapprox (s) 1.0 / (s+4) (s+8)
(d) Gapprox (s) 0.5(s+2) / (s+4) (s+8)
Plis help me I am not understand nothing
A sketch of line segment A'B' is shown on the graph below.
The length of line segment A'B' is equal to 3 units.
What is a reflection over the y-axis?In Geometry, a reflection over or across the y-axis is represented and modeled by this transformation rule (x, y) → (-x, y).
By applying a reflection over the y-axis to the coordinate of the given line segment AB, we have the following coordinates:
Coordinate A = (-4, 1) → Coordinate A' = (-(-4), 1) = (4, 1).
Coordinate B = (-1, 1) → Coordinate B' = (-(-1), 1) = (1, 1).
In Mathematics and Geometry, the distance between two (2) endpoints that are on a coordinate plane can be calculated by using the following mathematical equation (formula):
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance AB = √[(1 - 4)² + (1 - 1)²]
Distance AB = √[(-3)² + (0)²]
Distance AB = √9
Distance AB = 3 units.
Read more on reflection here: https://brainly.com/question/27912791
#SPJ1
(1) (10 pts each) Evaluate the integral. ∫ln x/ x^4 dx (hint: 1/x^4 = x^-4)
The evaluated integral is: (1/9) x^(-3) + C
Given the integral:
∫(ln x / x^4) dx
Using the hint provided, we can rewrite the integral as:
∫(ln x * x^(-4)) dx
Now, we will use integration by parts. Let u = ln x and dv = x^(-4) dx. Then, du = (1/x) dx and v = -1/(3x^3).
Using the integration by parts formula ∫u dv = uv - ∫v du, we have:
(-1/(3x^3)) * ln x - ∫(-1/(3x^3)) * (1/x) dx
Simplify the expression inside the second integral:
-∫(-1/(3x^4)) dx
Now, integrate:
-(-1/3) ∫ x^(-4) dx
Integrating x^(-4), we get:
-(-1/3) * (-1/3) x^(-3)
Simplifying, we get:
(1/9) x^(-3)
Now, add the constant of integration (C):
(1/9) x^(-3) + C
Know more about integral here:
https://brainly.com/question/18125359
#SPJ11
PART A: A circular swimming pool has a diameter of 10 m.The circular side of the pool is 4 m high, and the depth of thewater is 2.5 m. (The acceleration due to gravity is9.8 m/s2 and the density of water is1000 kg/m3.) PART B: A tank in the shape of an inverted right circularcone has height 17 meters and radius 12 meters.It is filled with 11 meters of hot chocolate.Find the work required to empty the tank by pumping the hotchocolate over the top of the tank. Note: the density of hotchocolate is 1510kg/m^3
PART A: To find the volume of the circular swimming pool, we can use the formula for the volume of a cylinder, which is V = πr^2h, where r is the radius of the pool and h is the height of the pool. Since the diameter of the pool is 10 m, the radius is 5 m. Therefore, the volume of the pool is:
V = π(5 m)^2(4 m)
V = 100π m^3
To find the weight of the water in the pool, we can use the formula W = mg, where m is the mass of the water and g is the acceleration due to gravity. The mass of the water is equal to its volume times its density, which is 1000 kg/m^3. Therefore, the weight of the water in the pool is:
W = (100π m^3)(1000 kg/m^3)(9.8 m/s^2)
W ≈ 3.09 x 10^6 N
PART B: To find the volume of the tank, we can use the formula for the volume of a cone, which is V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone. Since the tank is an inverted cone, we need to use the radius and height of the top of the cone, which are equal to 5 m and 6 m, respectively (since the height of the hot chocolate is 11 m, and the total height of the cone is 17 m). Therefore, the volume of the tank is:
V = (1/3)π(5 m)^2(6 m)
V ≈ 157.08 m^3
To find the mass of the hot chocolate, we can use its volume and density, which is 1510 kg/m^3. Therefore, the mass of the hot chocolate is:
m = (157.08 m^3)(1510 kg/m^3)
m ≈ 2.37 x 10^5 kg
To find the work required to empty the tank, we can use the formula W = mgh, where m is the mass of the hot chocolate, g is the acceleration due to gravity, and h is the height that the hot chocolate needs to be pumped to empty the tank. Since the hot chocolate needs to be pumped over the top of the tank, the height that it needs to be pumped is equal to the height of the tank, which is 17 m. Therefore, the work required to empty the tank is:
W = (2.37 x 10^5 kg)(9.8 m/s^2)(17 m)
W ≈ 3.89 x 10^7 J
Learn more about inverted cone here:
https://brainly.com/question/30712771
#SPJ11
classify each of the studies: the task is to match the lettered items with the correct numbered items. appearing below is a list of lettered items. following that is a list of numbered items. each numbered item is followed by a drop-down. select the letter in the drop down that best matches the numbered item with the lettered alternatives. a. occurrence of cancer was identified between april 1991 and july 2002 for 50,000 troops who served in the first gulf war (ended april 1991) and 50,000 troops who served elsewhere during the same period. b. subjects were children enrolled in a health maintenance organization. at 2 months, each child was randomly given one of two types of a new vaccine against rotavirus infection.
The occurrence of cancer was identified between april 1991 and july 2002 for 50,000 troops who served in the first gulf war (ended april 1991) and 50,000 troops who served elsewhere during the same period is observational study. subjects were children enrolled in a health maintenance organization is Randomized Controlled Trial.
In study A, two groups were identified based on their service during the Gulf War and elsewhere during the same period, and the occurrence of cancer was identified between April 1991 and July 2002. This study is an observational study, as the researchers did not assign any interventions or treatments to the participants, but rather just observed and collected data on the outcome of interest (cancer) in the two groups.
In study B, children enrolled in a health maintenance organization were randomly assigned to one of two types of a new vaccine against rotavirus infection. This study is a randomized controlled trial, as the researchers randomly assigned the intervention (vaccine type) to the participants and then observed the effect on the outcome of interest (rotavirus infection).
To know more about observational study:
https://brainly.com/question/15938198
#SPJ4
Use the variation of parameters technique to find a particular solution x, to the given system of differential equation. Find the general solution to the system of differential equations. Then, find the particular solution that satisfies the initial condition. x' = x+2y+3 y' = 2x + y +2 x(0)=1, y(0)=-1
The particular solution is:
Xp(t) = [(3/2)
To use the variation of parameters technique, we first need to find the general solution of the system of differential equations.
The system is:
x' = x + 2y + 3
y' = 2x + y + 2
We can write this system in matrix form as:
X' = AX + B
where
X = [x, y]
A = [1 2; 2 1]
B = [3, 2]
The characteristic equation of the matrix A is:
det(A - λI) = 0
(1-λ)(1-λ) - 4 = 0
λ^2 - 2λ - 3 = 0
(λ - 3)(λ + 1) = 0
So the eigenvalues of A are λ1 = 3 and λ2 = -1.
To find the eigenvectors of A, we solve the equations:
(A - λ1I)v1 = 0
(A - λ2I)v2 = 0
For λ1 = 3, we have:
(1-3)v1[0] + 2v1[1] = 0
2v1[0] + (1-3)v1[1] = 0
which gives v1 = [2, -1].
For λ2 = -1, we have:
(1+1)v2[0] + 2v2[1] = 0
2v2[0] + (1+1)v2[1] = 0
which gives v2 = [1, -1].
So the general solution of the system of differential equations is:
[tex]X(t) = c1e^{(\lambda1t)}v1 + c2e^{(\lambda2\times t)}\times v2[/tex]
where c1 and c2 are constants.
Substituting in the values of λ1, λ2, v1, and v2, we get:
[tex]X(t) = c1\times [2e^{(3t) }- e^{(-t)}, -e^{(3t)} + e^{(-t)}] + c2[e^{(-t)}, e^{(-t)}][/tex]
Now we can use the variation of parameters technique to find a particular solution.
We assume that the particular solution has the form:
Xp(t) = u1(t)× v1 + u2(t) × v2
where u1(t) and u2(t) are functions to be determined.
Taking the derivatives of Xp(t), we get:
Xp'(t) = u1'(t)× v1 + u2'(t) × v2 + u1(t)λ1v1 + u2(t)λ2v2
Substituting Xp(t) and Xp'(t) into the original system of differential equations, we get:
u1'(t) × v1 + u2'(t)v2 + u1(t)λ1v1 + u2(t)λ2v2 = Av1u1(t)v1 + Av2u2(t) × v2 + B
where Av1 and Av2 are the first and second columns of A.
Equating the coefficients of v1 and v2, we get the following system of equations:
u1'(t) + 3u1(t) = 3
u2'(t) - u2(t) = -2
The solutions of these equations are:
[tex]u1(t) = (3/4) - (1/4)\times e^{(-3t)}\\u2(t) = 2 - e^{(-t)[/tex]
for such more question on particular solution
https://brainly.com/question/22008756
#SPJ11
∫(1 to [infinity]) x2/(x3+2)2 dx is
A -1/2
B 1/9
C 1/3
D 1
E divergent
The answer is B, 1/9.
How to find an indefinite integral?To determine the result of the integral ∫(1 to ∞) x^2/(x^3+2)^2 dx, we can use the comparison test for improper integrals:
First, observe that x^2/(x^3+2)^2 < x^2/x^6 = 1/x^4 for x > 1, since the denominator in the original integrand is larger than x^6.
Now, consider the integral of 1/x^4 from 1 to ∞:
∫(1 to ∞) 1/x^4 dx = [(-1/3)x^(-3)](1 to ∞) = (-1/3)(0 - 1/3) = 1/9
Since 0 < x^2/(x^3+2)^2 < 1/x^4 for x > 1 and the integral ∫(1 to ∞) 1/x^4 dx converges, the given integral ∫(1 to ∞) x^2/(x^3+2)^2 dx also converges by the comparison test.
Hence, the answer is B, 1/9.
Learn more about indefinite integral.
brainly.com/question/28036871
#SPJ11
Find the antiderivative: k(x) = x⁻⁶ + 2x + 4
The antiderivative of k(x) is F(x) = x⁻⁵ / (-5) + x² + 4x + C. To find the antiderivative of k(x), we need to find a function F(x) such that F'(x) = k(x).Finding the antiderivative of x⁻⁶can be done using the power rule of integration:
∫ x⁻⁶ dx = x⁻⁵ / (-5) + C1, where C1 is the constant of integration.
We may locate the antiderivative of 2x by using the power rule once more:
∫ 2x dx = x² + C2, where C2 is another constant of integration.
Last but not least, here is the antiderivative of 4:
∫ 4 dx = 4x + C3, where C3 is yet another constant of integration.
Combining everything, we have:
F(x) = ∫ x⁻⁶ + 2x + 4 dx
= x⁻⁵ / (-5) + x² + 4x + C, where C = C1 + C2 + C3 is the overall constant of integration.
Therefore, the antiderivative of k(x) is F(x) = x⁻⁵ / (-5) + x² + 4x + C.
Learn more about antiderivative:
https://brainly.com/question/31385327
#SPJ4
A population of values has a normal distribution with = 22.5 and o = 17.7. a. Find the probability that a single randomly selected value is greater than 19.5. Round your answer to four decimal places. P(x > 19.5) = b. Find the probability that a randomly selected sample of size n= 154 has a mean greater than 19.5. Round your answer to four decimal places. PM > 19.5) =
a)The probability that a single randomly selected value is greater than 19.5 is 0.5668.
We must normalize the value and apply a typical normal distribution table in order to determine the likelihood that a single randomly chosen number is greater than 19.5. Z-score at x = 19.5 is as follows
z = (19.5 - 22.5) / 17.7 = -0.1695
According to the usual normal distribution table, there is a 0.4332 percent chance that a z-score will be less than -0.1695. Consequently, the likelihood that a single randomly chosen value will be higher than 19.5 is:
P(x > 19.5) = 1 - P(x ≤ 19.5) = 1 - 0.4332 = 0.5668
So, the probability is 0.5668.
b) the probability that a randomly selected sample of size n= 154 has a mean greater than 19.5 is 0.9822.
To find the probability that a randomly selected sample of size n = 154 has a mean greater than 19.5, we use the central limit theorem and standardize the sample mean. The sample mean's standard deviation is:
σX = σⁿ = 17.7 ¹⁵⁴ = 1.4278
The z-score for X> 19.5 is:
z = (19.5 - 22.5) / 1.4278 = -2.102
We determine that the chance of a z-score smaller than -2.102 is 0.0178 using a conventional normal distribution table. As a result, the likelihood that a randomly chosen sample of size n = 154 will have a mean higher than 19.5 is:
P(X> 19.5) = 1 - P(X ≤ 19.5) = 1 - 0.0178 = 0.9822
So, the probability is 0.9822.
Learn more about randomly selected sample:
https://brainly.com/question/14735856
#SPJ4
What is the range of the function y=3√x+8?
a) -infinity < y < infinity
b) -8 < y < infinity
c) 0 [tex] \leqslant [/tex] y < infinity
d) 2 [tex] \leqslant [/tex] y < infinity
The range of the function [tex]y=\sqrt[3]{x+8 }[/tex] is Real and option (a) -∞ ≤ y ≤ ∞
Define the term Range of function?The range of a function is the set of all possible output values (also known as the function's "y-values") that the function can produce for all possible input values (also known as the function's "x-values").
Here a function is given that;
[tex]y=\sqrt[3]{x+8 }[/tex]
Taking cube on both sides of above function,
[tex]y^{3}= x+8[/tex]
[tex]y^{3}-8= x[/tex]
⇒Since x is defined for all real values of y, we get real values of x for all real values of y, so the given function's range is y∈R.
Therefore, the range of the function [tex]y=\sqrt[3]{x+8 }[/tex] is option (a) -∞ ≤ y ≤ ∞
To know more about function, visit:
https://brainly.com/question/11624077
#SPJ1
Evaluate. ∫4 9 2t + 1/ √t dt . ∫4 9 2t + 1/ √t dt = ____. (Simplify your answer.)
To evaluate the integral ∫(2t + 1/√t) dt from 4 to 9, follow these steps:
Step 1: Break the integral into two separate integrals. ∫(2t + 1/√t) dt = ∫(2t) dt + ∫(1/√t) dt
Step 2: Find the antiderivatives of each function. For the first integral, the antiderivative of 2t is t^2.
For the second integral, rewrite 1/√t as t^(-1/2) and then find its antiderivative using the power rule: t^((-1/2)+1) / ((-1/2)+1) = 2√t. So, the antiderivatives are: ∫(2t) dt = t^2 ∫(1/√t) dt = 2√t
Step 3: Combine the antiderivatives. ∫(2t + 1/√t) dt = t^2 + 2√t
Step 4: Evaluate the antiderivative from 4 to 9. (t^2 + 2√t)|_4^9 = (9^2 + 2√9) - (4^2 + 2√4) = (81 + 6) - (16 + 4) = 87 - 20 = 67
So, the answer is ∫(2t + 1/√t) dt from 4 to 9 equals 67.
Learn more about integral,
https://brainly.com/question/30094386
#SPJ11
the temperature of a point $(x,y)$ in the plane is given by the expression $x^2 y^2 - 4x 2y$. what is the temperature of the coldest point in the plane?
The temperature of a point [tex]$(x,y)$[/tex] in the plane is given by the expression [tex]$x^2 y^2 - 4x 2y$[/tex] . The coldest point in the plane is[tex]$(0,1)$[/tex] with a temperature of[tex]$-4$[/tex]
To find the coldest point in the plane, we need to minimize the given expression of temperature [tex]$T(x,y) = x^2 y^2 - 4x 2y$[/tex] with respect to [tex]$x$[/tex] and [tex]$y$[/tex]. We can do this by taking partial derivatives of [tex]$T$[/tex] with respect to [tex]$x$[/tex] and [tex]$y$[/tex], and setting them equal to zero:
[tex]$\frac{\partial T}{\partial x}[/tex][tex] = 2xy^2 - 8y= 0$[/tex]
[tex]$\frac{\partial T}{\partial y}[/tex][tex] = 2x^2y - 8x = 0$[/tex]
Solving these equations simultaneously, we get [tex]$x=0$[/tex] or [tex]$x=2$[/tex] and [tex]$y=0$[/tex] or [tex]$y=1$[/tex] . We can then evaluate the temperature at each of these four points:
[tex]$T(0,0) = 0$[/tex]
[tex]$T(0,1) = -4$[/tex]
[tex]$T(2,0) = 0$[/tex]
[tex]$T(2,1) = 0$[/tex]
This makes intuitive sense as the expression for temperature is symmetric about the [tex]$x$[/tex] and [tex]$y$[/tex] axes, and the point [tex]$(0,1)$[/tex] corresponds to the "bottom" of the surface formed by the expression.
Learn more about derivatives here:
https://brainly.com/question/29020856
#SPJ4
Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A random sample of 24 fluorescent light bulbs has a mean life of 665 hours with a standard deviation of 24 hours.
It is important to note that this statement is about the process of constructing intervals, not about any particular interval we might construct.
To construct a confidence interval for the population mean, we need to know the sample mean, sample size, sample standard deviation, and the level of confidence. Given the problem statement, we have:
Sample mean (H) = 665 hours
Sample standard deviation (s) = 24 hours
Sample size (n) = 24
Level of confidence = 95%
We can use the formula for the confidence interval for the population mean as follows:
Confidence interval = H ± (tα/2) * (s/√n)
where X is the sample mean, s is the sample standard deviation, n is the sample size, tα/2 is the t-value from the t-distribution with n-1 degrees of freedom and a level of significance of α/2 (α/2 = 0.025 for a 95% confidence interval).
To find the t-value, we can use a t-table or a calculator. Using a t-table with 23 degrees of freedom (n-1), we find the t-value for α/2 = 0.025 to be 2.069.
Substituting the values into the formula, we get:
Confidence interval = X ± (tα/2) * (s/√n)
Confidence interval = 665 ± (2.069) * (24/√24)
Confidence interval = 665 ± 9.93
Therefore, the 95% confidence interval for the population mean, μ, is (655.07, 674.93).
This means that we are 95% confident that the true population mean falls within this interval. It is important to note that this statement is about the process of constructing intervals, not about any particular interval we might construct.
To learn more about Sample mean visit:
https://brainly.com/question/31101410
#SPJ11
