The null space solution can be written as the linear combination of columns 3 and 5 of the given matrix.
The given matrix is:
⎣⎡136701−211−20034⎦⎤
The row reduced echelon form of the matrix is:
⎣⎡1001000000000100⎦⎤
We can observe that the last row is already in row echelon form.
The remaining rows have leading 1s at different columns.
So we can create a basis for the null space by placing parameters for the non-pivot variables and then write the solutions in terms of the free variables.
x1 = -t3x2 = 0x3 = t1x4 = -t2x5 = 0
The null space of the given matrix can be represented as:
⎡⎢⎢−t300−t2000t1000⎤⎦⎥
Therefore, the solution of the null space of the given matrix is spanned by the set {A,B} where A and B are column vectors that can be used to write the solution of the null space in terms of these vectors.
So we need to represent the null space solution in the form: a A + b B
where a and b are scalars.
Let A and B be the standard basis vectors corresponding to the non-pivot columns of the row reduced echelon matrix.
So, A corresponds to the column with non-pivot column 3 and B corresponds to the column with non-pivot column 5.
The vectors A and B can be found as:
A = [tex]\begin{b matrix}-1 \\ 0 \\ 1 \\ 0 \\ 0\end{b matrix},[/tex]
B = \[tex]begin{b matrix}0 \\ 0 \\ 1 \\ 0 \\ 1\end{b matrix}[/tex]
The solution of the null space is obtained by taking linear combinations of A and B.
So the null space is given by:
\begin{aligned}\operator name{Null}(A)
&=\operator name{span}\{A,B\}\\&= \{a A + b B | a, b \in \math bb{R} \}\\
&=\Bigg\{a\begin{b matrix}-1 \\ 0 \\ 1 \\ 0 \\ 0\end{b matrix} + b \begin{b matrix}0 \\ 0 \\ 1 \\ 0 \\ 1\end{b matrix} \Bigg| a, b \in \math bb{R}\Bigg\} \end{aligned}
Hence the solution is spanned by {A,B}.
Since A corresponds to column 3, and B corresponds to column 5, we can say that the null space solution can be written as the linear combination of columns 3 and 5 of the given matrix.
Learn more about linear combination from this link:
https://brainly.com/question/8303882
#SPJ11
What is the formula for the circumference C of a circle of radius r ? What is the formula for the area A of a circle of radius r ? The formula for the circumference C of a circle of radius r is (Type an equation. Type an exact answer, using π as needed.) The formula for the area A of a circle of radius r is (Type an equation. Type an exact answer, using π as needed.) Complete the sentence below. On a circle of radius r, a central angle of θ radians subtends an arc of length s= the area of the sector formed by this angle θ is A= On a circle of radius r, a central angle of θ radians subtends an arc of length s= the area of the sector formed by this angle θ is A= 21πr2θ 21r2θ r2θ π2θ Complete the following sentence. If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d= If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d=
The circumference and the area of the circle of radius r is C=2πr and A = πr^2. If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d= rt.
The formula for the circumference C of a circle of radius r is given:
C = 2πr
The formula for the area A of a circle of radius r is given by:
A = πr^2
On a circle of radius r, a central angle of θ radians subtends an arc of length s = rθ.
The area of the sector formed by this angle θ is A = (1/2) r^2θ.
If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d = rt.
To learn more about circumference visit:
https://brainly.com/question/27447563
#SPJ11
Ex. 8 - Assumptions 2. Consider a regression model that uses 48 observations. Let e i
denote the residuals from the fitted regression and y
^
i
be the in-sample predicted values of the dependent variable. The least squares regression of e i
2
on y
^
i
has coefficient of determination 0.032. What can you conclude from this finding?
Expert Answer
The coefficient of determination of 0.032 suggests that the regression model has a weak fit to the data, as only a small proportion of the variation in the residuals can be explained by the predicted values of the dependent variable.
The coefficient of determination of 0.032 suggests that only a small proportion of the variation in the residuals (e i²) can be explained by the variation in the predicted values (y^i) of the dependent variable. This implies that the regression model does not adequately capture the relationship between the predictor variables and the dependent variable. In other words, the model does not provide a good fit to the data.
A coefficient of determination, also known as R-squared, measures the proportion of the total variation in the dependent variable that can be explained by the regression model. A value close to 1 indicates a strong relationship between the predictor variables and the dependent variable, while a value close to 0 suggests a weak relationship.
In this case, the coefficient of determination of 0.032 indicates that only 3.2% of the variability in the residuals can be explained by the predicted values. The remaining 96.8% of the variability is unaccounted for by the model. This low value suggests that the model is not capturing important factors or there may be other variables that are influencing the dependent variable but are not included in the model. It may be necessary to consider alternative models or gather additional data to improve the model's performance.
Learn more about regression model here: https://brainly.com/question/29855836
#SPJ11
If four years of college is expected to cost $150,000 18 years from now, how much must be deposited now into an account that will average 8% annually in order to save the $150,000? By how much would your answer change if you expected 11% annually? Use excel formulas to find the solution.
To save $150,000 for college expenses in 18 years, one would need to deposit approximately $46,356.90 with an 8% annual interest rate or $33,810.78 with an 11% annual interest rate.
To calculate the amount that must be deposited now into an account, we can use the future value of a lump sum formula in Excel.The formula to calculate the future value (FV) of an investment is: FV = PV * (1 + r)^n, where PV is the present value, r is the interest rate per period, and n is the number of periods.In this case, the future value (FV) is $150,000, the interest rate (r) is 8% or 0.08, and the number of periods (n) is 18.
Using the formula in Excel, the present value (PV) can be calculated as follows: PV = FV / (1 + r)^n
PV = $150,000 / (1 + 0.08)^18
PV = $46,356.90 , Therefore, approximately $46,356.90 must be deposited now into an account that will average 8% annually to save $150,000.If the expected annual interest rate is 11% instead of 8%, we can use the same formula to calculate the present value.
PV = $150,000 / (1 + 0.11)^18
PV = $33,810.78
Hence, if the expected annual interest rate is 11%, approximately $33,810.78 must be deposited now into the account to save $150,000.
To learn more about interest rate click here
brainly.com/question/13324776
#SPJ11
Determine the direction angle 0 of the vector, to the nearest degree. u= (-5, -7) 8= (Round to the nearest degree as needed.)
The direction angle of the vector u = (-5, -7) is approximately 50 degrees. To determine the direction angle of a vector, we can use the formula:
θ = arctan(y/x)
where (x, y) are the components of the vector.
Given the vector u = (-5, -7), we can calculate the direction angle as follows:
θ = arctan((-7)/(-5))
Using a calculator or trigonometric tables, we find:
θ ≈ 50.19 degrees
Rounding to the nearest degree, the direction angle of the vector u is 50 degrees.
To read more about angle, visit:
https://brainly.com/question/25716982
#SPJ11
A normal population has a mean of $88 and standard deviation of $7. You select random samples of 50. Required: a. Apply the central limit theorem to describe the sampling distribution of the sample mean with n=50. What condition is necessary to apply the central limit theorem? b. What is the standard error of the sampling distribution of sample means? (Round your answer to 2 decimal places.) c. What is the probability that a sample mean is less than $87 ? (Round z-value to 2 decimal places and final answer to 4 decimal places.)
a. The central limit theorem can be applied since the sample size is larger than 30.
b. The standard error of the sampling distribution of sample means is approximately $0.99.
c. The probability that a sample mean is less than $87 is approximately 0.1190.
a. The central limit theorem (CLT) states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, as long as the samples are selected randomly and independently.
To apply the central limit theorem, the sample size should typically be larger than 30. In this case, the sample size is 50, which satisfies the condition necessary to apply the central limit theorem.
b. The standard error of the sampling distribution of sample means can be calculated using the formula:
Standard Error = Standard Deviation / √(Sample Size)
Given that the population standard deviation is $7 and the sample size is 50, we can substitute these values into the formula:
Standard Error = 7 / √(50)
≈ 0.99
Therefore, the standard error of the sampling distribution of sample means is approximately $0.99.
c. To find the probability that a sample mean is less than $87, we need to standardize the value using the z-score formula:
z = (X - μ) / (σ / √n)
Where X is the value we want to find the probability for, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Substituting the given values:
z = (87 - 88) / (7 / √50)
≈ -1.18
To find the probability corresponding to the z-score, we can refer to the standard normal distribution table or use statistical software. Assuming a standard normal distribution, the probability can be found as P(Z < -1.18).
Using a standard normal distribution table or software, we find that the probability is approximately 0.1190.
Therefore, the probability that a sample mean is less than $87 is approximately 0.1190.
a. The central limit theorem can be applied since the sample size is larger than 30.
b. The standard error of the sampling distribution of sample means is approximately $0.99.
c. The probability that a sample mean is less than $87 is approximately 0.1190.
To know more about statistical software, visit
https://brainly.com/question/30368374
#SPJ11
Prove the following properties of sequences. a) Let a_n be a sequence such that a_2n→L and a_2n+1 →L, then a_n→L.
b) Let a_n and b_n be sequences such that lim_n→[infinity] a_n = L ≠ 0 and lim_n→[infinity] a_nb_n exists, then lim_n→[infinity] b_n exists.
c) Prove that every unbounded sequence contains a monotonic subsequence.
a) If a sequence a_n satisfies the condition a_2n → L and a_2n+1 → L, then the sequence a_n also converges to L.
b) If two sequences a_n and b_n satisfy the conditions lim_n→[infinity] a_n = L ≠ 0 and lim_n→[infinity] a_nb_n exists, then the sequence b_n also converges.
c) Every unbounded sequence contains a monotonic subsequence.
a) To prove this statement, we can consider the subsequence of a_n consisting of the even terms and the subsequence consisting of the odd terms. Since both subsequences converge to L, the original sequence a_n must also converge to L.
b) By the limit arithmetic property, if lim_n→[infinity] a_n = L ≠ 0 and lim_n→[infinity] a_nb_n exists, then lim_n→[infinity] (a_nb_n)/a_n = b_n exists. Since a_n tends to a non-zero value L, we can divide both sides of the equation by a_n to obtain the limit of b_n.
c) To prove this statement, we can use the Bolzano-Weierstrass theorem, which states that every bounded sequence contains a convergent subsequence. Since an unbounded sequence is not bounded, it must contain values that are arbitrarily large or small. By selecting a subsequence consisting of increasingly larger or smaller terms, we can obtain a monotonic subsequence. Therefore, every unbounded sequence contains a monotonic subsequence.
Hence, the properties a), b), and c) are proven to be true.
To learn more about original sequence: -brainly.com/question/17407461
#SPJ11
The following are the annual salaries of 19 chief executive officers of major companies. (The salaries are written in thousands of dollars.) 381,75,633,134,609,700,1250,362,790,814,157,586,542,743,271,676,495,405,653 Find 25 th and 90 th percentiles for these salaries. (If necessary, consult a list of formulas.) (a) The 25 th percentile: thousand dollars (b) The 90 th percentile: thousand dollars
90th percentile is $814,000.
To find the 25th and 90th percentiles for the given salaries, we need to first arrange the salaries in ascending order:
75, 157, 271, 362, 381, 405, 495, 542, 586, 609, 633, 653, 676, 700, 743, 790, 814, 1250
(a) The 25th percentile:
The 25th percentile represents the value below which 25% of the data falls. To find the 25th percentile, we need to calculate the position of the value in the ordered data.
The formula to find the position of the value is:
Position = (Percentile / 100) * (N + 1)
In this case, the 25th percentile corresponds to the position:
Position = (25 / 100) * (19 + 1) = 0.25 * 20 = 5
The 25th percentile will be the value at the 5th position in the ordered data, which is 405,000 dollars.
(b) The 90th percentile:
The 90th percentile represents the value below which 90% of the data falls. Similar to the 25th percentile, we need to calculate the position of the value in the ordered data.
The formula for the position remains the same:
Position = (Percentile / 100) * (N + 1)
In this case, the 90th percentile corresponds to the position:
Position = (90 / 100) * (19 + 1) = 0.9 * 20 = 18
The 90th percentile will be the value at the 18th position in the ordered data, which is 814,000 dollars.
Therefore, the 25th percentile is $405,000, and the 90th percentile is $814,000.
learn more about percentile
brainly.com/question/2416601
#SPJ11
Determine the exact value and include diagrams.
\( \# 1 . \) Determine the exact value for \( \cos \frac{7 \pi}{12} \). Include diagrams.
To determine the exact value for cos(7π/12) and to include diagrams, we should first understand the trigonometric ratios of 30°, 45°, and 60°.Consider the below image for understanding the trigonometric ratios:Trigonometric ratios.
From the above diagram, we know that sin let us look at the problem. We need to determine the exact value for cos(7π/12).For that, we will first convert 7π/12 into degrees. We know that cos is positive in the first and fourth quadrants and negative in the second and third quadrants. Thus, we need to determine the value of 105°/2 in which quadrant.
Let us divide 360° by 4, and we get 90°. Thus, 105°/2 lies between 90° and 180°, which is the second quadrant.So cos(105°/2) is negative.So, cos(7π/12) = - cos(105°/2)Now we will use the formula,
cos(2A) = cos²A - sin²A
and get cos(105°/2) in terms of 45° and 60°.
cos(2A) = cos²A - sin²A
cos(2A) = [2cos²A - 1]
Let's apply this to
cos(105°/2),105°/2 = 45° + 60°/2105°/2
= 15° + 90°/4 - 30°/2105°/2
= 30°/2 + 90°/4 - 30°/2105°/2
= (2 × 45° - 1) + 30°/2
cos(105°/2) = cos(2 × 45° - 1 + 30°/2)
cos(105°/2) = cos²45° - sin²45°
cos(105°/2) = [2cos²45° - 1]
cos(105°/2) = 2(√2/2)² - 1
cos(105°/2) = 2/2 - 1
cos(105°/2) = -1/2
cos(7π/12) = - cos(105°/2)
= - (-1/2) = 1/2
The exact value of cos(7π/12) is 1/2.
To know more about value visit :
https://brainly.com/question/29657983
#SPJ11
please explain
Separable Partial Differential Equations
What is the application?
Describe briefly how Separable Partial Differential Equations applies to the application.
Separable Partial Differential Equations refers to a type of differential equation that can be separated into two parts. One part consists of a function of one variable while the other part contains a function of another variable.
This means that the solution can be obtained by finding the integral of each of these parts separately.
The application of Separable Partial Differential Equations in mathematical modeling is useful in the development of computational models. These models are used to study various phenomena in physics, chemistry, biology, engineering, and many other fields.
Briefly, Separable Partial Differential Equations apply to the application in which the two functions in the differential equation can be separated and solved independently.
Afterward, the solutions are combined to form the final solution to the differential equation.
These types of equations are frequently used in modeling physical phenomena that are continuous and complex, which requires the use of a partial differential equation.
Learn more about Separable Partial Differential Equations from the given link
https://brainly.com/question/14747745
#SPJ11
what's n+15=-10 as a solution
Answer: n=-25
Step-by-step explanation:
n +15 = -10 >Subtract 15 from both sides
n = -25
Evaluate the sum. You mayNOT type it in your calculator. Show work! ∑ k=1
9
(k 2
−8)
Evaluating the given sum,
The answer is 165.
The given summation is to be evaluated.
The given summation is [tex]∑_{k=1}^{9} (k^{2}-8)[/tex].
First, we must expand k^{2}-8. [tex]$$k^{2}-8=k^{2}-2^{2}=(k+2)(k-2).$$[/tex]
Thus, we can write the sum as $$\begin{aligned} \sum_[tex]{k=1}^{9}(k^{2}-8[/tex])&
=\sum_[tex]{k=1}^{9}\{(k+2)(k-2)\}[/tex] \\ &
=\sum_[tex]{k=1}^{9}(k+2)(k-2)[/tex]. \end{aligned}$$
We'll expand $(k+2)(k-2)$ and rearrange the terms of the sum: $$\begin{aligned} \sum_{k=1}^{9}(k^{2}-8)&
=\sum_[tex]{k=1}^{9}\{(k+2)(k-2)\}[/tex]\\ &
=\sum_[tex]{k=1}^{9}(k^{2}-4k-2k+8)[/tex] \\ &
=\sum_[tex]{k=1}^{9}(k^{2}-6k+8)[/tex] \\ &
=\sum_[tex]{k=1}^{9}k^{2}-\sum_{k=1}^{9}[/tex] 6k+\sum_[tex]{k=1}^{9}[/tex]8 \\ &
=[tex]\frac{(9)(10)(19)}{6}-6\frac{(9)(10)}{2}+8(9)[/tex] \\ &
=\boxed{165}. \end{aligned}$$
Therefore, the answer is 165.
To know more about sum, visit:
https://brainly.com/question/31538098
#SPJ11
The value of the given sum is 70.5.
To evaluate the sum ∑(k^2 - 8) from k = 1 to 9, we can use the formula for the sum of squares:
∑(k^2) = n(n + 1)(2n + 1) / 6
∑(1) = n
∑(8) = 8n
Using these formulas, we can break down the given sum as follows:
∑(k^2 - 8) = ∑(k^2) - ∑(8)
Using the formula for the sum of squares:
∑(k^2 - 8) = [9(9 + 1)(2(9) + 1) / 6] - (8 * 9)
Simplifying the numerator:
∑(k^2 - 8) = [9(10)(19) / 6] - (72)
Calculating the numerator:
∑(k^2 - 8) = (90 * 19 / 6) - 72
Simplifying further:
∑(k^2 - 8) = (285 / 2) - 72
Now, subtracting:
∑(k^2 - 8) = 142.5 - 72
∑(k^2 - 8) = 70.5
Therefore, the value of the given sum is 70.5.
To know more about squares, visit:
https://brainly.com/question/28776767
#SPJ11
In your own words, explain how a repeated measures analysis of
variance can result in an increase in power to detect an effect
A repeated measures analysis of variance (ANOVA) is a statistical technique used to analyze data collected from the same subjects or participants at multiple time points or under different conditions.
It is commonly used when studying within-subject changes or comparing different treatments within the same individuals.
One way a repeated measures ANOVA can increase power to detect an effect is through the reduction of individual differences or subject variability. By using the same subjects in multiple conditions or time points, the variability among subjects is accounted for, and the focus shifts to the variability within subjects. This reduces the overall error variance and increases the power of the statistical test.
In other words, when comparing different treatments or time points within the same individuals, any individual differences that could confound the results are controlled for. This increases the sensitivity of the analysis, making it easier to detect smaller effects or differences between the conditions.
Additionally, the repeated measures design allows for increased statistical efficiency. Since each subject serves as their own control, the sample size required to achieve a certain level of power is often smaller compared to independent groups designs. This results in more precise estimates and higher statistical power.
Overall, the repeated measures ANOVA design provides greater statistical power by reducing subject variability and increasing statistical efficiency. It allows for a more precise evaluation of treatment effects or changes over time, making it a valuable tool in research and data analysis.
To learn more about statistic visit;
https://brainly.com/question/31538429
#SPJ11
Suppose we want to form four-digit numbers using the set of digits \( \{0,1,2,3\} \). For example, 3013 and 2230 are such numbers, but 0373 is not. How many of these numbers are multiples of 10 ?
There are 6 four-digit numbers that are multiples of 10 using the set of digits {0,1,2,3}.
Let's write down all four-digit numbers that we can form using the set of digits {0,1,2,3}. We can place any of the four digits in the first position, any of the remaining three digits in the second position, any of the remaining two digits in the third position, and the remaining digit in the fourth position.
So, the number of four-digit numbers we can form is:4 x 3 x 2 x 1 = 24Now, we want to count how many of these numbers are multiples of 10. A number is a multiple of 10 if its unit digit is 0. Out of the four digits in our set, only 0 is a possible choice for the unit digit.
Once we choose 0 for the units digit, we are free to choose any of the remaining three digits for the thousands digit, any of the remaining two digits for the hundreds digit, and any of the remaining one digits for the tens digit. So, the number of four-digit numbers that are multiples of 10 is:1 x 3 x 2 x 1 = 6
Therefore, there are 6 four-digit numbers that are multiples of 10 using the set of digits {0,1,2,3}.
We have a total of 24 four-digit numbers using the set of digits {0,1,2,3}. However, only 6 of these are multiples of 10. Thus, the probability that a randomly chosen four-digit number using the set of digits {0,1,2,3} is a multiple of 10 is:6/24 = 1/4
There are 6 four-digit numbers that are multiples of 10 using the set of digits {0,1,2,3}.
Learn more about units digit from the given link
https://brainly.com/question/32380970
#SPJ11
Suppose that 20% of articles produced by a machine are defective, the defective occurring at random during the production process. Use a Gaussian approximation to approximate the following probabilities. (a) Find the probability that, if a sample of 500 items is taken from the production, more than 120 will be defective. (b) For what value of K is the probability that the number of defectives in a sample of 500 lie within 100±K is 0.95 ?
Given that 20% of articles produced by a machine are defective, the defective occurring at random during the production process. We have to use a Gaussian approximation to approximate the following probabilities.
Probability that more than 120 will be defective when a sample of 500 items is taken from the production We have, Mean = np
= 500 × 0.2
= 100
Standard deviation,σ = √np(1 - p)
= √500 × 0.2 × 0.8
≈ 8.944
Therefore, Probability of selecting a defective item from a batch of items, p = 0.2
Probability of selecting a non-defective item from a batch of items = q = 0.8
Let X be the number of defective items in a sample of 500 items taken from the production. Then X follows a normal distribution with mean μ = np = 100 and
variance σ² = npq
= 500 × 0.2 × 0.8
= 80.
Let Z be the standard normal variable. Then, If X follows a normal distribution, then Z follows a standard normal distribution (mean = 0 and variance = 1).
We are to find, P(X > 120) = P(Z > (120 - 100) / 8.944)
= P(Z > 2.236)
Therefore, we can say that the area under the standard normal distribution curve between -K / 8.944 and K / 8.944 is 0.95.Now, from the Z table, we can say that the area under the standard normal distribution curve between -1.96 and 1.96 is 0.95. Therefore, K / 8.944 = 1.96
⇒ K = 1.96 × 8.944 / 1
= 17.68Hence, the value of K is 17.68 (approximately).Therefore, the probability that the number of defectives in a sample of 500 lie within 100±K is 0.95 if K is equal to 17.68.
To know more about probabilities visit:
https://brainly.com/question/29381779
#SPJ11
You are given that the matrix A= ⎝
⎛
1
2
1
−2
1
k
2
9
5
⎠
⎞
is non-invertible (singular). What is the value of the constant k ? Select one: A. 2 B. 1 C. 3 D. 4 E. 0 Let u 1
=(1,1) and u 2
=(1,−1). What are the coordinates of y=(3,5) with respect to the ordered basis B={u 1
,u 2
}? Select one: The matrices M and N are given by M
and N
=( 1
1
1
0
)
=( 1
3
2
4
)
You are now told that A=M T
N What is det(A) ? Select one: A. 2 B. 4 C. 3 D. 0 E. 1 Let v 1
,v 2
,v 3
be vectors in R 3
. Which of the following statements is TRUE? I II III :dim(Span{v 1
,v 2
,v 3
})=3
:Span{v 1
,v 2
,v 3
}=R 3
:Span{v 1
,v 2
,v 3
}=R 3
iff v 1
,v 2
The first statement is about the dimension of the span of the three vectors, the second statement is about the span being equal to \(\mathbb{R}^3\), and the third statement is the same as the second but includes the condition that \(v_1\), \(v_2\), and \(v_3\) are linearly independent.
Let's go through each question one by one:
1. Given the matrix \(A\), we are told that it is non-invertible. To find the value of the constant \(k\), we can examine the determinant of \(A\). If the determinant is zero, then \(A\) is non-invertible. Therefore, we need to calculate the determinant of \(A\) and set it equal to zero to find \(k\).
2. The coordinates of \(y=(3,5)\) with respect to the ordered basis \(B=\{u_1,u_2\}\) can be found by expressing \(y\) as a linear combination of \(u_1\) and \(u_2\). We need to find scalars \(c_1\) and \(c_2\) such that \(y = c_1u_1 + c_2u_2\).
3. We are given two matrices, \(M\) and \(N\), and told that \(A = M^TN\). To find \(\text{det}(A)\), we can use the property that the determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore, we need to calculate \(\text{det}(A)\) using the given matrices \(M\) and \(N\).
4. In this question, we have vectors \(v_1\), \(v_2\), and \(v_3\) in \(\mathbb{R}^3\). We need to determine which of the given statements are true. The first statement is about the dimension of the span of the three vectors, the second statement is about the span being equal to \(\mathbb{R}^3\), and the third statement is the same as the second but includes the condition that \(v_1\), \(v_2\), and \(v_3\) are linearly independent.
Please provide the options for each question, and I'll be able to provide you with the correct answers.
Learn more about dimension here
https://brainly.com/question/30997214
#SPJ11
construct a polynomial with 3,-1, and 2 as the only zeros
The polynomial function of least degree with the given zeros is P(x) = (x - 3)(x + 1)(x - 2)
How to determine the polynomialFrom the question, we have the following parameters that can be used in our computation:
Zeros = 3,-1, and 2
We assume that the multiplicites of the zeros are 1
So, we have
P(x) = (x - zeros)
This gives
P(x) = (x - 3)(x + 1)(x - 2)
Hence,, the polynomial function is P(x) = (x - 3)(x + 1)(x - 2)
Read more about polynomials at
brainly.com/question/11536910
#SPJ1
direction of ⎣
⎡
4
3
2
3
1
⎦
⎤
. A unit vector in the direction of the given vector is (Type an exact answer, using radicals as needed.)
A unit vector in the direction of the given vector is [tex]\(\begin{bmatrix} \frac{12}{\sqrt{61}} \\ \frac{6}{\sqrt{61}} \\ \frac{4}{\sqrt{61}} \end{bmatrix}\).[/tex].
To find a unit vector in the direction of a given vector, we divide the vector by its magnitude.
The given vector is [tex]\(\mathbf{v} = \begin{bmatrix} \frac{3}{4} \\ \frac{3}{2} \\ 1 \end{bmatrix}\)[/tex].
To find the magnitude of the given vector, we calculate:
[tex]\(|\mathbf{v}| = \sqrt{\left(\frac{3}{4}\right)^2 + \left(\frac{3}{2}\right)^2 + 1^2}\)[/tex]
[tex]\(= \sqrt{\frac{9}{16} + \frac{9}{4} + 1}\)[/tex]
[tex]\(= \sqrt{\frac{9}{16} + \frac{36}{16} + \frac{16}{16}}\)[/tex]
[tex]\(= \sqrt{\frac{61}{16}}\)[/tex]
[tex]\(= \frac{\sqrt{61}}{4}\)[/tex]
Now, we can divide the vector by its magnitude to obtain a unit vector in the same direction:
[tex]\(\frac{\mathbf{v}}{|\mathbf{v}|} = \begin{bmatrix} \frac{3}{4} \\ \frac{3}{2} \\ 1 \end{bmatrix} \cdot \frac{4}{\sqrt{61}}\)[/tex]
[tex]\(= \begin{bmatrix} \frac{12}{\sqrt{61}} \\ \frac{6}{\sqrt{61}} \\ \frac{4}{\sqrt{61}} \end{bmatrix}\)[/tex]
Therefore, a unit vector in the direction of the given vector is [tex]\(\begin{bmatrix} \frac{12}{\sqrt{61}} \\ \frac{6}{\sqrt{61}} \\ \frac{4}{\sqrt{61}} \end{bmatrix}\).[/tex]
To know more about unit vector, refer here:
https://brainly.com/question/32231746
#SPJ4
Complete Question:
Given vector: [tex]\(\mathbf{v} = \begin{bmatrix} \frac{3}{4} \\ \frac{3}{2} \\ 1 \end{bmatrix}\)[/tex]. A unit vector in the direction of the given vector is __ (Type an exact answer, using radicals as needed.)
Solve the equation \( 4-x=3-2(6 x+7) \) algebraically. Ansuer as a ureduced proper or improper fraction"
The solution to the equation
4
−
�
=
3
−
2
(
6
�
+
7
)
4−x=3−2(6x+7) is
�
=
−
20
49
x=−
49
20
.
To solve the equation algebraically, we will simplify both sides of the equation and isolate the variable, x.
Starting with the given equation
4
−
�
=
3
−
2
(
6
�
+
7
)
4−x=3−2(6x+7), let's simplify the right-hand side first:
4
−
�
=
3
−
12
�
−
14
4−x=3−12x−14
4
−
�
=
−
12
�
−
11
4−x=−12x−11
Now, we can combine like terms by adding
12
�
12x to both sides:
12
�
+
4
−
�
=
−
12
�
−
�
−
11
12x+4−x=−12x−x−11
11
�
+
4
=
−
11
11x+4=−11
Next, we'll subtract 4 from both sides:
11
�
=
−
11
−
4
11x=−11−4
11
�
=
−
15
11x=−15
To solve for x, divide both sides by 11:
�
=
−
15
11
x=
11
−15
However, the question specifies that the answer should be in the form of an unreduced proper or improper fraction. So, let's express
−
15
11
−
11
15
as a reduced fraction:
The greatest common divisor (GCD) of 15 and 11 is 1, so the fraction is already in reduced form. Therefore, the solution to the equation is
�
=
−
15
11
x=−
11
15
.
The solution to the equation
4
−
�
=
3
−
2
(
6
�
+
7
)
4−x=3−2(6x+7) is
�
=
−
15
11
x=−
11
15
.
To know more about improper fraction, visit;
https://brainly.com/question/21449807
#SPJ11
if
it is estimated that 80% people recieve a call back after an
interview and 20% dont. in a random sample of 100, how many recieve
a call back
If it is estimated that 80% people receive a call back after an interview and 20% don't in a random sample of 100, then 80 people receive a call back.
To find the number of people who get a call back, follow these steps:
It is given that the total number of people= 100 and 80% of people receive a call back. So, the number of people who get a call back can be found by multiplying the percentage of people who get a call back by the total number of people.So, the number of people who get a call back = 80% of 100= (80/100) × 100 = 80.So, we can estimate that 80 people will receive a call back after the interview.
Learn more about percentage:
brainly.com/question/24877689
#SPJ11
Which of the following condition is evaluated to
False:
a.
"Vb".ToLower() < "VB"
b.
All of the Options
c.
"ITCS".subString(0,1) <> "I"
d.
"Computer".IndexOf ("M") = 1
The condition that is evaluated to False is `"Vb".
Option a. ToLower() < "VB"`.
a. "Vb".ToLower() < "VB"
Here, `"Vb".ToLower()` converts the string "Vb" to lower case and returns "vb". So the condition becomes "vb" < "VB". Since in ASCII, the uppercase letters have lower values than the lowercase letters, this condition is True.
b. All of the Options
This option cannot be the answer as it is not a specific condition. It simply states that all options are True.
c. "ITCS".subString(0,1) <> "I"
Here, `"ITCS".subString(0,1)` returns "I". So the condition becomes "I" <> "I". Since the two sides are equal, the condition is False.
d. "Computer".IndexOf ("M") = 1
Here, `"Computer".IndexOf ("M")` returns 3. So the condition becomes 3 = 1. Since this is False, this condition is not the answer.
Therefore, the condition that is evaluated to False is `"Vb".
Option a. ToLower() < "VB"`.
Learn more About ASCII from the given link
https://brainly.com/question/13143401
#SPJ11
Let y=⎣⎡4−10−10⎦⎤,u1=⎣⎡3−41⎦⎤,u2=⎣⎡−2422⎦⎤ Compute the distance d from y to the plane in R3 spanned by u1 and u2. d=
The distance from the vector y to the plane in R^3 spanned by u1 and u2 is found to be 0. This means that the vector y lies exactly on the plane defined by u1 and u2.
The distance from the vector y to the plane in R^3 spanned by u1 and u2 is computed as d = 3.
To explain the solution in more detail, we start by considering the plane in R^3 spanned by u1 and u2. This plane can be represented by the equation Ax + By + Cz + D = 0, where A, B, C are the coefficients of the plane's normal vector and D is a constant.
In this case, the normal vector of the plane is the cross product of u1 and u2. We calculate the cross product as follows:
u1 x u2 = (3)(4) - (-4)(-2)i + (1)(-2) - (3)(4)j + (-2)(3) - (4)(-4)k
= 12i - 6j + 2k + 6i - 24k + 16j
= 18i + 10j - 22k
So the equation of the plane becomes 18x + 10y - 22z + D = 0.
To find the value of D, we substitute the coordinates of y into the equation and solve for D:
18(4) + 10(-10) - 22(-10) + D = 0
72 - 100 + 220 + D = 0
D = -192
Thus, the equation of the plane becomes 18x + 10y - 22z - 192 = 0.
Now, we can compute the distance d from y to the plane using the formula:
d = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2)
Plugging in the coordinates of y and the coefficients of the plane, we get:
d = |18(4) + 10(-10) - 22(-10) - 192| / sqrt(18^2 + 10^2 + (-22)^2)
= |72 - 100 + 220 - 192| / sqrt(648 + 100 + 484)
= 0 / sqrt(1232)
= 0
Therefore, the distance from y to the plane spanned by u1 and u2 is 0.
To learn more about normal vector click here: brainly.com/question/31980415
#SPJ11
Use the given information to find the critical values X and X2. (Use technology or the attached Chi-Square table.) Platelet Counts of Women 80% confidence n=26 s=65.3 ChiSquare.pdf A. 16.473 and 34.382 B. 15.308 and 44.461 C. 9.542 and 40.289 O D. 11.808 and 49.645
The critical value X2 that leaves 10% of the area in the left tail is approximately 15.308. The correct answer is B. 15.308 and 34.382.
To find the critical values for a chi-square distribution, we need to determine the degrees of freedom and the confidence level.
In this case, the degrees of freedom can be calculated as (n - 1), where n is the sample size. Thus, degrees of freedom = 26 - 1 = 25.
For an 80% confidence level, we want to find the critical values that enclose 80% of the area under the chi-square distribution curve.
Since the chi-square distribution is right-skewed, we need to find the critical value that leaves 10% of the area in the right tail (80% + 10% = 90%) and the critical value that leaves 10% of the area in the left tail (80% - 10% = 70%).
Using a chi-square table or a chi-square calculator, we find:
The critical value X1 that leaves 10% of the area in the right tail is approximately 34.382.
The critical value X2 that leaves 10% of the area in the left tail is approximately 15.308.
to learn more about degrees of freedom.
https://brainly.com/question/32093315
#SPJ11
For the differential equation dx
dy
= y 2
−81
does the existence/uniqueness theorem guarantee that there is a solution to this equation through the point 1. (−4,84)? 2. (−2,90)? 3. (−3,9) ? 4. (−1,−9) ?
The existence/uniqueness theorem guarantees a solution to the given differential equation through the points[tex]\((-3, 9)\) and \((-1, -9)\).[/tex]
How to find the differential equationThe existence/uniqueness theorem states that if a differential equation is of the form [tex]\(dy/dx = f(x, y)\) and \(f(x, y)\)[/tex]is continuous in a region containing the point [tex]\((x_0, y_0)\),[/tex] then there exists a unique solution to the differential equation that passes through the point [tex]\((x_0, y_0)\).[/tex]
Let's check the given points one by one:
1.[tex]\((-4, 84)\):[/tex]
Plugging in the values [tex]\((-4, 84)\)[/tex] into the equation [tex]\(y = \frac{1}{3}x^3 - 9x + C\),[/tex] we get[tex]\(84 = \frac{1}{3}(-4)^3 - 9(-4) + C\)[/tex], which simplifies to[tex]\(84 = 104 + C\)[/tex]. This equation has no solution, so the existence/uniqueness theorem does not guarantee a solution through this point.
2. [tex]\((-2, 90)\):[/tex]
Plugging in the values [tex]\((-2, 90)\)[/tex] into the equation[tex]\(y = \frac{1}{3}x^3 - 9x + C\),[/tex] we get [tex]\(90 = \frac{1}{3}(-2)^3 - 9(-2) + C\),[/tex] which simplifies to[tex]\(90 = \frac{8}{3} + 18 + C\).[/tex] This equation has no solution, so the existence/uniqueness theorem does not guarantee a solution through this point.
3. [tex]\((-3, 9)\):[/tex]
Plugging in the values[tex]\((-3, 9)\)[/tex] into the equation [tex]\(y = \frac{1}{3}x^3 - 9x + C\)[/tex], we get [tex]\(9 = \frac{1}{3}(-3)^3 - 9(-3) + C\),[/tex] which simplifies to[tex]\(9 = -\frac{9}{3} + 27 + C\).[/tex] This equation has a unique solution, so the existence/uniqueness theorem guarantees a solution through this point.
4. [tex]\((-1, -9)\):[/tex]
Plugging in the values \((-1, -9)\) into the equation [tex]\(y = \frac{1}{3}x^3 - 9x + C\), we get \(-9 = \frac{1}{3}(-1)^3 - 9(-1) + C\)[/tex] , which simplifies to[tex]\(-9 = -\frac{1}{3} + 9 + C\)[/tex]. This equation has a unique solution, so the existence/uniqueness theorem guarantees a solution through this point.
Therefore, the existence/uniqueness theorem guarantees a solution to the given differential equation through the points[tex]\((-3, 9)\) and \((-1, -9)\).[/tex]
Learn more about differential equation at https://brainly.com/question/1164377
#SPJ4
Question 4. The amount of caffeine ingested, c in mg, is a function of the amount of coffee drank, D, in ounces. (A) Write a sentence that interprets the following: f(1)=15 (B) Represent the following statement in function notation: "After drinking 20 oz of coffee, the participant ingested 200mg of caffeine." (C) Your classmate made the case that C=f(D) is a linear function. Do you agree or disagree? Clearly explain your reasoning.
Answer:
The function relating caffeine ingestion (C) to coffee consumption (D) is not linear but rather a nonlinear function.
(A) The sentence that interprets the given function f(1) = 15 is: "After consuming 1 ounce of coffee, the participant ingested 15 mg of caffeine."
(B) The statement "After drinking 20 oz of coffee, the participant ingested 200 mg of caffeine" can be represented in function notation as f(20) = 200.
(C) I disagree with the claim that C = f(D) is a linear function. A linear function has a constant rate of change, meaning that the amount of caffeine ingested would increase or decrease by the same amount for every unit increase or decrease in coffee consumed. However, in the case of caffeine ingestion, this assumption does not hold true.
Caffeine content is not directly proportional to the amount of coffee consumed. While there is a relationship between the two, the rate at which caffeine is ingested is not constant. The caffeine content in coffee can vary based on factors such as the type of coffee bean, brewing method, and the strength of the coffee. Additionally, individual differences in metabolism can also affect how the body processes and absorbs caffeine.
Therefore, the function relating caffeine ingestion (C) to coffee consumption (D) is not linear but rather a nonlinear function.
Learn more about caffeine ingestion from below link
https://brainly.com/question/29608734
#SPJ11
GRE scores, Part II: Assume that scores on the verbal portion of the GRE (Graduate Record Exam) follow the normal distribution with mean score 151 and standard deviation 7 points, while the quantitative portion of the exam has scores following the normal distribution with mean 153 and standard deviation 7.67. Use this information to answer the following. USE THE TI CALCULATOR FUNCTIONS (or similar method) TO COMPUTE YOUR ANSWER. a) Find the score of a student who scored in the 80th percentile on the Quantitative Reasoning section of the exam. (please round to two decimal places, XXX.XX ) b)Find the score of a student who scored worse than 70% of the test takers in the Verbal Reasoning section of the exam. (please round to two decimal places, XXX.XX)
Using the TI Calculator Answer is 162.20 and 155.09
a) We know that scores on the quantitative portion of the GRE follow the normal distribution with mean score 153 and standard deviation 7.67 points, and we need to find the score of a student who scored in the 80th percentile on this section of the exam.
Using the TI calculator, we can find this score as follows:
Press 2nd VARS (DISTR) to access the distribution menu, then scroll down to invNorm and press enter.
Enter the area to the left of the desired percentile as a decimal (in this case, 0.80).
Enter the mean score as 153 and the standard deviation as 7.67.
Press enter to find the score corresponding to the 80th percentile, which is 162.20 (rounded to two decimal places).
Therefore, the score of a student who scored in the 80th percentile on the Quantitative Reasoning section of the GRE is 162.20.
b) We know that scores on the verbal portion of the GRE follow the normal distribution with mean score 151 and standard deviation 7 points, and we need to find the score of a student who scored worse than 70% of the test takers in this section of the exam.
Using the TI calculator, we can find this score as follows:
Press 2nd VARS (DISTR) to access the distribution menu, then scroll down to invNorm and press enter.
Enter the area to the left of the desired percentile as a decimal (in this case, 0.70).Enter the mean score as 151 and the standard deviation as 7.
Press enter to find the score corresponding to the 70th percentile, which is 155.09 (rounded to two decimal places).
Therefore, the score of a student who scored worse than 70% of the test takers in the Verbal Reasoning section of the GRE is 155.09.
Learn more about: TI calculator
https://brainly.com/question/32809051
#SPJ11
2 apples cost 2 dabloons.
How much does 1 apple cost
In the problems, obtain the general solution of the DE. If you cannot find yp by inspection, use the method of undetermined coefficients.
y" = 1
y" + y' - 2y = 3 - 6t
y" - y' - 2y = 6et
The general solution of the given differential equation is y(t) = c1e^t + c2e^(-2t) - 3/2 + (1/4)e^t, obtained by combining the complementary and particular solutions.
To find the general solution of the given differential equation, we can use the method of undetermined coefficients. By assuming a particular solution and solving for the unknown coefficients, we can combine it with the complementary solution to obtain the complete general solution.
The given differential equation is:
y'' + y' - 2y = 3 - 6t
Step 1: Find the complementary solution
To find the complementary solution, we solve the associated homogeneous equation by setting the right-hand side of the equation to zero:
y'' + y' - 2y = 0
The characteristic equation of the homogeneous equation is:
r^2 + r - 2 = 0
Solving this quadratic equation, we find two distinct roots: r = 1 and r = -2.
Hence, the complementary solution is:
y_c(t) = c1e^t + c2e^(-2t)
Step 2: Find the particular solution
For the particular solution, we use the method of undetermined coefficients.
Particular solution 1: 3 - 6t
Since the right-hand side of the equation is a polynomial of degree 0, we assume a particular solution of the form: yp1(t) = A
Substituting this into the original equation, we get:
0 + 0 - 2A = 3 - 6t
Comparing coefficients, we find A = -3/2.
Hence, the particular solution is:
yp1(t) = -3/2
Particular solution 2: 6et
Since the right-hand side of the equation is an exponential function, we assume a particular solution of the form: yp2(t) = Be^t
Substituting this into the original equation, we get:
e^t + e^t - 2Be^t = 6et
Comparing coefficients, we find B = 1/4.
Hence, the particular solution is:
yp2(t) = (1/4)e^t
Step 3: Find the general solution
Combining the complementary and particular solutions, we obtain the general solution of the differential equation:
y(t) = y_c(t) + yp(t)
= c1e^t + c2e^(-2t) - 3/2 + (1/4)e^t
Therefore, the general solution of the given differential equation is:
y(t) = c1e^t + c2e^(-2t) - 3/2 + (1/4)e^t
To learn more about differential equation click here: brainly.com/question/32645495
#SPJ11
In 2015, the mean number of books that college students would buy was 48 . With the use of Open Educational Resources, it is believed that this number has been decreasing. A recent sample of 80 college students found a sample mean of 47 books and a sample standard deviation of 5 books. We will perfrom a hypothesis test to determine if this is strong evidence that the mean number of books that college students purchase is decreasing and the use of Open Educational Resources is increasing. Define your variable. Let μ be the mean number of books that college students in the sample buy. Let p be the proprotion of college students in the sample that buy books. Let μ be the mean number of books that college students buy. Let p be the proportion of students that buy books. Write your hypotheses: H 0
:
H A
:?∨ <
>
>
Find the P-value: Type your answer as a decimial rounded to three decimal places. Do not type a percentage or a percent sign. P-value = Using an α level of 5%, you should Fail to Reject H 0
Reject H 0
and Accept H A
Accept H 0
State your conclusions: We do do not mean of book purchased by college students in now less than 47 mean number of book purchased by college students in now less than mean of book purchased by the sampled college students in now less t mean number of book purchased by college students in now greater th Is this evidence that the use of Open Educational Resources is increasing? No Yes
Alternative Hypothesis, the opposite of the null hypothesis. It is what we want to prove to be true based on our evidence. μ < 48, which means that the mean number of books that college students buy is less than 48. P-value = 0.105. We do not have sufficient evidence that the use of Open Educational Resources is increasing. Hence, the answer is No.
Let μ be the mean number of books that college students buy. Let p be the proportion of students that buy books. H0: μ ≥ 48HA: μ < 48 H0: Null Hypothesis; that is what we assume to be true before collecting any data. μ ≥ 48, which means that the mean number of books that college students buy is greater than or equal to 48.HA: Alternative Hypothesis, the opposite of the null hypothesis. It is what we want to prove to be true based on our evidence. μ < 48, which means that the mean number of books that college students buy is less than 48. P-value = 0.105 (rounded to three decimal places)
We fail to reject H0. We do not have enough evidence to suggest that the mean number of books that college students purchase is decreasing, and the use of Open Educational Resources is increasing. The mean number of books purchased by college students is now less than 47. Therefore, we do not have sufficient evidence that the use of Open Educational Resources is increasing. Hence, the answer is No.
To learn more about null hypothesis: https://brainly.com/question/25263462
#SPJ11
In a mayoral election, candidate A is facing two opposing candidates. In a preselected poll of 100 residents, 22 supported candidate B and 14 supported candidate C. Can we conclude that more than 60% of residents in the population supported candidate A? Conduct the test with a=0.05. Which of the following statements is (are) correct? This is a multiple-answer question. It may have more than one correct answers. i. The proportion of residents supported candidate A based on this sample is (100−22−14)/100=0.64. ii. The null hypothesis is p>0.6 and the alternative hypothesis is p=0.6. iii. The rejection region is (1.645, infinity). iv. The resulting statistic is z ∗= 0.64−0.6/rootover0.204/100=0.82. The p-value is P(z>0.82)=1−0.7939=0.2061. v. Since 0.2061>0.05, we reject the null hypothesis. We conclude that the population proportion is greater than 0.6.
The conclusion that more than 60% of residents in the population supported candidate A cannot be made based on the given information and analysis. The correct statements are i, ii, and iv.
In order to determine whether more than 60% of residents supported candidate A, a hypothesis test needs to be conducted. The null hypothesis (H0) assumes that the population proportion (p) is greater than 0.6, while the alternative hypothesis (Ha) assumes that p is equal to or less than 0.6.
Statement i is correct as it calculates the proportion of residents who supported candidate A based on the given sample, which is (100 - 22 - 14) / 100 = 0.64, or 64%.
Statement ii is correct in describing the null and alternative hypotheses. The null hypothesis assumes p > 0.6, while the alternative hypothesis assumes p ≤ 0.6.
Statement iii is incorrect. The rejection region for a hypothesis test with a significance level (α) of 0.05 should be based on the critical value of the z-statistic. For a one-tailed test (as implied by the alternative hypothesis), the critical value is approximately 1.645.
Statement iv is correct in calculating the z-statistic using the given sample proportion, the assumed population proportion, and the sample size. However, the calculated z-value is incorrect. The correct calculation is (0.64 - 0.6) / √(0.6 * 0.4 / 100) = 0.2357.
Statement v is incorrect. The p-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true. In this case, the p-value is P(z > 0.2357) ≈ 0.4098, which is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis. The correct conclusion is that there is insufficient evidence to conclude that more than 60% of residents support candidate A.
Learn more about population here:
https://brainly.com/question/29095323
#SPJ11
Solve the initial value problem, u'= -2tu², u(0)=1 with h = 0.2 on the interval [0,0.4]. Use the 4th order Runge Kutta (R – K) method and compare with exact solution.
A. The 4th order Runge-Kutta method is employed to solve u' = -2tu^2, u(0) = 1 with h = 0.2 on [0, 0.4], and the obtained numerical solution will be compared with the exact solution.
To solve the initial value problem u' = -2tu^2, u(0) = 1 on the interval [0, 0.4] using the 4th order Runge-Kutta (R-K) method with step size h = 0.2, we can follow these steps:
1. Define the function f(t, u) = -2tu^2.
2. Initialize t = 0 and u = 1.
3. Iterate from t = 0 to t = 0.4 with a step size of h = 0.2 using the R-K method.
Calculate k1 = h * f(t, u).Calculate k2 = h * f(t + h/2, u + k1/2).Calculate k3 = h * f(t + h/2, u + k2/2).Calculate k4 = h * f(t + h, u + k3).Update u = u + (k1 + 2k2 + 2k3 + k4)/6.Update t = t + h4. Repeat step 3 until t reaches 0.4.
5. Compare the obtained numerical solution with the exact solution for evaluation.
Exact Solution:
The given differential equation is separable. We can rewrite it as du/u^2 = -2tdt and integrate both sides:
∫(du/u^2) = ∫(-2tdt)
Solving the integrals, we get:
-1/u = -t^2 + C,
where C is a constant of integration.
Applying the initial condition u(0) = 1, we find C = -1.
Therefore, the exact solution is given by:
-1/u = -t^2 - 1
u = -1 / (-t^2 - 1).
Now, we can compare the numerical solution obtained using the R-K method with the exact solution.
To learn more about function click here:
https://brainly.com/question/31062578
#SPJ11
