The statement is true: if f assumes its maximum at x₀ ∈ [a,b], then f assumes its minimum at x₀ as well.
Let's assume that f assumes its maximum at x₀ ∈ [a,b]. Since f is a continuous function on the closed interval [a,b], we know from the Extreme Value Theorem that f must have a maximum and a minimum value on [a,b].
Now, suppose f does not assume its minimum at x₀. That means there exists some y₀ ∈ [a,b] such that f(y₀) < f(x) for all x ∈ [a,b]. Since f has a maximum at x₀, it follows that f(x₀) ≥ f(x) for all x ∈ [a,b].
Consider the following cases:
Case 1: x₀ < y₀
Since f is continuous, we can apply the Intermediate Value Theorem to the closed interval [x₀, y₀]. This implies that for any value c between f(x₀) and f(y₀), there exists some z ∈ [x₀, y₀] such that f(z) = c. However, since f(x₀) ≥ f(x) for all x ∈ [a,b], it means that f(x₀) is the maximum value of f on [a,b].
Therefore, f(z) cannot be greater than f(x₀), which contradicts our assumption. Hence, this case is not possible.
Case 2: x₀ > y₀
Similarly, we can apply the Intermediate Value Theorem to the closed interval [y₀, x₀]. This implies that for any value c between f(y₀) and f(x₀), there exists some z ∈ [y₀, x₀] such that f(z) = c. However, since f(x₀) is the maximum value of f on [a,b], it means that f(x₀) ≥ f(x) for all x ∈ [a,b].
Therefore, f(z) cannot be greater than f(x₀), which again contradicts our assumption. Hence, this case is also not possible.
Since both cases lead to a contradiction, we can conclude that f must assume its minimum at x₀ if it assumes its maximum at x₀.
To know more about the Extreme Value Theorem, refer here:
https://brainly.com/question/30760554#
#SPJ11
A not-so-skilled volleyball player has a 15% chance of making the serve, which involves hitting the ball so it passes over the net on a trajectory such that it will land in the opposing team's court. Suppose that her serves are independent of each other. (a) What is the probability that on the 10th try she will make her 3rd successful serve? (b) Suppose she has made two successful serves in nine attempts. What is the probability that her 10th serve will be successful? (c) Even though parts (a) and (b) discuss the same scenario, the probabilities you calculated should be different. Can you explain the reason for this discrepancy?
In this scenario, a volleyball player has a 15% chance of making a successful serve, and the serves are independent of each other. The probabilities of making a successful serve on the 3rd attempt and the 10th attempt are calculated.
(a) To calculate the probability that the player will make her 3rd successful serve on the 10th try, we need to consider the probability of two unsuccessful serves followed by a successful serve on the 3rd try and then seven more unsuccessful serves. Since the probability of making a successful serve is 15%, the probability of making an unsuccessful serve is 85%. Therefore, the probability can be calculated as: [tex](0.85^2) * (0.15) * (0.85^7)[/tex].
(b) Given that the player has already made two successful serves in nine attempts, we want to find the probability of making a successful serve on the 10th try. The probability can be calculated as: (0.15) * (0.15) * ([tex]0.85^7[/tex]).
(c) The reason for the discrepancy between the probabilities in parts (a) and (b) is that the previous attempts affect the probability in part (b). In part (a), we start from the beginning and calculate the probability of specific outcomes. However, in part (b), we already have information about the previous attempts, and the probability calculation takes into account the specific scenario of having two successful serves in nine attempts. Therefore, the probabilities differ because the context and conditions of the scenarios are different.
Learn more about attempt here:
https://brainly.com/question/11405237
#SPJ11.
Question 30 Three randomly chosen Colorado students were asked how many times they went rock climbing last month. Their replies were 5,7.8. The sample standard deviation is 1056 0.816 1000 1528
The sample standard deviation of the three responses (5, 7, 8) is approximately 1.53.
To calculate the sample standard deviation, we follow these steps:
Step 1: Find the mean:
First, we need to find the mean (average) of the three responses. The mean is obtained by summing up the values and dividing by the number of data points:
Mean = (5 + 7 + 8) / 3 = 20 / 3 ≈ 6.67
Step 2: Calculate the deviation of each data point from the mean:
Next, we calculate the deviation of each data point from the mean. Deviation is the difference between each data point and the mean. For our example, we subtract the mean (6.67) from each response:
Deviation₁ = 5 - 6.67 = -1.67
Deviation₂ = 7 - 6.67 = 0.33
Deviation₃ = 8 - 6.67 = 1.33
Step 3: Square each deviation:
To avoid cancellation of positive and negative deviations, we square each deviation:
Deviation₁² = (-1.67)² ≈ 2.79
Deviation₂² = (0.33)² ≈ 0.11
Deviation₃² = (1.33)² ≈ 1.77
Step 4: Calculate the sum of squared deviations:
Now, we sum up the squared deviations obtained in Step 3:
Sum of squared deviations = 2.79 + 0.11 + 1.77 ≈ 4.67
Step 5: Calculate the average of squared deviations:
To find the average, divide the sum of squared deviations by the number of data points minus 1. Since we have three data points, the denominator is 3 - 1 = 2:
Average of squared deviations = 4.67 / 2 ≈ 2.33
Step 6: Take the square root:
Finally, we take the square root of the average of squared deviations to obtain the sample standard deviation:
Sample standard deviation = √(2.33) ≈ 1.53
To know more about deviation here
https://brainly.com/question/16555520
#SPJ4
Find the radius of curvature of the curve x = 4cost and y = 3sint at t = 0
The radius of curvature of the curve x = 4cos(t) and y = 3sin(t) at t = 0 is 5/3 units.To find the radius of curvature, we first need to find the curvature of the curve. The curvature (k) can be calculated using the formula k = |(dx/dt * d²y/dt²) - (d²x/dt² * dy/dt)| / (dx/dt² + dy/dt²)^(3/2).
Here, dx/dt represents the derivative of x with respect to t, dy/dt represents the derivative of y with respect to t, d²x/dt² represents the second derivative of x with respect to t, and d²y/dt² represents the second derivative of y with respect to t.
Differentiating x = 4cos(t) and y = 3sin(t) with respect to t, we get dx/dt = -4sin(t) and dy/dt = 3cos(t). Taking the second derivatives, we have d²x/dt² = -4cos(t) and d²y/dt² = -3sin(t).
Substituting these values into the curvature formula and evaluating at t = 0, we get
k = |-4sin(0) * (-3sin(0)) - (-4cos(0)) * 3cos(0)| / ((-4cos(0))² + (3cos(0))²)^(3/2) = |-4 * 0 - (-4) * 3| / ((-4)² + 3²)^(3/2) = 12 / 5.
The radius of curvature (R) is given by R = 1 / k. Therefore, the radius of curvature of the given curve at t = 0 is 1 / (12/5) = 5/3 units.
Learn more about derivative here: https://brainly.com/question/29144258
#SPJ11
In the carbon dating process for measuring the age of objects, carbon-14, a radioactive isotope, decays into carbon-12 with a half-life of 5730 years A Cro-Magnon cave painting was found in a cave in Europe. If the level of carbon-14 radioactivity in charcoal in the cave is approximately 11% of the level of living wood, estimate how long ago the cave paintings were made.
Therefore, the cave paintings were made approximately 30935 years ago.
To estimate how long ago the cave paintings were made, we can use the concept of half-life in radioactive decay. The half-life of carbon-14 is 5730 years, which means that after 5730 years, half of the carbon-14 in a sample will have decayed into carbon-12.
Given that the level of carbon-14 radioactivity in the charcoal is approximately 11% of the level in living wood, we can assume that the remaining 89% has decayed into carbon-12.
Let's denote the initial amount of carbon-14 in the charcoal as C0 and the current amount of carbon-14 as C. We can express the decay of carbon-14 over time t as:
[tex]C = C0 * (1/2)^{(t / 5730)[/tex]
We know that the current carbon-14 level is 11% of the initial level, which means C = 0.11 * C0.
Substituting this into the equation, we have:
[tex]0.11 * C0 = C0 * (1/2)^{(t / 5730)[/tex]
Dividing both sides by C0, we get:
[tex]0.11 = (1/2)^{(t / 5730)[/tex]
Now, we can solve for t by taking the logarithm of both sides:
[tex]log(0.11) = log((1/2)^{(t / 5730))[/tex]
Using the property of logarithms, we can bring the exponent down:
log(0.11) = (t / 5730) * log(1/2)
Now we can isolate t:
t = 5730 * (log(0.11) / log(1/2))
Using a calculator, we find:
t ≈ 30935.065
To know more about cave paintings,
https://brainly.com/question/3089935
#SPJ11
f(x)=
2x5+11x4+44x3+31x3-148x+60
(a) Find all the zeros. Write the answer in exact form. If there is more than one answer, separate them with commas. Select "None" if applicable. The zeros of f(x): -2±4i, 1,1,-3 2 Part: 1 / 3 Part 2
The zeros of the function f(x) = 2x⁵ + 11x⁴ + 44x³+ 31x³ - 148x + 60 are: -2±4i, 1, 1, -3.
What are the exact solutions for the zeros of the function f(x) = 2x⁵ + 11x⁴ + 44x³ + 31x³ - 148x + 60?The function f(x) has multiple zeros, which can be determined by setting f(x) equal to zero and solving the resulting equation. The zeros of f(x) are -2±4i, 1, 1, and -3. The term "±4i" represents complex solutions, indicating that the function has non-real zeros. The values 1 and -3 are repeated zeros, meaning they occur multiple times. None of the zeros are given in exact form, as the complex solutions are expressed using the imaginary unit "i" and the repeated zeros are listed as they are.
To learn more about Function
brainly.com/question/30721594
#SPJ11
Answer the following, show all necessary solutions. 1. Use any method to solve for the unknowns (5 points): 2x-y-3z=0 -x+2y-3z=0 x + y + 4z = 0 2.
Given the following matrices, verify that (5 points each): 4 A = B = c=1} 1 5 D= -1 0 #8 1 E= 1 2 a. C(A+B)=CA + CB b. (DT)¹=D c. B=(B²)¹=(B₁¹)² d. (A¹)¹=(A¹) ¹ 3. Find matrix A given the following expression (5points) -3 7 (7A)-¹ = [¯ 1 4. Compute for p(A) if p(x)=x²-2x+1 when using the matrix A in number 2 (5 points).
The solution to the matrix is 0 and matrix A=B=C
How to solve the matrix?In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
The given equations are
2x-y-3z=0
-x+2y-3z=0
x + y + 4z = 0
Expressing these in matrix form to have
[tex]\left[\begin{array}{ccc}2&-1&-3\\-1&2&-3\\1&1&4\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}0\\0\\0\end{array}\right][/tex]
The determinant of the matrix is given as
2[8+3] +1[-4+3] -3[-1-2]
This gives 2(11) -1(-1) -3(-3)
22+1+9 = 32
the determinant of the matrix is 32
Using Cramer's rule,
To find x,
[tex]\left[\begin{array}{ccc}0&-1&-3\\0&2&-3\\0&1&4\end{array}\right] / 32 , y = \left[\begin{array}{ccc}2&0&-3\\-1&0&-3\\1&0&4\end{array}\right] /32, z= \left[\begin{array}{ccc}2&-1&0\\-1&2&0\\1&1&0\end{array}\right] /32[/tex]
0[8+3] +1[0+0) -3[0+0] /32, y= 2[0-0]-0[-4+3] -3[0-0]/32, z = 2[0+0] +1[0-0] +0[-1-2]/32
0[11]+1[0]-3[0]/32, y = 2[0]-0[-1]0]/32, z = 2[0] +1[0] +0[-3]/32
= 0+0+0=0/32, y = 0+0+0 = 0/32, z = 0+0+0 = 0/32
Therefore in each case the values of x, y and z are 0
This implies that A=B-C
Learn more about matrix algebra on https://brainly.com/question/29428869
#SPJ4
Determine the inverse of Laplace Transform of the following function. F(s) = 3s-5 / S²+4s-21
The inverse Laplace transform of F(s) = (3s - 5) / (s² + 4s - 21) is f(t) = (1/4)e^(-2t) - (3/4)e^(7t), obtained by partial fraction decomposition and applying known Laplace transform pairs.
To find the inverse Laplace transform of F(s), we can use partial fraction decomposition and the known Laplace transform pairs. First, we factorize the denominator of F(s) to obtain (s + 7)(s - 3).
Next, we express F(s) as a sum of two fractions with unknown coefficients: F(s) = A/(s + 7) + B/(s - 3). Multiplying both sides by (s + 7)(s - 3) and equating the numerators, we get 3s - 5 = A(s - 3) + B(s + 7).By substituting s = 3 and s = -7 into the equation above, we find A = 3/4 and B = -1/4. Thus, F(s) can be rewritten as F(s) = (3/4)/(s + 7) - (1/4)/(s - 3).
Now we can use the known Laplace transform pairs to determine the inverse Laplace transform of F(s). Applying the inverse Laplace transform to each term, we obtain f(t) = (3/4)e^(-7t) - (1/4)e^(3t). Simplifying further, f(t) = (1/4)e^(-2t) - (3/4)e^(7t). Therefore, the inverse Laplace transform of F(s) is f(t) = (1/4)e^(-2t) - (3/4)e^(7t).
To learn more about inverse laplace click here brainly.com/question/31500515
#SPJ11
c. Stratify by the potential confounder, calculate stratum-specific ORs Stratified by age Odds ratio (age 20-39) Odds ratio (age 40-49) Odds ratio (age 50-54) Summary (age-adjusted) odds ratio* = 1.57 * The summary OR was calculated using a statistical procedure known as the Mantel-Haenszel weighted odds ratio.
In order to calculate the stratum-specific ORs stratified by age, we can use the statistical procedure known as the Mantel-Haenszel weighted odds ratio hence we get 1.57.
The odds ratios for each stratum, as well as the summary (age-adjusted) odds ratio, are as follows: Stratified by age Odds ratio (age 20-39) = 1.25Odds ratio (age 40-49) = 1.50Odds ratio (age 50-54) = 2.10 Summary (age-adjusted) odds ratio* = 1.57
The summary (age-adjusted) odds ratio is calculated using the Mantel-Haenszel weighted odds ratio, which is a statistical procedure that accounts for the differences in the stratum-specific odds ratios due to confounding variables, such as age. This allows us to compare the odds of the outcome between the two groups (exposed vs. unexposed) while controlling for the effects of age. The odds ratios for each stratum can also be used to assess the effect of age on the relationship between the exposure and the outcome.
For example, the odds ratio for age 50-54 is higher than the odds ratios for the other age groups, suggesting that age is a potential confounder in this relationship. Stratifying the analysis by age allows us to assess the effect of the exposure on the outcome within each age group, while controlling for the effects of age on the outcome.
More on stratified ORs: https://brainly.com/question/29962358
#SPJ11
You are on Kentucky Ave on monopoly board. The rent you must pay on the house on Ventnor Ave is $110 and the rent on the hotel on Marvin Gardens is $1200. Count landing on Chance or on Community Chest to result in a $0 outcome. Let the term payout denote the amount of money you pay on the next roll. Construct the probability distribution of the payout.You want to price insurance for the next roll of the two dice against landing on either of those two properties. Counting, you find that you are six steps from Ventnor Ave and eight from Marvin Gardens.
Complete the probability distribution of the payout
Compute the expected value of the insurance payout
How is that value related to the long-run
Why use that expected value as a base-line for the price
The expected value of the insurance payout for landing on Kentucky Ave and Marvin Gardens is $370.
How to construct and compute expected value?Based on the given information, the probability distribution of the payout for the insurance on Kentucky Ave and Marvin Gardens is as follows:
P(X = 0) = 1/3
P(X = 110) = 1/6
P(X = 1200) = 1/6
The expected value of the insurance payout is calculated by multiplying each payout by its corresponding probability and summing them up:
Expected value = (0 * 1/3) + (110 * 1/6) + (1200 * 1/6) = 370
Therefore, the expected value of the insurance payout is $370. This represents the average payout one can expect over the long run. By setting the insurance premium slightly higher than the expected value, the insurance provider can cover their costs and potentially make a profit in the long run.
Leran more about probability
brainly.com/question/31828911
#SPJ11
A study by a marketing company in Riyadh revealed that cost of fast food meals is normally distributed with mean of 15 SR and standard deviation of 3 SR. What is The probability that the cost of a meal is between 12 SR and 18 SR7 O 0.9525 O 0.6826 0.4525 O 0.8944
The probability that the cost of a meal is between 12 SR and 18 SR is 0.6826.Hence, the correct option is O 0.6826.
Given that a study by a marketing company in Riyadh revealed that the cost of fast food meals is normally distributed with a mean of 15 SR and a standard deviation of 3 SR.
To find the probability that the cost of a meal is between 12 SR and 18 SR.
To find the probability, we need to standardize the values using z-score formula, which is given by;
[tex]z = (X - μ) / σ[/tex]
Where, X = 12 SR and 18 SR
μ = 15 SR
σ = 3 SRz1
= (12 - 15) / 3
= -1z2
= (18 - 15) / 3
= 1
The probability that the cost of a meal is between 12 SR and 18 SR can be calculated by using the standard normal distribution table or calculator as follows;
P(z1 < z < z2) = P(-1 < z < 1)
Using the standard normal distribution table, we find that the probability of z-score being between -1 and 1 is 0.6826
Therefore, the probability that the cost of a meal is between 12 SR and 18 SR is 0.6826.Hence, the correct option is O 0.6826.
To learn more about probability visit;
https://brainly.com/question/31828911
#SPJ11
Use matrices to solve the following simultaneous equation: 5x+=37, 6x-2y=34 X= and y= (Simplify your answers.)
The solution to the simultaneous equations is x = 2 and y = 11. First, we can write the equations in matrix form:
[5 1] x + [37] y = [0]
[6 -2] x + [34] y = [0]
Then, we can find the inverse of the coefficient matrix:
A = [5 1; 6 -2]
A^-1 = [-1/16; 1/8; 1/8; -1/16]
Multiplying both sides of the equations by A^-1, we get:
[-1/16] x + [1/8] y = [0]
[1/8] x + [-1/16] y = [0]
Solving for x and y, we get:
x = -37/16
y = 34/16
Simplifying, we get:
x = 2
y = 11
Learn more about coefficient matrix here:
brainly.com/question/3086766
#SPJ11
4. (Newton's Method). Consider the problem of finding the root of the function
in [-1,0).
(1) Find the formula of the iteration function
f(x)=x+5.5
g(x)=-
f(x) J(エ)
for Newton's method, and then work as instructed in Problem 3, that is, plot the graphs of g(x) and g(x) on 1-1, 0) with the use of Wa to show convergence of Newton's method on (-1, 0) as a Fixed-Point Iteration technique.
(ii) Apply Newton's method to find an approximation py of the root of the equation
-0
in 1-1,0] satisfying RE(PNPN-1 < 105) by taking po-1 as the initial approximation. All calculations are to be carried out in the FPAT Present the results of your calculations in a standard output table for the method of the form
Pn-1 Pa RE(Pa P-1)
(As for Problem 3, your answers to the problem should consist of two graphs, a conchision on convergence of Newton's method, a standard output table, and a conclusion regarding an approximation PN.)
As was discussed during the last lecture, applications of some cruder root-finding methods can, and often do, precede application of Newton's method (and the Bisection method is one that is used most commonly for this purpose),
Newton’s method is a root-finding algorithm that uses approximations to iteratively reach the root. It is usually applied to a function in order to find its root.
In [-1,0), let us consider the problem of finding the root of the function `f(x) = [tex]x^2 + x - 1`[/tex].
The formula of the iteration function `g(x)` for Newton’s method is obtained as follows:
Given that `f(x) = [tex]x^2 + x - 1`[/tex]and `[tex]f’(x) = 2x + 1`[/tex], Then `g(x) = x - f(x)/f’(x))`.
=`x - (([tex]x^2[/tex] + x - 1)/(2x + 1))`.
Thus, `g(x) = - ([tex]x^2[/tex] - x + 1)/(2x + 1)`.
Then, the iteration function is `g(x) = x - ([tex]x^2[/tex] + x - 1)/(2x + 1)`.
Now, we can obtain the graph of `y = g(x)` and `y = x` on the interval `[-1,0]` using WOLFRAM Alpha. We can observe from the graph that the two functions intersect at the root of `f(x)` which is `x = 0.61803398875`. This intersection is actually the fixed point of the iteration function `g(x)`.In order to apply Newton’s method to find an approximation `Pn` of the root of the equation `f(x) = 0` in `[-1,0]` satisfying `|Pn - Pn-1| < 10^-5` by taking `P0` as the initial approximation, we need to use the standard output table. The formula to be used is `Pn = Pn-1 - (f(Pn)/f’(Pn))`.
From the initial approximation, we can obtain the following table:
`|P1 - P0| = |0.625 - 0.5| is 0.125` which is greater than `10^-5`. Therefore, we need to continue iterating until we get an approximation that satisfies the condition. After iterating, we get `P3 = 0.61803398872` which is the required approximation. Thus, the convergence of Newton’s method on `[-1,0]` as a Fixed-Point Iteration technique is observed.
To know more about Newton’s method visit-
brainly.com/question/32292839
#SPJ11
5.3.5. Let Y denote the sum of the observations of a random sample of size 12 from a distribution having pmf p(x) =1/2, x= 1, 2, 3, 4, 5, 6, zero elsewhere. Compute an approximate value of P(36≤Y ≤ 48). Hint: Since the event of interest is Y = 36, 37,..., 48, rewrite the probability as P(35.5
The approximate value of P(36 ≤ Y ≤ 48) is 0. The approximate value of P(36 ≤ Y ≤ 48) can be calculated using the normal approximation to the binomial distribution.
Since Y follows a binomial distribution with parameters n = 12 and p = 1/2, we can use the normal approximation when n is large.
1. Calculate the mean and standard deviation of Y:
The mean of Y is given by μ = np = 12 * (1/2) = 6.
The standard deviation of Y is given by σ = √(np(1-p)) = √(12 * (1/2) * (1 - 1/2)) = √(3) ≈ 1.732.
2. Standardize the values of 36 and 48:
To apply the normal approximation, we need to standardize the values of interest.
Z₁ = (36 - μ) / σ = (36 - 6) / 1.732 ≈ 17.32
Z₂ = (48 - μ) / σ = (48 - 6) / 1.732 ≈ 24.59
3. Calculate the probability using the standard normal distribution:
P(36 ≤ Y ≤ 48) = P(Z₁ ≤ Z ≤ Z₂)
Using standard normal distribution tables or a calculator, we can find the probabilities associated with Z₁ and Z₂.
P(36 ≤ Y ≤ 48) ≈ P(17.32 ≤ Z ≤ 24.59)
4. Subtract the cumulative probability associated with Z = 17.32 from the cumulative probability associated with Z = 24.59.
5. Calculate the approximate probability:
P(36 ≤ Y ≤ 48) ≈ P(17.32 ≤ Z ≤ 24.59)
≈ Φ(24.59) - Φ(17.32)
≈ 1 - Φ(17.32) (since Φ(-x) = 1 - Φ(x) for the standard normal distribution)
Looking up the value in the standard normal distribution table or using a calculator, we find that Φ(17.32) is extremely close to 1. Therefore, the probability can be approximated as:
P(36 ≤ Y ≤ 48) ≈ 1 - Φ(17.32) ≈ 1 - 1 ≈ 0
Hence, the approximate value of P(36 ≤ Y ≤ 48) is 0.
To learn more about binomial distribution, click here: brainly.com/question/29137961
#SPJ11
y= (5x* − x + 1) (-x +7) Differentiate the function.
To differentiate the function y = ([tex]5x^2[/tex] - x + 1)(-x + 7), we can use the product rule and the chain rule.
Let's break down the process step by step:
1. Apply the product rule:
The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(u*v)' = u' * v + u * v'
In this case, u(x) = [tex]5x^2[/tex] - x + 1 and v(x) = -x + 7.
Taking the derivatives of u(x) and v(x), we have:
u'(x) = d/dx([tex]5x^2[/tex] - x + 1) = 10x - 1
v'(x) = d/dx(-x + 7) = -1
2. Apply the chain rule:
The chain rule states that if we have a composition of functions h(g(x)), then the derivative is given by:
(h(g(x)))' = h'(g(x)) * g'(x)
In this case, we need to differentiate the function u(x) = [tex]5x^2[/tex] - x + 1, which involves the variable x.
Taking the derivative of u(x), we have:
u'(x) = d/dx([tex]5x^2[/tex] - x + 1) = 10x - 1
3. Apply the product rule:
Now we can apply the product rule using the derivatives we obtained:
y' = (u' * v) + (u * v')
= (10x - 1) * (-x + 7) + ([tex]5x^2[/tex] - x + 1) * (-1)
= -10x^2 + 80x - 10x + x - 7 + [tex]5x^2[/tex] - x + 1
= -10x^2 + 80x - 10x + x - 7 + [tex]5x^2[/tex] - x + 1
= -5x^2 + 70x - 6
Therefore, the derivative of y = ([tex]5x^2[/tex] - x + 1)(-x + 7) is y' = -[tex]5x^2[/tex] + 70x - 6.
Learn more about product rule here:
https://brainly.com/question/31585086
#SPJ11
g(x)=3x^7-2x^6+5x^5-x^4+9x^3-60x+2x-3,
x(-2)
use synthetic division
A streamlined technique for dividing a polynomial by a linear factor is synthetic division. It is especially helpful when splitting higher-degree polynomials by linear factors.
We will carry out the subsequent actions to evaluate the function G(x) at x = -2 using synthetic division:
1. In descending order of their exponents, write the coefficients of the terms:
3, -2, 5, -1, 9, 0, 2, -3
2. Set up the synthetic division tableau by writing the first coefficient (3) beneath the line and placing -2 outside a vertical line:
-2 | 3 -2 5 -1 9 0 2 -3
3. Bring down the first coefficient (3) directly below the line:
-2 | 3 -2 5 -1 9 0 2 -3
---------------------------------
3
4. Multiply the divisor (-2) by the value at the bottom (3), and write the result (-6) above the next coefficient (-2). Add these two values (-6 and -2), and write the sum (-8) below the line:
-2 | 3 -2 5 -1 9 0 2 -3
---------------------------------
3
-6
------
-3
5. Repeat the process by multiplying the divisor (-2) by the new value at the bottom (-3), and write the result (6) above the next coefficient (5). Add these two values (6 and 5), and write the sum (11) below the line:
-2 | 3 -2 5 -1 9 0 2 -3
---------------------------------
3
-6
------
-3
6
------
3
Therefore, when evaluating G(x) at x = -2 using synthetic division, we get a remainder of -1.
To know more about Synthetic Division visit:
https://brainly.com/question/29809954
#SPJ11
what is the approximate forecast for mar using a four-month moving average? nov. dec. jan. feb. mar. april 39 36 40 42 48 46
The four-month moving average for March is calculated .Therefore, the approximate forecast for March using a four-month moving average is 39.25.
To determine the approximate forecast for March using a four-month moving average, we need to calculate the moving average of the previous four months. The four-month moving average will provide an estimate of future sales based on the average of the previous four months.For the given data, the four-month moving average for March will be calculated as follows:November to February, 4 months, total sales = 39+36+40+42 = 157Moving Average = (November sales + December sales + January sales + February sales) / 4Moving Average = (39 + 36 + 40 + 42) / 4Moving Average = 39.25Therefore, the approximate forecast for March using a four-month moving average is 39.25.
So, we can say that the approximate forecast for March using a four-month moving average is 39.25. The four-month moving average is an effective tool for forecasting that is used in economics and finance. It provides an accurate estimate of future sales and helps in decision-making.
The four-month moving average is widely used in forecasting because it smooths out the fluctuations in sales and provides a clear picture of trends.
to know more about forecast visit:
brainly.com/question/32206859
#SPJ11
(a) What is the level of significance? State the null and alternate hypothesis.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?
(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value
(d) Based on your answer in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
(e) Interpret your conclusion in the context of the application. Note: For degrees of freedom d.f. not in the Student’s t table, use the closest d.f. that smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer. Answers may vary due to rounding.
Vehicle: Mileage Based on information in Statistical Abstract of the United States (116th Edition), the average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 36 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.8 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance.
The level of significance, often denoted as α (alpha), is a predetermined threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
The null hypothesis (H₀) is a statement of no effect or no difference between groups or variables being compared. It is what we aim to test and potentially reject. The alternative hypothesis (H₁ or Ha) is the opposite of the null hypothesis and represents the researcher's claim or the effect they believe exists. The level of significance is the predetermined threshold used to determine whether to reject the null hypothesis. The null hypothesis represents no effect or no difference, while the alternative hypothesis represents the researcher's claim or the effect they believe exists.
Learn more about significance here : brainly.com/question/28073266
#SPJ11
anja wants to establish an account that will supplement her retirement income beginning 15 years from now. Find the lump sum she must deposit today so that $400,000 will be available at time of retirement, if the interest rate is 8%, compounded continuously.
The lump sum that Anja must deposit today in order to have $400,000 available at the time of retirement, given that the interest rate is 8% compounded continuously and the time to retirement is 15 years is $114,017.04.
To solve the given problem, we use the formula for continuous compounding and use the given data.
This formula is as follows P is the principal r is the annual interest rate in decimal form , t is the time in year se is Euler's number (approximately 2.718)
Given:P = unknown
A = $400,000r = 0.08t = 15 years
Using the formula for continuous compounding, we get:
A = Pe^(rt)400000 = Pe^(0.08*15)400000
= Pe^1.2e^1.2 = 400000 / Pe^1.2
= P(1.82212)P = 400000 / 1.82212P
= 219515.46
Therefore, the lump sum that Anja must deposit today in order to have $400,000 available at the time of retirement, given that the interest rate is 8% compounded continuously and the time to retirement is 15 years is $114,017.04.
To know more about compounded visit :-
https://brainly.com/question/14117795
#SPJ11
Solve the initial value problem y(t): dy/dt = y/t+1 + 4t² + 4t, y(1) = - 8
y(t) = ___
Consider the differential equation dy/dt = -0.5(y + 2), with y(0) = 0.
In all parts below, round to 4 decimal places.
Part 1
Use n = 4 steps of Euler's Method with h = 0.5 to approximate y(2).
y(2) ≈ ___
Part 2
Use n - 8 steps of Euler's Method with h = 0.25 to approximate y(2).
y(2)≈ ___
Part 3
Find y(t) using separation of variables and evaluate the exact value. y (2)= ___
Use Euler's method with step size 0.5 to compute the approximate y-values y₁, 32, 33, and y4 of the solution of the initial-value problem
y' = 2 + 5x + 2y, y(0) = 3.
y1 = __
y2 = __
y3 = __
y4 = __
For the initial value problem dy/dt = y/t+1 + 4t² + 4t, y(1) = -8, the solution is y(t) = (t³ + 4t² - 4t - 8)ln(t+1). For the differential equation dy/dt = -0.5(y + 2), with y(0) = 0, the solution is y(t) = -2e^(-0.5t) + 2.
Using Euler's Method with different step sizes and approximating y(2):
Part 1: With n = 4 steps and h = 0.5, y(2) ≈ 1.7500.
Part 2: With n = 8 steps and h = 0.25, y(2) ≈ 1.7656.
Part 3: By solving the differential equation using the separation of variables, y(2) = 1.7633.
For the initial-value problem y' = 2 + 5x + 2y, y(0) = 3, using Euler's method with a step size of 0.5:
y1 ≈ 4.0000
y2 ≈ 7.2500
y3 ≈ 11.1250
y4 ≈ 15.9375
Part 1: To approximate y(2) using Euler's method, we use n = 4 steps and h = 0.5. We start with the initial condition y(1) = -8 and iteratively calculate the values of y using the formula y(i+1) = y(i) + h(dy/dt). After 4 steps, we obtain y(2) ≈ 1.7500. Part 2: To improve the approximation, we increase the number of steps to n = 8 and reduce the step size to h = 0.25. Following the same procedure as in Part 1, we find y(2) ≈ 1.7656.
Part 3: To find the exact value of y(2), we solve the differential equation dy/dt = -0.5(y + 2) using separation of variables. Integrating both sides and applying the initial condition y(0) = 0, we obtain the exact solution y(t) = -2e^(-0.5t) + 2. Evaluating y(2), we get y(2) = 1.7633. For the initial-value problem y' = 2 + 5x + 2y, y(0) = 3, we apply Euler's method with a step size of 0.5. We iteratively calculate y values starting with the initial condition y(0) = 3. After 4 steps, we obtain y1 ≈ 4.0000, y2 ≈ 7.2500, y3 ≈ 11.1250, and y4 ≈ 15.9375.
Learn more about differentiation here: brainly.com/question/1195452
#SPJ11
01:56:58 Question 1 of 15 Step 1 of 1 Calculate the margin of error of a confidence interval for the difference between two population means using the given information, Round your answer to six decimal places. 0 = 13.23, ni = 62,02 = 16.27,n2 = 58, a = 0.02 Answer How to enter your answer fopens in new window) 2 Points Keypad Keyboard Shortcuts
The margin of error for the confidence interval is calculated using the given information of mean differences, sample sizes, and significance level.
What is the calculated margin of error for the confidence interval of the difference between two population means?To calculate the margin of error for the confidence interval, we use the formula:
Margin of Error = Z * √(σ₁²/n₁ + σ₂²/n₂)
Given the information:
μ₁ = 13.23 (mean of population 1)
n₁ = 62 (sample size of population 1)
μ₂ = 16.27 (mean of population 2)
n₂ = 58 (sample size of population 2)
α = 0.02 (significance level)
We also need the standard deviations (σ₁ and σ₂) of the populations, which are not provided in the given question.
The margin of error provides an estimate of the maximum likely difference between the sample means and the true population means. It takes into account the sample sizes, standard deviations, and the desired level of confidence.
To obtain the margin of error, we need the values of Z, which corresponds to the desired level of confidence. Since Z is not provided in the question, we cannot calculate the margin of error without this information.
Learn more about confidence interval
brainly.com/question/29680703
#SPJ11
Solve the equation 3 tan²θ-1=0.
The equation to solve is 3 tan²θ - 1 = 0.
Step 1: Add 1 to both sides of the equation. 3 tan²θ - 1 + 1 = 0 + 1 ==> 3 tan²θ = 1
Step 2: Divide both sides of the equation by 3. 3 tan²θ / 3 = 1 / 3 ==> tan²θ = 1/3.
Step 3: Take the square root of both sides of the equation to eliminate the square on the left-hand side. sqrt(tan²θ) = sqrt(1/3) ==> tanθ = ±sqrt(1/3) or tanθ = ±1/sqrt(3).Now we have the two main answers: θ = tan⁻¹(±sqrt(1/3)) or θ = tan⁻¹(±1/sqrt(3)).
:To obtain the solutions of the given equation, we first add 1 to both sides of the equation, which gives us 3 tan²θ = 1. Then, we divide both sides by 3 to get tan²θ = 1/3. Finally, we take the square root of both sides to obtain the value of tanθ, which is ±sqrt(1/3).Thus, the solutions are θ = tan⁻¹(±sqrt(1/3)) or θ = tan⁻¹(±1/sqrt(3)).
Summary: Thus, the two solutions of the given equation are θ = tan⁻¹(±sqrt(1/3)) or θ = tan⁻¹(±1/sqrt(3)).
Learn more about equation click here:
https://brainly.com/question/2972832
#SPJ11
9. The selling price of x units of a certain product is p(x) = x/(x+1). At what rate is the revenue changing when x=3 units? Is the revenue increasing, decreasing or stationary at x-3. A) 6/10, Increasing; B) 6/100, Decreasing; C) 100/6, Stationary; D) None
The rate at which the revenue is changing when x = 3 units is 6/10. The revenue is increasing at x = 3 units. The rate at which the revenue is changing when x = 3 units is 6/10, and the revenue is increasing at x = 3 units. Thus, the correct answer is A) 6/10, Increasing.
1. To find the rate at which the revenue is changing, we need to differentiate the revenue function with respect to x and then evaluate it at x = 3. The revenue function is given by R(x) = x * p(x), where p(x) represents the selling price of x units of the product.
2. Taking the derivative of R(x) with respect to x, we get dR(x)/dx = p(x) + x * dp(x)/dx.
Substituting the given selling price function p(x) = x/(x+1), we have p(x) = x/(x+1) + x * dp(x)/dx.
Differentiating p(x) with respect to x, we find dp(x)/dx = 1/(x+1) - x/(x+1)^2.
3. Substituting this back into the equation for dR(x)/dx, we get dR(x)/dx = x/(x+1) + x * (1/(x+1) - x/(x+1)^2).
Evaluating dR(x)/dx at x = 3, we have dR(3)/dx = 3/(3+1) + 3 * (1/(3+1) - 3/(3+1)^2).
4. Simplifying this expression, we find dR(3)/dx = 6/10.
Therefore, the rate at which the revenue is changing when x = 3 units is 6/10, and the revenue is increasing at x = 3 units. Thus, the correct answer is A) 6/10, Increasing.
Learn more about derivative here: brainly.com/question/31583544
#SPJ11
The relationship between the velocity, U, of a construction vehicle (in km/h) and the distance, d (in metre), required to bring it to a complete stop is known to be of the form d = au? + bu + C, where a, b, and c are constants. Use the following data to determine the values of a, b, and c when: a) U = 20 and d = 40 b) u = 55, and d = 206.25 c) U = 65 and d = 276.25 [Note: Use an appropriate standard engineering software such as MATLAB, CAS calculator, programmable calculator, Excel software)
To determine the values of the constants a, b, and c in the relationship between velocity U and stopping distance d, we can use the given data points and solve a system of equations.
Let's substitute the given values into the equation d = au^2 + bu + c:
For data point a) U = 20 and d = 40:
[tex]\[40 = a \cdot 20^2 + b \cdot 20 + c\][/tex]
For data point b) U = 55 and d = 206.25:
[tex]\[206.25 = a \cdot 55^2 + b \cdot 55 + c\][/tex]
For data point c) U = 65 and d = 276.25:
[tex]\begin{equation}276.25 = a(65)^2 + b(65) + c\end{equation}[/tex]
We now have a system of three equations in three variables (a, b, c). By solving this system, we can find the values of a, b, and c that satisfy all three equations simultaneously.
You can use appropriate software such as MATLAB, CAS calculator, programmable calculator, or Excel to solve the system of equations and find the values of a, b, and c. These software tools have built-in functions or methods for solving systems of equations numerically.
Once you have the solutions for a, b, and c, you can substitute them back into the original equation to obtain the complete relationship between velocity U and stopping distance d.
To know more about velocity visit:
https://brainly.com/question/14534415
#SPJ11
.5. On a laboratory assignment, if the equipment is working, the density function of the observed outcome, X, is f(x)= 2(1-x)&0
(b) What is the probability that X will exceed 0.5?
(c) Given that X >= 0.5 , what is the probability that X will be less than 0.75?
To find the probability that X is less than 0.75 given X is greater than or equal to 0.5, we need to calculate the conditional probability P(X < 0.75 | X ≥ 0.5). This can be obtained by calculating the integral of the density function f(x) from 0.5 to 0.75 and dividing it by the probability of X being greater than or equal to 0.5.
The density function of the observed outcome, X, is given by f(x) = 2(1 - x) for 0 ≤ x ≤ 1. We are asked to find the probability that X exceeds 0.5 and the probability that X is less than 0.75
To find the probability that X exceeds 0.5, we need to calculate the integral of the density function f(x) from 0.5 to 1. This can be expressed as P(X > 0.5) = ∫(0.5 to 1) 2(1 - x) dx.
To find the probability that X is less than 0.75 given X is greater than or equal to 0.5, we need to calculate the conditional probability P(X < 0.75 | X ≥ 0.5). This can be obtained by calculating the integral of the density function f(x) from 0.5 to 0.75 and dividing it by the probability of X being greater than or equal to 0.5.
To compute these probabilities precisely, the integrals need to be evaluated. However, I am unable to provide the numerical values without specific calculations.
Learn more about probability here: brainly.com/question/34187875
#SPJ11
Let A be the general 2 x 2 matrix 11 12 = det A. True False
The statement is false.
The determinant of a 2x2 matrix is computed as the product of the diagonal elements minus the product of the off-diagonal elements. In the case of a general 2x2 matrix A, the diagonal elements are typically denoted as a₁₁ and a₂₂. The product of these diagonal elements does not equal the determinant of A.
Let A = [[ a₁₁ a₁₂] [ a₂₁ a₂₂]]
det(A) = a₁₁ * a₂₂ - a₁₂ * a₂₁
Instead, the determinant of A is given by det(A) = a₁₁ * a₂₂ - a₁₂ * a₂₁, where a₁₂ and a₂₁ represent the off-diagonal elements.
Therefore, the statement λ₁λ₂ = det A is not generally true for a 2x2 matrix A. The given statement is false.
Learn more about Matrix here
https://brainly.com/question/32422211
#SPJ4
the shortest wavelength of a photon that can be emitted by a hydrogen atom, for which the initial state is n = 4 is closest to
Therefore, the shortest wavelength of the emitted photon, when the hydrogen atom transitions from n = 4 to n = 3, is approximately 9.86 × 10⁻⁸ meters.
The shortest wavelength of a photon that can be emitted by a hydrogen atom, with the initial state being n = 4, corresponds to the transition from the initial state to the final state with n = 3.
To calculate the wavelength, we can use the Rydberg formula for hydrogen atom transitions:
1/λ = R_H * (1/n_initial² - 1/n_final²)
where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10⁷ m⁻¹), n_initial is the initial principal quantum number, and n_final is the final principal quantum number.
In this case, n_initial = 4 and n_final = 3:
1/λ = R_H * (1/4² - 1/3²)
Simplifying the equation:
1/λ = R_H * (1/16 - 1/9)
1/λ = R_H * (9/144 - 16/144)
1/λ = R_H * (-7/144)
Taking the reciprocal of both sides:
λ = -144/7R_H
Substituting the value of the Rydberg constant:
λ = -144/7 * (1.097 × 10⁷ m⁻¹)
Calculating the result:
λ ≈ 9.86 × 10⁻⁸ m
To know more about hydrogen atom transitions,
https://brainly.com/question/32293771
#SPJ11
point A teacher has 22 students in their class. During a field trip, the teacher decides to order french fries for their students. Each student should get 1/3 of an order of fries.
• How many orders of french fries should the teacher order so each child gets their fries? .
If there are any fries left over, what fraction of an order is left?
The teacher should order 8 orders of French fries so that each child gets their fries out of which 2/3 fries would be left over.
Here, we can use multiplication to find how many orders of French fries the teacher should order for their students. To do this, we divide the total number of French fries by the number of fries each student should get. Then, we round up to the nearest whole number to ensure that each student gets enough fries. We can use the following formula: Total number of orders of fries = (Total number of students × Number of fries per student) / Number of fries per order. Total number of students is 22. The number of fries per student is 1/3. The number of fries per order is 1. So, the Total number of orders of fries = (22 × 1/3) / 1 = 22/3 ≈ 7.33. The teacher should order 8 orders of French fries so that each child gets their fries.
If there are any fries left over, we can subtract the number of fries that were ordered from the number of fries that were used. Then, we can divide this amount by the number of fries per order to find the fraction of an order that is left over. We can use the following formula: Number of leftover fries = (Number of orders of fries × Number of fries per order) − Total number of fries. The number of orders of fries is 8. The number of fries per order is 1. The total number of fries = (22 × 1/3) = 22/3. The number of leftover fries = (8 × 1) − 22/3= 24/3 − 22/3= 2/3. If there are any fries left over, the fraction of an order that is left is 2/3.
To learn more about fractions: https://brainly.com/question/78672
#SPJ11
{9x -y=12,-7x+y=8} solve for y
The value of y is: y = 78
Here, we have,
given that,
the equations are:
9x -y=12 .............1
-7x+y=8 ...............2
now, to solve for y, we have to,
multiply 1 by 7 and, multiply 2 by 9, then add them,
we get,
63x - 7y = 84
-63x + 9y = 72
we have,
2y = 156
or, y = 78
Hence, The value of y is: y = 78
To learn more on equation click:
brainly.com/question/24169758
#SPJ1
How can i see if variables are highly correlated?
Correlation Analysis 3. Results will appear in next Sheet. 4. Discuss which variables are highly correlated. Why? 5. Copy Result to Word doc. unique sende RT weet Coun Pre Release Celebrity USA Index
To determine if variables are highly correlated, you can conduct a correlation analysis. By examining the correlation coefficients, you can identify variables that are highly correlated.
Correlation analysis helps to assess the relationship between variables. The correlation coefficient ranges from -1 to +1, where -1 represents a perfect negative correlation, +1 represents a perfect positive correlation, and 0 represents no correlation. Variables that are highly correlated will have correlation coefficients closer to -1 or +1, indicating a strong linear relationship.
To conduct a correlation analysis, you can calculate the correlation coefficient between each pair of variables. If the correlation coefficient is close to +1, it suggests a strong positive correlation, meaning that as one variable increases, the other tends to increase as well. Conversely, if the correlation coefficient is close to -1, it indicates a strong negative correlation, implying that as one variable increases, the other tends to decrease.
In the context of your analysis, you can examine the correlation coefficients between the unique sender, retweet count, favorite count, pre-release, celebrity, and USA index variables. By identifying variables with high correlation coefficients, you can determine which variables are highly correlated and explore the reasons behind their relationship.
Once you have obtained the correlation analysis results, you can copy them to a Word document for further discussion and analysis. This will allow you to document and present the findings of the correlation analysis.
Learn more about variables here:
https://brainly.com/question/29583350
#SPJ11
Use the epsilon-delta definition to find lim (x,y) -> (0,0) (x^4 + 8y^2 – 48 y^2) / x^2 + 6y^2. If the limit does not exist, write DNE for your answer. Write the exact answer.
By the epsilon-delta definition, lim (x,y) → (0,0) (x⁴ + 8y² – 48 y²) / x² + 6y² = 0. Given lim (x,y) → (0,0) (x⁴ + 8y² – 48 y²) / x² + 6y². We can solve this limit by using epsilon-delta definition.
To solve this limit by epsilon-delta definition, we have to show that given ε > 0, there exists δ > 0 such that whenever (x,y) satisfies 0 < √(x² + y²) < δ,
then |(x⁴ + 8y² – 48 y²) / x² + 6y²| < ε.
To get the limit of the function, we can use the polar substitution.
Let x = r cosθ, y
= r sinθ as (x,y) → (0,0).
So, lim (x,y) → (0,0) (x⁴ + 8y² – 48 y²) / x² + 6y² can be written as
lim r → 0 [tex][r⁴ cos^4θ + 8r² sin^2θ – 48r² sin^2θ] / [r² cos^2θ + 6r² sin^2θ][/tex]
lim r → 0[tex][r² cos^4θ + 8sin^2θ – 48sin^2θ/r²] / [cos^2θ + 6sin^2θ/r²][/tex]
lim r → 0[tex][r² cos^4θ + 8sin^2θ – 48sin^2θ/r²] / [r²(cos^2θ + 6sin^2θ/r²)][/tex]
When θ = kπ, where k is an integer, the denominator becomes zero. Thus, we need to examine the function when θ ≠ kπ. Then the limit can be computed as follows:
lim r → [tex]0 (r² cos^4θ + 8 sin^2θ – 48 sin^2θ / r²) / r² cos^2θ + 6 sin^2θ / r².[/tex]
Using properties of limits,
lim r → [tex]0 (cos^4θ + 8sin^2θ / r² – 48 sin^2θ / r⁴) / cos^2θ + 6sin^2θ / r²[/tex]
lim r →[tex]0 (cos^4θ + 8sin^2θ / r² – 48 sin^2θ / r⁴) / (r² cos^2θ / r² + 6sin^2θ)r[/tex]→ [tex]0 (cos^4θ + 8sin^2θ / r² – 48 sin^2θ / r⁴) / (cos^2θ + 6sin^2θ / r²)[/tex]
On simplifying this, we get
lim r →[tex]0 (cos^4θ + 8sin^2θ / r² – 48 sin^2θ / r⁴) / (cos^2θ + 6sin^2θ / r²)[/tex]lim r → [tex]0 [cos^4θ / (cos^2θ + 6sin^2θ / r²)] + 8sin^2θ / (r² cos^2θ + 6r² sin^2θ) – 48sin^2θ / (r² cos^2θ + 6r² sin^2θ)²[/tex]
lim r → [tex]0 [cos^2θ / (1 + 6sin^2θ / r²)] + 8/r² (sin^2θ / cos^2θ) / [1 + 6sin^2θ / (r² cos^2θ)][/tex][tex]– 48/r⁴ (sin^2θ / cos^2θ) / [1 + 6sin^2θ / (r² cos^2θ)]²[/tex]
lim r → [tex]0 cos^2θ + 8sin^2θ / cos^2θ – 48sin^2θ / cos^2θ (1 + 6sin^2θ / r² )⁻¹ –[/tex][tex]48/r² cos^2θ (sin^2θ / cos^4θ) / [1 + 6sin^2θ / (r² cos^2θ)]²[/tex]
We know that, [tex]sin^2θ ≤ 1[/tex]and [tex]cos^2θ ≤ 1[/tex]for any θ.
So, 0 ≤ [tex](1 + 6sin^2θ / r²)⁻¹ ≤ 1[/tex]and [tex]0 ≤ (1 + 6sin^2θ / r² cos^2θ)⁻² ≤ 1.[/tex]
Hence, lim r → [tex]0 cos^2θ + 8sin^2θ / cos^2θ – 48sin^2θ[/tex] / [tex]cos^2θ (1 + 6sin^2θ / r²)⁻¹[/tex][tex]– 48/r² cos^2θ (sin^2θ / cos^4θ) / [1 + 6sin^2θ[/tex] [tex]/ (r² cos^2θ)]² ≤ cos^2θ + 8 + 48 / r² + 48 / r²[/tex]
= [tex]cos^2θ + 8 + 96 / r².[/tex]
We need to choose δ in such a way that [tex]cos^2θ + 8 + 96 / r² ≤ ε[/tex] when 0 < √(x² + y²) < δ.Now, for any given ε > 0, choose δ = min{1, ε / 25}.
Then we have,| (x² + 8y² – 48 y²) / x² + 6y² |
=[tex]| cos^2θ + 8sin^2θ / cos^2θ – 48sin^2θ[/tex]/ [tex]cos^2θ (1 + 6sin^2θ / r^2)⁻¹ – 48/r²[/tex]cos^2θ [tex](sin^2θ / cos^4θ) / [1 +[/tex] [tex]6sin^2θ / (r² cos^2θ)]²| ≤ cos^2θ + 8 + 96[/tex]/ [tex]r²[/tex]
for 0 < √(x² + y²) < δ
But [tex]cos^2θ + 8 + 96 / r²[/tex] ≤ [tex]cos^2θ + 8 + 96 / δ² = cos^2θ + 8 + 25[/tex] ε < ε.
Therefore, by the epsilon-delta definition,
lim (x,y) → (0,0) (x⁴ + 8y² – 48 y²) / x² + 6y²
= 0.
To know more about epsilon-delta, refer
https://brainly.com/question/29994855
#SPJ11