Hello, Please solve/find the final answer to those functions
a,b, and c with steps appreciate to solve in word format.
differentiation / derivative
5) (a) If f(x) = (2x7 + 7x5)³(5x2 + 2x )3, then find f'(x) 8(x) = (3x2 - 94 (4x-6)7 (b) Differentiate: (c) If y = u³ - 4u² + 2u - 1 and u = √√x + 6, find dy dx when x = -2

1. The matrix of linear transformation L with respect to the basis[tex](u_1, u_2, u_3)[/tex] is L =[tex]\left|\begin{array}{ccc}3&2&1\\3&2&0\\0&0&0\end{array}\right|[/tex]  2. The vector v = ⎝⎛​752​⎠⎞​ can be written as the linear combination of [tex]u_1, u_2, u_3[/tex]​ as [tex]v = 7u_1+5u_2+2u_3[/tex] .3.  The result of applying the linear transformation L to v is [tex]L(v) = 23u_1+19u_2[/tex].

To find the matrix of linear transformation L with respect to the basis (u1​, u2​, u3​), we need to determine the images of the basis vectors under L and represent them as linear combinations of the basis vectors.

Applying L to each basis vector, we have:

[tex]L(u_1) = (1+1+1)u_1 + (2(1)+1)u_2 - (2(1)+1)u_3\\\\= 3u_1+3u_2- 3u_3[/tex]

[tex]L(u_2) = (1+1+0)u_1+(2(1)+0)u_2+(2(0)+0)u_3\\=2u_1+2u_2[/tex]

[tex]L(u_3)= (1+0+0)u_1+(2(0)+0)u_2+(2(0)+0)u_3\\=u_1[/tex]

Now, we can express the images of the basis vectors in terms of the basis vectors themselves:

[tex]L(u_1) = 3u_1+3u_2-3u_3\\-3u_1+3u_2+0u_3\\L(u_2) = 2u_1+2u_2+0u_3\\L(u_3) = 1u_1+0u_2+0u_3\\[/tex]

The matrix of L with respect to the basis ([tex]u_1, u_2, u_3[/tex]) is formed by arranging the coefficients of the basis vectors as columns:

Matrix of L =[tex]\left|\begin{array}{ccc}3&2&1\\3&2&0\\0&0&0\end{array}\right|[/tex]

To write the vector v = ⎝⎛​752​⎠⎞​ as a linear combination of the vectors  [tex]u_1, u_2, u_3[/tex], we need to find the scalars [tex]c_1, c_2, c_3[/tex]​ such that [tex]v= c_1u_1+c_2u_2+c_3u_3[/tex]

Let's solve the following system of equations:

[tex]c_1u_1+c_2u_2+c_3u_3 = v[/tex]

c1​⎝⎛​111​⎠⎞​ + c2​⎝⎛​110​⎠⎞​ + c3​⎝⎛​100​⎠⎞​ = ⎝⎛​752​⎠⎞​

This can be written as a matrix equation:

[tex]\left|\begin{array}{ccc}1&1&1\\1&1&0\\1&0&0\end{array}\right|\left|\begin{array}{ccc}c_1\\c_2\\c_3\end{array}\right|= \left|\begin{array}{ccc}7\\5\\2\end{array}\right|[/tex]

By solving this system of equations, we can find the values of [tex]c_1, c_2, c_3[/tex], which represent the linear combination of [tex]u_1, u_2, u_3[/tex] that gives v.

To determine L(v), we can simply apply the linear transformation L to the vector v using the given formula:

[tex]L(v) = L(c_1u_1+c_2u_2+c_3u_3)[/tex]

Substituting the values of [tex]c_1, c_2, c_3[/tex]  obtained from part 2, we can evaluate L(v) using the given formula for L.

Hence, the solutions are:

1. The matrix of linear transformation L with respect to the basis[tex](u_1, u_2, u_3)[/tex] is L =[tex]\left|\begin{array}{ccc}3&2&1\\3&2&0\\0&0&0\end{array}\right|[/tex]  2. The vector v = ⎝⎛​752​⎠⎞​ can be written as the linear combination of [tex]u_1, u_2, u_3[/tex]​ as [tex]v = 7u_1+5u_2+2u_3[/tex] .3.  The result of applying the linear transformation L to v is [tex]L(v) = 23u_1+19u_2[/tex].

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The interest Grace has to pay on the loan after 120 days is $514.50.

The correct option is letter C.

Grace borrowed $12,500 for 120 days, with the interest charged on the loan at 12.5%.

We can determine the interest to be paid by using the following formula:

Interest = Principal × Rate × Time

Interest = $12,500 × 0.125 × (120 / 365)

Interest = $12,500 × 0.125 × 0.3288

Interest = $514.50

Therefore, the interest Grace has to pay on the loan after 120 days is $514.50.

The correct option is letter C.

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The solution of the given initial value problem x^2y' − xy = 2, y(1) = 1 is given by:

y(x) = 2/x^3 − 2/x, x ≠ 0

Given initial value problem:

x^2y' − xy = 2and y(1) = 1

Multiplying both sides of the given differential equation by x, we have:

x^3y' − x^2y = 2x......(i)

Let us find the integrating factor of the above differential equation.

The integrating factor is given by exp{∫P(x)dx}, where P(x) = -1/x.

Thus, we have:

integrating factor = exp{∫(-1/x)dx} = exp{-ln|x|} = 1/x

Therefore, multiplying both sides of (i) by the integrating factor, we obtain:

d/dx [x^3y(1/x)] = 2/x^2Now integrating both sides of the above equation, we get:

x^3y(1/x) = -2/x + C1 where C1 is the constant of integration.

Substituting x = 1 and y = 1 in the above equation, we get:

C1 = 2Hence, we have:

x^3y(1/x) = 2/x − 2

Substituting x = 1 in the above equation, we get:

y(1) = 0

Hence, the solution of the given initial value problem:

x^2y' − xy = 2, y(1) = 1 is given by:

y(x) = 2/x^3 − 2/x, x ≠ 0

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The given differential equation is a Cauchy-Euler equation.

To solve it, we assume a solution of the form y = x^r. Substituting this into the differential equation and simplifying, we get the indicial equation r(r-1) + r - 2 = 0. Solving for r, we find that the roots are r1 = 2 and r2 = -1.

For the larger root r1 = 2, we can use the recurrence relationship a_n = -a_(n-2)/(n(n+1)) to find the first three terms of the solution. Since y = x^r * (a_0 + a_1*x + a_2*x^2 + ...), we have y = x^2 * (a_0 + a_1*x + a_2*x^2 + ...). Substituting n=2 into the recurrence relationship, we find that a_2 = -a_0/6. Substituting n=3, we find that a_3 = -a_1/12. Thus, the first three terms of the solution corresponding to the larger root are x^2 * (a_0 + a_1*x - (a_0/6)*x^2).

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(a) The number of 3-combinations of X is 4C3 = 4.(b) The 3-combinations of X are {a,b,c}, {a,b,d}, {a,c,d}, and {b,c,d}.(c) The number of 3-permutations of X is 4P3 = 24.(d) The 3-permutations of X are {a,b,c}, {a,b,d}, {a,c,b}, {a,c,d}, {a,d,b}, {a,d,c}, {b,a,c}, {b,a,d}, {b,c,a}, {b,c,d}, {b,d,a}, {b,d,c}, {c,a,b}, {c,a,d}, {c,b,a}, {c,b,d}, {c,d,a}, {c,d,b}, {d,a,b}, {d,a,c}, {d,b,a}, {d,b,c}, {d,c,a}, and {d,c,b}.

(a) If repetitions are allowed, there will be 26 choices of letters for each of the three blanks and 10 choices of digits for each of the two blanks. Therefore, the total number of different car license plates is as follows:26 x 26 x 26 x 10 x 10 = 175,760(b) If repetitions are not allowed, there will be 26 choices of letters for the first blank, 25 for the second blank, and 24 for the third blank. There will be 10 choices of digits for the fourth blank and 9 choices of digits for the fifth blank. Therefore, the total number of different car license plates is as follows:26 x 25 x 24 x 10 x 9 = 14,0408. Basic Principles. How many strings of length 5 formed using the letters ABCDEFG without repetitions (a) begin with AC or DB in that order? (b) contain letters B and D consecutively in either order (i.e., BD or DB)?(a) For the first case, we will find the number of ways we can order the other three letters. For the second case, we will find the number of ways we can order the other two letters.

For the third case, we will find the number of ways we can order the other three letters. First case = 5 x 4 x 3 = 60Second case = 3 x 2 = 6Third case = 5 x 4 x 3 = 60 Total number of strings = 60 + 6 + 60 = 126(b) First, we will count the number of ways we can have BD. Then, we will count the number of ways we can have DB. Finally, we will add the two counts.BD = 4 x 3 x 2 = 24DB = 4 x 3 x 2 = 24Total number of strings = 24 + 24 = 489. Basic Principles. To find the number of eight-bit strings that either start with a 1 or end with a 1 or both, we need to find the total number of eight-bit strings that start with a 1, the total number of eight-bit strings that end with a 1, and the number of eight-bit strings that both start with a 1 and end with a 1.

Total number of eight-bit strings that start with a 1:2^7 = 128Total number of eight-bit strings that end with a 1:2^7 = 128Total number of eight-bit strings that both start with a 1 and end with a 1:2^6 = 64Therefore, the total number of eight-bit strings that either start with a 1 or end with a 1 or both is:128 + 128 - 64 = 19210.

The number of ways to arrange eight Jovians in a line is 8!. There are nine spaces between the Jovians, including the two ends. We need to choose five of these spaces for the five Martians. The number of ways to choose five spaces from nine is 9C5. For each arrangement of the Jovians and the Martians, there are 5! ways to arrange the Martians in their spaces. Therefore, the total number of arrangements is as follows:8! x 9C5 x 5! = 24,024,00011. Permutations and Combinations.

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a) The PDE xu_xx + u_t = 0 can be solved using separation of variables. We can assume a solution of the form u(x, t) = X(x)T(t), substitute it into the PDE, and separate the variables to obtain two ordinary differential equations, one involving X(x) and the other involving T(t).

These equations can then be solved separately using standard techniques.

b) The PDE u_xx + (x+y)u_yy = 0 cannot be solved using separation of variables. This is because the presence of the term (x+y)u_yy introduces a coupling between the variables x and y, making it impossible to separate them into independent equations.

c) The PDE u_xx + u_yy + xu = 0 cannot be solved using separation of variables. Similar to the previous case, the presence of the term xu introduces a coupling between the variables x and y, preventing their separation into independent equations.

In summary, only the PDE in part a) can be solved using separation of variables.

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The interest paid during the fifth year will be $7,805.54. Payment per month = $375.58. Principal paid at the end of the fifth year = $10,701.62

a. Calculation of monthly payment:

Given, Principal amount = $41,700Interest rate = 9.1% per annum = 0.91/12 = 0.07583 per month

Time period = 15 years = 180 months Using the formula of monthly payment, we get:

PMT = P × r / [1 - (1+r)-n]

Here, P = Principal amount

r = rate of interest per month

n = time period in months

Putting the given values in the formula:

PMT = 41700 × 0.07583 / [1 - (1+0.07583)-180]PMT ≈ $375.58

Payment per month = $375.58 (rounded to 2 decimal places)

Thus, the monthly payment will be $375.58.

b. Calculation of payment made at the end of the fifth year:

Time period = 5 years = 60 months

Using the PMT formula, we have already discovered that the monthly payment is $375.58.

The total payment made during five years = 60 × $375.58 = $22,534.80

Using the formula of monthly payment to calculate the portion of each payment going towards the principal:

Principal paid = PMT × [(1+r)n - (1+r)t] / [(1+r)n - 1]

Here, PMT = $375.58

r = 0.07583

n = 180 months (total time period) and

t = 60 months (time period after which calculation is to be done)Putting the values in the formula:

Principal paid = $375.58 × [(1+0.07583)180 - (1+0.07583)60] / [(1+0.07583)180 - 1]Principal paid

≈ $10,701.62

Interest paid during fifth year = $9,307.86 - $1,502.32

= $7,805.54Thus, the interest paid during the fifth year will be $7,805.54. Principal paid at the end of the fifth year = $10,701.62

Interest paid at the end of the fifth year = $11,833.18 - $10,701.62

= $1,131.56

Balance after five years = $30,998.38

Therefore, Interest paid during the fifth year = $9,307.86 - $1,502.32 = $7,805.54

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The probability that a randomly selected student is female or a physics major is 0.7.

To find the probability, we need to determine the number of students who are either female or physics majors and divide it by the total number of students.

Let's break down the given information:

Total number of students (n) = 40

Number of physics majors (P) = 12

Number of female students (F) = 16

Number of female physics majors (F ∩ P) = 3

To calculate the probability of selecting a student who is female or a physics major, we can use the principle of inclusion-exclusion:

P(F ∪ P) = P(F) + P(P) - P(F ∩ P)

P(F ∪ P) = F/n + P/n - (F ∩ P)/n

P(F ∪ P) = 16/40 + 12/40 - 3/40

P(F ∪ P) = 28/40

P(F ∪ P) = 0.7

The probability that a randomly selected student is female or a physics major is 0.7. This means that there is a 70% chance of selecting a student who is either female or a physics major out of the total population of 40 students.

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The transition matrix from basis B to basis B' is:

[1 2 2]

[3 7 9]

[-1 -4 7]

To find the dimension and a basis of the solution space W of the system of equations:

x + 2y + 2z - 5 + 3t = 0,

x + 2y + 3z + 5 + t = 0,

3x + 6y + 8z + 5 + 5t = 0.

We can write the augmented matrix and perform row operations to obtain the reduced row-echelon form:

[1 2 2 -5 3 | 0]

[1 2 3 5 1 | 0]

[3 6 8 5 5 | 0]

Performing row operations:

R2 = R2 - R1

R3 = R3 - 3R1

[1 2 2 -5 3 | 0]

[0 0 1 10 -2 | 0]

[0 0 2 20 -4 | 0]

R3 = R3 - 2R2

[1 2 2 -5 3 | 0]

[0 0 1 10 -2 | 0]

[0 0 0 0 0 | 0]

Now, we can see that there are two leading variables (x and z) and two free variables (y and t). We can express the leading variables in terms of the free variables:

x = -2y - 2z + 5 - 3t

z = -10y + 2t

Therefore, the solution space W can be expressed as a linear combination of the free variables:

W = {(x, y, z, t) | x = -2y - 2z + 5 - 3t, z = -10y + 2t, y, t ∈ ℝ}

The dimension of the solution space W is 2, corresponding to the number of free variables (y and t).

To find a basis for W, we can express the solution space in terms of the free variables:

W = {(x, y, z, t) | x = -2y - 2z + 5 - 3t, z = -10y + 2t, y, t ∈ ℝ}

Choosing y = 1 and t = 0, we get a particular solution:

W1 = {(-2, 1, -10, 0)}

Choosing y = 0 and t = 1, we get another particular solution:

W2 = {(5, 0, 2, 1)}

Therefore, a basis for the solution space W is B' = {(-2, 1, -10, 0), (5, 0, 2, 1)}.

Now, let's find the transition matrix from basis B to basis B':

To find the transition matrix, we express the vectors in B' in terms of the vectors in B:

(1, 3, -1) = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1)

= a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1)

Solving the system of equations, we find a = 1, b = 3, and c = -1. Similarly, we can find the coefficients for the other vectors in B'.

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The amount of interest paid so far (over the last 10 years) is $78,636 and the amount of interest you will pay over the life of the new loan is $99,999.17.

In order to find out how much interest has been paid so far, we need to find out how much the initial loan was. The down payment made on the home was 10%, so:

Down payment = 10% of $110,000

Down payment = 0.10 × $110,000 = $11,000

So the initial loan was the difference between the price of the home and the down payment:

Initial loan = $110,000 - $11,000

Initial loan = $99,000

Now we can use the loan balance and the initial loan to find out how much of the loan has been paid off in 10 years:

Amount paid off so far = Initial loan - Loan balance

Amount paid off so far = $99,000 - $88,536

Amount paid off so far = $10,464

Now we can find out what percentage of the initial loan that is:

Percent paid off so far = (Amount paid off so far / Initial loan) × 100

Percent paid off so far = ($10,464 / $99,000) × 100

Percent paid off so far = 10.56%

So the amount of interest paid so far is the total payments made minus the amount paid off:

Interest paid so far = Total payments - Amount paid off

Interest paid so far = ($99,000 × 0.09 × 10) - $10,464

Interest paid so far = $89,100 - $10,464

Interest paid so far = $78,636

So the interest paid so far is $78,636.

The life of the new loan is the remaining 20 years of the original 30-year loan. The interest rate is still 9%. To find out how much interest will be paid over the life of the new loan, we can use an online loan calculator or a spreadsheet program like Microsoft Excel.

Using an online loan calculator with a loan amount of $88,536, a term of 20 years, and an interest rate of 9%, we get the following result:

Total payments over life of loan = $188,535.17

Principal paid over life of loan = $88,536.00

Interest paid over life of loan = $99,999.17

So the interest paid over the life of the new loan is $99,999.17.

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The life expectancy of females in 1998 is estimated to be 80.6 years.

We are to find a linear equation of the form `E(t) = mt + b` that models the data, and then estimate the life expectancy of a female in 1998.

The equation of a line is given by

`y = mx + b`, where `m` is the slope and `b` is the y-intercept.

Here, we want to use the equation

`E(t) = mt + b`.

We are given that the life expectancy for a female in 1990 was 79.4 years, and in 1992, it was 79.7 years. We want to estimate the life expectancy in 1998, which is 8 years after 1990.

To find `m` and `b`, we first need to find the slope.

Using the two points (0, 79.4) and (2, 79.7) as (t, E(t)) coordinates, we can find the slope `m`:

`m = (E(2) - E(0)) / (2 - 0)

= (79.7 - 79.4) / 2 = 0.15`.

Using the point (0, 79.4) and the slope `m = 0.15`, we can find the y-intercept `b`: `79.4 = 0.15(0) + b`, so `b = 79.4`.

Thus, the linear function `E(t)` that fits the data is

`E(t) = 0.15t + 79.4`.

To predict the life expectancy of females in 1998, we evaluate `E(8)`:

`E(8) = 0.15(8) + 79.4 = 80.6`.

Therefore, the life expectancy of females in 1998 is estimated to be 80.6 years.

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At the end of the 12-year period, Emily would have approximately $19,739.18 in the RRSP.

To calculate the final amount in Emily's RRSP at the end of the 12-year period, we can use the formula for compound interest:

[tex]A = P(1 + r/n)^(^n^t^)[/tex]

Where:

A = Final amount

P = Initial investment or principal ($90)

r = Annual interest rate (3.30% or 0.033)

n = Number of times interest is compounded per year (quarterly, so n = 4)

t = Number of years (12)

Plugging in the values, we have:

[tex]A = 90(1 + 0.033/4)^(^4^*^1^2^)[/tex]

Simplifying the equation, we get:

A ≈ 90(1.00825)^(48)

A ≈ 90(1.437827977)

Calculating the final amount, we have:

A ≈ $19,739.1793

Rounding this amount to the nearest cent, we get $19,739.18.

Therefore, at the end of the 12-year period, Emily would have approximately $19,739.18 in the RRSP.

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1. The solution of equation dy/dx + y = e^(-x) is y(x) = -1/2 * e^(-x) + 1/2 * e^(2x)

2. The series  [tex]\sum n=1[infinity](-1)^n-1/(n(n+1))[/tex] is conditionally convergent

4. The power series expansion of f(x) = e^(2x) * (e^x + 1) is ∑(2^n * x^(2n) / (n!)^2) from n = 0 to infinity

5. The general solution of the differential equation y'' = y' + x is [tex]y = e^x (x - 1) + C1[/tex]

6. Therefore, [tex]dx^2/d^2y = (1+t^2)^2 / [-4t^2 + 2][/tex]

1. First, we will consider the homogeneous equation. Let us assume that the solution is of the form

y = e^mx

Now, differentiating this, we get:

y' = me^mx

Now, the given equation is

dy/dx + y = e^(-x)

So, substituting the value of y = e^mx, we get

me^mx + e^mx = e^(-x)

on simplification, we get me^(2x) + e^x = 1 So

y(x) = e^(mx) - 1/m

which is the solution of the homogeneous equation. Now, we will consider the non-homogeneous part of the equation. We have y(x) = 1 for x = 0. Using the method of variation of parameters, we assume that y = u(x) * v(x). So,

u(x) = exp[(-∫(p(x) dx)]dx   Let p(x) = 1 Therefore,

u(x) = exp[-x]dx = -exp[-x]

Substituting this, we get:

Y = [-exp[-x] * ∫(e^(-x) * e^x dx)] - [∫(e^(-x) * -exp[-x] * e^(-x) dx)]

on simplification, we get

y(x) = -1/2 * e^(-x) + 1/2 * e^(2x)

2. [tex]\sum n=1[infinity](-1)^n-1/(n(n+1))[/tex]

The series is convergent by Alternating Series Test. The absolute value of each term in the series is

|(-1)^(n-1)/(n(n+1))| = 1/(n(n+1))

Therefore, we can say that the series is conditionally convergent.

4. To expand the function f(x) = e^(2x) * (e^x + 1) into a power series, we can start by expanding e^(2x) and e^x using their respective power series:

e^(2x) = ∑(2^n * x^n / n!) from n = 0 to infinity

e^x = ∑(x^n / n!) from n = 0 to infinity

Multiplying the two power series:

f(x) = e^(2x) * e^x = ∑((2^n * x^n / n!) * (x^n / n!)) from n = 0 to infinity

Simplifying the expression:

f(x) = ∑(2^n * x^n * x^n / (n!)^2) from n = 0 to infinity

f(x) = ∑(2^n * x^(2n) / (n!)^2) from n = 0 to infinity

Therefore, the power series expansion of f(x) = e^(2x) * (e^x + 1) is ∑(2^n * x^(2n) / (n!)^2) from n = 0 to infinity.

5. Given: y'' = y' + x On rearranging, we get: y'' - y' = x This is a first-order linear differential equation. We can solve this using the method of integrating factors, where the integrating factor is given by:

e^-dx/dx - e^-x * y = xe^-x

On multiplying both sides by the integrating factor, we get:

d/dx(ye^-x) = xe^[tex]e^-dx/dx - e^-x * y = xe^(-x)[/tex]-x

Therefore, [tex]ye^-x = ∫(x e^(-x) dx) + C_1.[/tex]Therefore, the general solution of the differential equation is:

[tex]y = e^x (x - 1) + C1[/tex]

6. We have: y = ln(1+t^2)x = t - arctan(t) Differentiating y with respect to x, we get:

dy/dx = [1/(1+t^2)] * [2t/(1+t^2)] - [1/(1+t^2)]

Therefore,

[tex]d^2y/dx^2 = -2[2t/(1+t^2)]^2 - 2[1/(1+t^2)]^2 + 4t/(1+t^2)^2 = [-4t^2 + 2]/(1+t^2)^2[/tex]

Therefore, [tex]dx^2/d^2y = (1+t^2)^2 / [-4t^2 + 2][/tex]

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The approximate value of the given integral using the Midpoint Rule with n = 3 is -4372.

The given integral is, ∫ 10 18(2x−4x 2)dx

We need to use the Midpoint Rule with n = 3 to approximate the value of the given integral.

Midpoint Rule:

The midpoint rule is a numerical integration method that estimates the value of an integral by approximating the function as a straight line.

It divides the integration interval into equally spaced intervals and uses the midpoints of these intervals as approximations to the true value of the function at those points.

The formula for the midpoint rule is given by: M = (b-a)/n

where M is the midpoint of each sub-interval, b and a are the upper and lower limits of integration, and n is the number of sub-intervals.

We have, a = 10, b = 18, and n = 3 (given)

∴ The width of each subinterval = (b-a)/n = (18-10)/3 = 2

Therefore, the three subintervals are as follows: [10, 12], [12, 14], [14, 16], and [16, 18]

Midpoints of each subinterval are: x1 = 11, x2 = 13, x3 = 15.

Using the Midpoint Rule with n=3, the approximate value of the integral is given by: ∆x = (b-a)/n = (18-10)/3 = 2I ≈ ∆x[f(x1) + f(x2) + f(x3)]

where f(x) = 2x - 4x²

∴ f(x1) = 2(11) - 4(11)² = -438f(x2) = 2(13) - 4(13)² = -728f(x3) = 2(15) - 4(15)² = -1020I ≈ ∆x[f(x1) + f(x2) + f(x3)]I ≈ 2[(-438) + (-728) + (-1020)]I ≈ 2(-2186)I ≈ -4372

Therefore, the approximate value of the given integral using the Midpoint Rule with n = 3 is -4372.

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To find the value of x where h'(x) = 0 in the interval [1, 10], we need to solve the equation obtained by setting the derivative of the difference function h(x) equal to zero.

To find the value of x in the interval [1, 10] where h'(x) = 0, we first need to determine the difference function h(x) between f(x) and g(x). The difference function h(x) is defined as h(x) = f(x) - g(x).

f(x) = ln(x² - 5x - 14)

g(x) = ln(x + 2) - 26(x + 7)²

To find h(x), we subtract g(x) from f(x):

h(x) = f(x) - g(x) = ln(x² - 5x - 14) - (ln(x + 2) - 26(x + 7)²)

Now, to find the critical points of h(x) where h'(x) = 0, we differentiate h(x) with respect to x:

h'(x) = [ln(x² - 5x - 14)]' - [ln(x + 2)]' + [26(x + 7)²]'

To simplify, we apply the chain rule and power rule:

h'(x) = [(x² - 5x - 14)' / (x² - 5x - 14)] - [1 / (x + 2)] + [26(2)(x + 7)(1)]

h'(x) = [(2x - 5) / (x² - 5x - 14)] - [1 / (x + 2)] + [52(x + 7)]

Now, we need to find the value of x where h'(x) = 0. Setting h'(x) = 0 and solving for x:

[(2x - 5) / (x² - 5x - 14)] - [1 / (x + 2)] + [52(x + 7)] = 0

After solving this equation, we can find the value of x in the interval [1, 10] for which h'(x) = 0.

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A student with a z-score of 1.80 on a test with a mean score of 90 and a standard deviation of 15 would have a score of approximately 118.5 on the test.

1. A z-score measures how many standard deviations a particular value is away from the mean. In this case, the z-score of 1.80 indicates that the student's score is 1.80 standard deviations above the mean. To find the actual score on the test, we can use the formula:

z = (x - mean) / standard deviation

Rearranging the formula to solve for x (the score), we have:

x = (z * standard deviation) + mean

Substituting the given values into the formula, we get:

x = (1.80 * 15) + 90 = 118.5

Therefore, the student's score on the test would be approximately 118.5.

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The required value of the given integral is [tex]$(2x+4) \ln\left|\frac{x-2}{x-3}\right| + 2x - 4 + 2(x-2)\ln|x-2| - 2(x-3)\ln|x-3| + 2C_2 - 2C_1$[/tex].

To integrate the given expression using integration by parts, we need to choose two functions: one to differentiate and one to Integrate. The standard formula for integration by parts is:

[tex]$$\int u \, dv = u \, v - \int v \, du$$[/tex]

Let's apply this formula to the given integral: [tex]$$I = \int \frac{2x+4}{x^2-5x+6} \, dx$$[/tex]

We can rewrite the integrand as: [tex]$$I = \int \frac{2x+4}{(x-2)(x-3)} \, dx$$[/tex]

Now, let's choose: [tex]$$u = 2x+4 \quad \Rightarrow \quad du = 2 \, dx$$[/tex]

and [tex]$$dv = \frac{dx}{(x-2)(x-3)} \quad \Rightarrow \quad v = \int \frac{dx}{(x-2)(x-3)}$$[/tex]

To find v, we can use partial fraction decomposition: [tex]$$\frac{1}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}$$[/tex]

Multiplying through by (x-2)(x-3) gives: [tex]$$1 = A(x-3) + B(x-2)$$[/tex]

Expanding and equating coefficients: [tex]$$1 = (A+B)x - (3A+2B)$$[/tex]

We have the system of equations:

\begin{align*}

A+B &= 0 \\

3A+2B &= -1

\end{align*}

Solving this system, we find A = 1 and B = -1. Therefore: [tex]$$v = \int \frac{dx}{(x-2)(x-3)} = \int \left(\frac{1}{x-2} - \frac{1}{x-3}\right) \, dx = \ln|x-2| - \ln|x-3| + C$$[/tex]

Now, we can apply the integration by parts formula:

[tex]$$I = u \, v - \int v \, du$$[/tex]

Substituting the values we found:

\begin{align*}

I &= (2x+4) \cdot \left(\ln|x-2| - \ln|x-3|\right) - \int \left(\ln|x-2| - \ln|x-3|\right) \cdot 2 \, dx \\

&= (2x+4) \cdot \left(\ln|x-2| - \ln|x-3|\right) - 2 \int \left(\ln|x-2| - \ln|x-3|\right) \, dx

\end{align*}

We can simplify the integral inside the expression:

[tex]$$\int \left(\ln|x-2| - \ln|x-3|\right) \, dx = \int \ln|x-2| \, dx - \int \ln|x-3| \, dx$$[/tex]

Integrating each term separately:

[tex]$$\int \ln|x-2| \, dx = (x-2) \ln|x-2| - (x-2) + C_1$$[/tex]

[tex]$$\int \ln|x-3| \, dx = (x-3) \ln|x-3| - (x-3) + C_2$$[/tex]

Where [tex]$C_1$[/tex] and [tex]$C_2$[/tex] are constants of integration. Now we can substitute these back into the expression for [tex]$I$[/tex]:

\begin{align*}

I &= (2x+4) \cdot \left(\ln|x-2| - \ln|x-3|\right) - \int \left(\ln|x-2| - \ln|x-3|\right) \cdot 2 , dx \

&= (2x+4) \cdot \left(\ln|x-2| - \ln|x-3|\right) - 2 \int \left(\ln|x-2| - \ln|x-3|\right) , dx

\end{align*}

So, the integral of [tex]$\frac{2x+4}{x^2-5x+6}$[/tex] is given by [tex]$(2x+4) \ln\left|\frac{x-2}{x-3}\right| + 2x - 4 + 2(x-2)\ln|x-2| - 2(x-3)\ln|x-3| + 2C_2 - 2C_1$[/tex], where [tex]$C_1$[/tex] and [tex]$C_2$[/tex] are constants of integration.

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For odd primes p ≥ 7, the value of the expression ∑(i=1 to p+5) i^(p-1) (mod p) is always 0, since (p+5)(p+6) is divisible by 2.



To find the value of the given expression, we need to evaluate the summation:

∑ i=1

p+5

i p−1 (modp)

Let's break down the expression and evaluate it step by step.

First, we have:

∑ i=1

p+5

​i p−1 (modp)

Since we're taking the modulo p at each step, we can simplify the expression by removing the modulo operation:

∑ i=1

p+5

​i p−1

Now, let's expand the summation:

(i p−1 + i p−2 + ... + i + 1)

We can use the formula for the sum of an arithmetic series to simplify this expression:

(i p−1 + i p−2 + ... + i + 1) = (p+5)(i) + (p+4)(i) + ... + (2)(i) + (1)(i)

Next, we can factor out the common factor of i:

(i)((p+5) + (p+4) + ... + 2 + 1)

Now, we can simplify the sum of consecutive integers using the formula for the sum of an arithmetic series:

((p+5) + (p+4) + ... + 2 + 1) = ((p+5)(p+6))/2

Therefore, the simplified expression becomes:

(i)((p+5)(p+6))/2

Finally, we can evaluate this expression modulo p:

(i)((p+5)(p+6))/2 (modp)

Since p is an odd prime, we know that p+5 and p+6 are both even, and their product is divisible by 2. Therefore, we can simplify further:

(i)((p+5)(p+6))/2 ≡ 0 (modp)

So, the value of the given expression ∑ i=1

p+5

​i p−1 (modp) is 0 for any odd prime p greater than or equal to 7.

For odd primes p ≥ 7, the value of the expression ∑(i=1 to p+5) i^(p-1) (mod p) is always 0, since (p+5)(p+6) is divisible by 2.

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The possible values of k for the given system are:

(a) k ≠ 5 → The system has no solution.

(b) k = 5 → The system has infinitely many solutions.

(c) k can take any value except 5 → The system has a unique solution.

(a) To write down the augmented matrix of the given linear system, we can arrange the coefficients of the variables and the constant terms in a matrix form. The augmented matrix has the form [A|B], where A represents the coefficient matrix and B represents the constant matrix.

The given system of equations is:

2x + 2y + 12z = 8

3x - 2y + z = -10

x + y + (k² - 3)z = k + 1

The corresponding augmented matrix [A|B] is:

[  2   2   12  |  8 ]

[  3  -2    1   | -10 ]

[  1   1  (k²-3)| k+1 ]

(b) To find the possible values of k for which the system has no solution, infinitely many solutions, or a unique solution, we can use Gaussian elimination or row reduction.

Performing Gaussian elimination, we'll reduce the augmented matrix to row-echelon form or row-reduced echelon form.

[  2   2   12  |  8 ]

[  3  -2    1   | -10 ]

[  1   1  (k²-3)| k+1 ]

R2 = R2 - (3/2)R1

R3 = R3 - (1/2)R1

[  2   2   12  |  8 ]

[  0  -5  -17  | -26 ]

[  0  -1  (k²-15)/2 | (k-5)/2 ]

R3 = R3 - (1/5)R2

[  2   2   12  |  8 ]

[  0  -5  -17  | -26 ]

[  0   0  (k²-15)/2 - (k-5)/10 | (5-k)/10 ]

Now, we have the row-echelon form of the augmented matrix.

For the system to have a unique solution, the row-echelon form must not have any row of the form [0 0 0 | b] where b ≠ 0. So, we need to analyze the last row of the row-echelon form.

The last row can be written as:

0 * x + 0 * y + 0 * z = (5 - k) / 10

Now, let's consider the three cases:

Case 1: The system has no solution.

If (5 - k) / 10 ≠ 0, i.e., (5 - k) ≠ 0, the system has no solution. This implies k ≠ 5.

Case 2: The system has infinitely many solutions.

If (5 - k) / 10 = 0, i.e., (5 - k) = 0, the system has infinitely many solutions. This implies k = 5.

Case 3: The system has a unique solution.

If (5 - k) / 10 can take any non-zero value, the system has a unique solution. This implies k can be any value except 5.

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The solution approximation as, U(x) = Bsin(2πx). Value of A is 0 but the value of B and C cannot be determined using the orthogonality condition as well.

The given boundary value problem is -u''(x) + u(x) = x, 0 < x < 1, Subjected to boundary conditions:

u(0) = 0, u(1) = sinh 1. b

Let, U(x) be the solution approximation. Therefore, U(x) = Asin(πx) + Bsin(2πx) + Csin(3πx). The R(x) function would be-R(x) = - U''(x) + U(x) - x

Differentiating the U(x) two times with respect to x, we get,

U'(x) = Aπcos(πx) + 2Bπcos(2πx) + 3Cπcos(3πx)` and U''(x) = - Aπ^2sin(πx) - 4Bπ^2sin(2πx) - 9Cπ^2sin(3πx)

Therefore, the R(x) function is,

R(x) = -U''(x) + U(x) - x = [Aπ^2sin(πx) + 4Bπ^2sin(2πx) + 9Cπ^2sin(3πx)] - [Asin(πx) + Bsin(2πx) + Csin(3πx)] - xc.

To determine the values of A, B and C, we will use the given orthogonality condition:

∫_0^1 R(x)sin(πnx) dx = 0, n = 1, 2, 3

Using the values of R(x), we get the following equation for A:

∫_0^1 R(x)sin(πx) dx = 0``⇒ ∫_0^1 [Aπ^2sin^2(πx) - Asin(πx)sin(πx)] dx + ∫_0^1 [4Bπ^2sin(πx)sin(2πx)] dx + ∫_0^1 [9Cπ^2sin(πx)sin(3πx)] dx = 0``⇒ [(A/2) - (A/4)] + 0 + 0 = 0

Therefore, we have A = 0.

Now, let's find out the values of B and C. Using the values of R(x), we get the following equation for B:

∫_0^1 R(x)sin(2πx) dx = 0``⇒ ∫_0^1 [4Bπ^2sin^2(2πx) - Bsin(2πx)sin(πx)] dx + ∫_0^1 [9Cπ^2sin(2πx)sin(3πx)] dx = 0``⇒ 0 + 0 = 0

Therefore, we can't determine the value of B using the orthogonality condition.

Now, let's find out the value of C. Using the values of R(x), we get the following equation for C:

∫_0^1 R(x)sin(3πx) dx = 0``⇒ ∫_0^1 [9Cπ^2sin^2(3πx) - Csin(3πx)sin(πx)] dx + ∫_0^1 [4Bπ^2sin(2πx)sin(3πx)] dx = 0``⇒ 0 + 0 = 0

Therefore, we can't determine the value of `C` using the orthogonality condition as well. Finally, we have the solution approximation as, U(x) = Bsin(2πx).

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It is not reasonable to conclude that more than 10% of customers wait for more than 5 minutes.

The null hypothesis H0 is that less than or equal to 10 percent of the bank customers wait more than five minutes for a teller. The alternate hypothesis HA is that more than 10 percent of bank customers wait more than five minutes for a teller. In order to determine whether the data supports the alternative hypothesis, the z-test is the best approach to use. Here, the sample size (n) is 100, and the number of customers who wait for over 5 minutes (x) is 13.

The standard error of proportion is:SEp = sqrt [(p * (1 - p))/n] where p is the proportion of customers who wait for over 5 minutes and is assumed to be equal to 10 percent, which is 0.1. SEp = sqrt [(0.1 * (1 - 0.1))/100] = 0.03162

The test statistic, which follows the standard normal distribution, is calculated as:z = (x/n - p)/SEp = (13/100 - 0.1)/0.03162 = 0.6317

The critical value of z at the 0.01 level of significance is 2.33. Since the calculated test statistic (0.6317) is less than the critical value (2.33), we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that more than 10 percent of bank customers wait more than five minutes for a teller.

Therefore, the branch manager's request for additional part-time tellers should not be approved.

The answer is, it is not reasonable to conclude that more than 10% of customers wait for more than 5 minutes.

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P(X > 56) = 1 - P(Z ≤ 2/3).To solve the given problems, we can use the properties of the normal distribution.

(a) To determine P(X ≤ 50), we need to find the cumulative probability up to the value of 50.

Using the properties of the standard normal distribution, we can standardize the value 50:

Z = (X - μ) / σ

Z = (50 - 54) / 3

Z = -4 / 3

Now, we can look up the cumulative probability associated with Z = -4/3 in the standard normal distribution table or use a calculator to find P(Z ≤ -4/3).

P(X ≤ 50) = P(Z ≤ -4/3)

(b) To determine P(X > 56), we need to find the cumulative probability beyond the value of 56.

Again, we standardize the value 56:

Z = (X - μ) / σ

Z = (56 - 54) / 3

Z = 2 / 3

Now, we can find P(Z > 2/3) by subtracting the cumulative probability associated with Z ≤ 2/3 from 1.

P(X > 56) = 1 - P(Z ≤ 2/3)

You can use a standard normal distribution table or a calculator to find the cumulative probabilities associated with the standardized values and obtain the final probabilities.

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The interval of convergence is :

a)   (-1,1).

b)  (-1,1).

c)   (-1,1).

d)   (-sqrt(3)/3, sqrt(3)/3).

e)  (-5,5).

f) (-1/4,1/4).

(a) For f(x), we start with 1/(1-x) and differentiate both sides, then multiply by (1+x) to get the given expression. Thus, we have:

f(x) = (1+x)/(1-x) = (1+x) * ∑ n=0 [infinity] (-1)^n x^n

The interval of convergence is (-1,1).

(b) For g(x), we again start with 1/(1-x^4) and use the formula given to get:

g(x) = (1-x^4)^(-3/4) = ∑ n=0 [infinity] 3x^(4n)

The interval of convergence is also (-1,1).

(c) For h(x), we start with 1/(1-x^3) and use the formula to get:

h(x) = (1-x^3)^(-1) = ∑ n=0 [infinity] x^(3n)

The interval of convergence is (-1,1).

(d) For r(x), we start with 1/(1+9x^2) and use the formula to get:

r(x) = (1+9x^2)^(-1) = ∑ n=0 [infinity] (-1)^n 9^n x^(2n)

The interval of convergence is (-sqrt(3)/3, sqrt(3)/3).

(e) For f(x), we can rewrite it as x^-5 - ∑ n=0 [infinity] 5^(n+1) x^n, and use the formula to get:

f(x) = x^-5 - 5x/(1-5x)

The interval of convergence is (-5,5).

(f) For g(x), we can rewrite it as (4x+1)/x * 1/(1-4x), and use the formula to get:

g(x) = (4x+1)/x * ∑ n=0 [infinity] (-1)^n 4^n x^(n+1)

The interval of convergence is (-1/4,1/4).

In summary, we can use the building block formula to find power series representations for each function, and then determine the interval of convergence using the known criteria.

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 The values of cos 20 and sin 20 are frac{3sqrt{2}}{7}) and (frac{2}{7}) respectively.

Given that \(\sin \theta = \frac{\sqrt{2}}{7}\) and \(\cos \theta > 0\), we can find the values of \(\cos 20\) and \(\sin 20\).  The values of \(\cos 20\) and \(\sin 20\) are \(\frac{3\sqrt{2}}{7}\) and \(\frac{2}{7}\) respectively.

The value of \(\cos 20\) is \(\frac{3\sqrt{2}}{7}\), and the value of \(\sin 20\) is \(\frac{2}{7}\).

We are given \(\sin \theta = \frac{\sqrt{2}}{7}\) and \(\cos \theta > 0\). To find the values of \(\cos 20\) and \(\sin 20\), we can use trigonometric identities.

Since \(\cos^2 \theta + \sin^2 \theta = 1\), we can solve for \(\cos \theta\). Given \(\sin \theta = \frac{\sqrt{2}}{7}\), we can substitute this value into the identity:

\[\cos^2 \theta + \left(\frac{\sqrt{2}}{7}\right)^2 = 1\]

\[\cos^2 \theta + \frac{2}{49} = 1\]

\[\cos^2 \theta = 1 - \frac{2}{49}\]

\[\cos^2 \theta = \frac{47}{49}\]

Since \(\cos \theta > 0\), we can take the positive square root:

\[\cos \theta = \frac{\sqrt{47}}{7}\]

Now, we can find \(\cos 20\) by substituting \(\theta = 20\) degrees:

\[\cos 20 = \frac{\sqrt{47}}{7}\]

Similarly, we can find \(\sin 20\) using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[\sin^2 \theta = 1 - \cos^2 \theta\]

\[\sin^2 \theta = 1 - \frac{47}{49}\]

\[\sin^2 \theta = \frac{2}{49}\]

\[\sin \theta = \frac{\sqrt{2}}{7}\]

Substituting \(\theta = 20\) degrees:

\[\sin 20 = \frac{\sqrt{2}}{7}\]

Therefore, the values of \(\cos 20\) and \(\sin 20\) are \(\frac{3\sqrt{2}}{7}\) and \(\frac{2}{7}\) respectively.

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Where we take the Laurent series expansion of the given expression about the point z0. For this, we note that:We can write the expression of the given function in the standard integral form.

The integral to be calculated is given by:2πi
1
​
∫ ∣a∣=r
​
z−z 0
​
z+z 0
​
​
⋅ z
dz
​
+where 0 < ∣z 0 ∣ < r.Let's try to put it in the standard integral form:Where we take the Laurent series expansion of the given expression about the point z0. For this, we note that:We can write the expression of the given function in the standard integral form:2πi
1
​
∫ ∣a∣=r
​
z−z 0
​
z+z 0
​
​
⋅ z
dz
​
+
=
2πi
1
​
∫ ∣a∣=r
​
z−z 0
​
z+z 0
​
​
⋅
(
∑
n=−∞
∞
c
n
(z−z
0
) n
)
dz
​
=
2πi
1
​
∑
n=−∞
∞
c
n
∫ ∣a∣=r
​
(
z−z
0
)
n+1
(
z+z
0
)
n
dz
​
Now, we need to calculate the residue at z=z0. For this, we observe that:(z-z0)2(z+z0) = [(z0+z)-(z-z0)]/[2(z+z0)] = (1/2)[(z0+z)/(z+z0) - (z-z0)/(z+z0)]. Using partial fractions, we can write this expression as:1/2[1 + (2z0)/(z0+z)] - 1/2[(z-z0)/(z0+z)]. The first term is a constant, and does not contribute to the residue. The second term is of the standard form for calculation of residue, and we can write its residue as:(lim
z→z
0
z−z 0
​
)
(1/2)(−1/(2z
0
)) = −1/4z0. Thus, the final expression of the integral is given by:2πi
1
​
∫ ∣a∣=r
​
z−z 0
​
z+z 0
​
​
⋅ z
dz
​
+ = −πi/(2z0).Using the given result, we need to prove that:2π
1
​
∫ 0
2π
​
∣re iθ
−z 0
∣ 2
dθ
​
= r 2
−∣z 0
∣ 2
1
​
Now, let us convert the integral on LHS in terms of z. Using z = re iθ, we get dz = i z dθ. Also, |z-z0| = |re iθ - z0| = sqrt(r^2 + z0^2 - 2rz0cosθ). Substituting these values, we get:LHS = 2π
0
​
∣re iθ
−z 0
∣ 2
dθ
​
= 2π
0
​
[r^2 + z0^2 - 2rz0cosθ]
dθ
​
= 2π(r^2 + z0^2) - 4πrz0[(1/2π)∫ 0
2π
​
cosθ dθ
​
] = 2π(r^2 + z0^2) - 4πrz0[0] = 2π(r^2 + z0^2).Thus, LHS is proved. Also, using the given expression, we can write that the RHS is given by:(r^2 - z0^2)/(2z0).Substituting this value in the expression of LHS, we get that the given identity is indeed true.

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The probability of winning $11 in the given game can be calculated as follows: Since the player draws balls without replacement, there are a total of 4 balls in the urn, with one of them being a ten-dollar bill.

The player needs to draw the ten-dollar bill last, which means the order of the draws matters. The probability of drawing the ten-dollar bill last can be calculated as the product of the probabilities of drawing the other three bills first.  To calculate the probability, we start with the probability of drawing a one-dollar bill on the first draw, which is 2/4 (since there are two one-dollar bills out of four balls). On the second draw, the player needs to draw the remaining one-dollar bill out of the three remaining balls, so the probability is 1/3. On the third draw, the player needs to draw the five-dollar bill out of the two remaining balls, so the probability is 1/2. Finally, on the fourth and last draw, the player will draw the ten-dollar bill, so the probability is 1/1.

To find the overall probability, we multiply these individual probabilities together:

[tex]\[\frac{2}{4} \times \frac{1}{3} \times \frac{1}{2} \times \frac{1}{1} = \frac{1}{12}\][/tex]

Therefore, the probability of winning $11 in this game is 1/12.

In summary, the probability of winning $11 by drawing the ten-dollar bill last from the urn is 1/12. This is obtained by multiplying the probabilities of drawing the one-dollar bills, the five-dollar bill, and the ten-dollar bill in the specified order.

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The product of \((-3+5i)(3+i)\) is \(-14 + 12i\) in standard form, where the real part is \(-14\) and the imaginary part is \(12\).

Finding the product of \((-3+5i)(3+i)\) and writing the result in standard form.

To begin, we use the distributive property to expand the expression:

\((-3+5i)(3+i) = -3 \cdot 3 + (-3) \cdot i + 5i \cdot 3 + 5i \cdot i\)

Simplifying the multiplication:

\((-3+5i)(3+i) = -9 - 3i + 15i - 5\)

Next, we combine like terms:

\((-3+5i)(3+i) = -14 + 12i\)

Therefore, the product of \((-3+5i)(3+i)\) is \(-14 + 12i\) in standard form. The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. In this case, the real part is \(-14\) and the imaginary part is \(12\).

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The domain of the function can be expressed as, {(x, y): y < 1-x}.

Given the function,  [f(x, y)=ln (1-x-y)].

To find the domain of the function, observe that the natural logarithmic function is only defined for positive numbers (i.e. greater than zero).

Thus, we have the following restriction:\[1-x-y > 0\].

Rearranging the above inequality:\[1 > x+y\]Therefore, the domain of the function can be expressed as,\[\{(x, y): y < 1-x\}\]

Given a function f(x, y) = ln (1-x-y).

To find the domain of the function, we can observe that the natural logarithmic function is only defined for positive numbers (i.e. greater than zero).

This means that we have the following restriction: 1-x-y > 0. By rearranging the above inequality, we get 1 > x+y.

Therefore, the domain of the function can be expressed as, {(x, y): y < 1-x}.

Hence, we can conclude that the domain of the given function is the set of all ordered pairs (x, y) such that y is less than 1 subtracted by x.

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1. The solution to the initial value problem (IVP) t^2y' - y = t, t > 0, y(1) = 4 is y(t) = t^2 + 3t - 3 + (C / t), where C is a constant determined by the initial condition.

2. The interval in which the IVP (sin(t))y' + (1 - t^2)y = t^(-2/8), y(2) = 5 is guaranteed to have a unique solution is (-∞, ∞).

1. To solve the IVP t^2y' - y = t, we can first rewrite the equation in standard form as y' - (1/t^2)y = 1/t. This is a linear first-order differential equation. The integrating factor is given by μ(t) = e^(∫(-1/t^2)dt) = e^(1/t).

Multiplying the original equation by the integrating factor, we have e^(1/t)y' - (1/t^2)e^(1/t)y = e^(1/t)/t. By the product rule, we can rewrite the left-hand side as (e^(1/t)y)' = e^(1/t)/t.

Integrating both sides with respect to t, we obtain e^(1/t)y = ∫(e^(1/t)/t)dt. This integral is not elementary, so we express the solution in integral form as y(t) = e^(-1/t)∫(e^(1/t)/t)dt + Ce^(-1/t), where C is a constant of integration.

2. The interval in which an IVP is guaranteed to have a unique solution is determined by the Lipschitz condition. In this case, the given IVP has the form y' + p(t)y = g(t), where p(t) = 1 - t^2 and g(t) = t^(-2/8).

Since p(t) and g(t) are continuous on the interval (-∞, ∞), the Lipschitz condition is satisfied, and the IVP is guaranteed to have a unique solution for any value of t in the interval (-∞, ∞).

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Using the Law of Cosines, we have found the following angles:

Angle A ≈ 5.72°, Angle B ≈ 167° 38' 0", Angle C ≈ 1.14°. And the length of side b is approximately 49.86.

To solve the triangle using the Law of Cosines, we can use the following formula:

c^2 = a^2 + b^2 - 2ab * cos(C)

B = 18° 25'

a = 30

c = 20

A = C = 6° 11' 0"

Let's find angle B in degrees:

B = 180° - A - C

B = 180° - 6° 11' 0" - 6° 11' 0"

B = 167° 38' 0"

Now we can apply the Law of Cosines to find side b:

b^2 = a^2 + c^2 - 2ac * cos(B)

b^2 = 30^2 + 20^2 - 2 * 30 * 20 * cos(167° 38' 0")

Now we can calculate the value of b:

b^2 ≈ 900 + 400 - 2 * 30 * 20 * cos(167.6333°)

b^2 ≈ 1300 - 1200 * (-0.988)

b^2 ≈ 1300 + 1185.6

b^2 ≈ 2485.6

b ≈ √(2485.6)

b ≈ 49.86

Now let's find angle A in degrees:

A = cos^(-1)((b^2 + c^2 - a^2) / (2 * b * c))

A = cos^(-1)((49.86^2 + 20^2 - 30^2) / (2 * 49.86 * 20))

A ≈ cos^(-1)((2485.6 + 400 - 900) / (2 * 49.86 * 20))

A ≈ cos^(-1)(1985.6 / 1997.2)

A ≈ cos^(-1)(0.9947)

A ≈ 5.72°

Similarly, angle C can be found as:

C = cos^(-1)((a^2 + b^2 - c^2) / (2 * a * b))

C = cos^(-1)((30^2 + 49.86^2 - 20^2) / (2 * 30 * 49.86))

C ≈ cos^(-1)((900 + 2485.6 - 400) / (2 * 30 * 49.86))

C ≈ cos^(-1)(2985.6 / 2989.2)

C ≈ cos^(-1)(0.9988)

C ≈ 1.14°

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