Replace? with an expression that will make the equation valid. d ₂ x³ + 3 = ex³ +3 +3 ? dx

The general solution of the differential equation is y(x) = C₁x¹/² + 1/4 cos(x).

Given a differential equation:

4x²y" + y = x³/² sin(x)

and y(x) = x¹/²

We need to verify whether the given function is the solution of the differential equation or not.

Therefore, we will substitute the value of y(x) in the differential equation.

Let's start by finding the first and second derivatives of y(x) which will be used further.

y(x) = x¹/²y'(x)

= d/dx (x¹/²)y'(x)

= (1/2)x^(-1/2)y''(x)

= d/dx[(1/2)x^(-1/2)]y''(x)

= (-1/4)x^(-3/2)

Therefore, substituting y(x) and y" (x) in the differential equation:

4x² (-1/4)x^(-3/2) + x¹/² = x³/²sin(x)

Thus, the above equation simplifies as:-

x^(-1) + x¹/² = x³/²sin(x)

Here, we can see that the given function is not a solution of the differential equation.

However, we can find the particular solution of the differential equation by the method of variation of parameters.

Where we write the given equation in the standard form:

y'' + [1/4x⁴]y = [x¹/² sin(x)]/4x⁻²

On comparing with the standard form:

y'' + p(x) y' + q(x) y = g(x) where p(x) = 0, q(x) = 1/4x⁴ and g(x) = [x¹/² sin(x)]/4x⁻²

Now, let's calculate the Wronskian for the differential equation as:

W(y₁, y₂) = | y₁    y₂ |-1/4x²               1/4x²-1/2W(y₁, y₂)

= 1/4x³

The particular solution y₂(x) will be:

y₂(x) = -y₁(x) ∫[g(x) y₁(x)] / W(x) dx

Substituting the given value in the above equation, we get:

y₂(x) = -x¹/² ∫[x¹/² sin(x)] / (x³/² 4x⁻²) dx

y₂(x) = -1/4 ∫sin(x) dx

y₂(x) = -1/4 [-cos(x)] + C₁

y₂(x) = 1/4 cos(x) + C₁

Hence, the general solution of the differential equation is:

y(x) = C₁x¹/² + 1/4 cos(x)

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To show that -(pq) is logically equivalent to p↔q, we can use a truth table to compare the two expressions. The truth table will have columns for p, q, pq, -(pq), p↔q.

The expression -(pq) represents the negation of the conjunction (AND) of p and q. This means that -(pq) is true when pq is false, and vice versa.

The expression p↔q represents the biconditional (IF and ONLY IF) between p and q. It is true when p and q have the same truth value, and false when they have different truth values.

By comparing the truth values of -(pq) and p↔q for all possible combinations of truth values for p and q, we can determine if they are logically equivalent.

The truth table shows that -(pq) and p↔q have the same truth values for all combinations of p and q. Therefore, -(pq) is logically equivalent to p↔q.

b) To show that -p → (q→r) is logically equivalent to q→ (pvr), we can again use a truth table to compare the two expressions. The truth table will have columns for p, q, r, -p, q→r, -p → (q→r), pvr, and q→ (pvr).

The expression -p represents the negation of p, so -p is true when p is false, and false when p is true.

The expression q→r represents the conditional (IF...THEN) statement between q and r. It is true when q is false or r is true, and false otherwise.

By comparing the truth values of -p → (q→r) and q→ (pvr) for all possible combinations of truth values for p, q, and r, we can determine if they are logically equivalent.

The truth table shows that -p → (q→r) and q→ (pvr) have the same truth values for all combinations of p, q, and r. Therefore, -p → (q→r) is logically equivalent to q→ (pvr).

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The row operations are Add 1 times row 1 to row 2,

R2 = R2 + R1 for the first two matrices and Add 1 times row 1 to row 2,

R2 = R2 + R1, R3 = R3 + R1 for the last matrix.

For each of the given matrices, we are to give the row operation that has the same result as left-multiplication by the given matrix.

Below is the solution to the given problem:

1000 0300

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.0 0 1 0 0 0 1 0 0 becomes

0 0 1 1 0 0 1 0 0 00001 1050

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.

0 0 1 0 0001 1 0 0 0 becomes

0001 1 0 1 0 00001 0 0 1 0 0

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.

0 0 10 0 0 1 1 0 0 becomes

0 0 1 0 0 10 1 0 0 00001 0 0 10

The row operations are Add 1 times row 1 to row 2,

R2 = R2 + R1 for the first two matrices and Add 1 times row 1 to row 2,

R2 = R2 + R1, R3 = R3 + R1 for the last matrix.

Note: There are several ways to do this problem but this is one of the simplest and quickest ways.

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Let A be a non-singular nx n matrix. Show that A is not similar to 2.A singular matrix is one in which the determinant is zero, whereas a non-singular matrix has a non-zero determinant.

This implies that the inverse matrix of the non-singular matrix exists and is also unique for a given matrix.In matrix algebra, two matrices A and B are said to be similar if there is a nonsingular matrix P such that

$$A=PBP^{-1}$$

Now let us suppose that A is similar to a 2 × 2 matrix.

We can write, $$A=PBP^{-1}$$

Taking determinants on both sides, $$det(A)=det(PBP^{-1})$$

Expanding the determinant, we obtain

:$$det(A)=det(P)det(B)det(P^{-1})$$$$det(A)=det(P)det(B)\frac{1}{det(P)}$$$$det(A)=det(B)$$Since B is a 2 × 2 matrix, then det(B) is either the product of its diagonal elements or the determinant of a 2 × 2 matrix,

which is of the form$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$Now, if B is similar to A and A is non-singular, then B must also be non-singular.

Therefore, the determinant of B is non-zero.But we have just seen that det(A) = det(B), so det(A) ≠ 0 and hence A is non-singular.

This implies that the matrix A cannot be similar to a 2 × 2 singular matrix. Thus, we can conclude that A is not similar to 2.

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A is similar to the identity matrix. This implies that A is diagonalizable. A is not similar to matrix 2.

Let A be a non-singular nx n matrix.

Show that A is not similar to 2

Suppose A is similar to matrix 2, which means there exists a matrix P such that P²=2 and A = P²  = PP.

Let λ be an eigenvalue of A, then A = PDP⁻¹,

where D is a diagonal matrix with the diagonal entries being the eigenvalues of A.

Since A is non-singular, λ ≠ 0.

Let D be the diagonal matrix of the eigenvalues of A.

Then the similarity condition A = PDP⁻¹ can be written as PDP⁻¹ = PP.

Multiplying both sides by P, we get DP⁻¹ = P.

Therefore, P² = D, which is a diagonal matrix with the diagonal entries being the squares of the eigenvalues of A.

This implies that P is also diagonal.

Therefore, if A is similar to a diagonal matrix, then A is diagonalizable.

Let A be a non-singular nx n matrix.

If A is similar to a diagonal matrix, then A is diagonalizable.

The proof of this fact is quite simple.

If A is similar to a diagonal matrix, then A = PDP⁻¹, where D is a diagonal matrix.

Therefore, the columns of P are eigenvectors of A.

Since the eigenvectors of A form a basis for the vector space, A is diagonalizable.

The determinant of A is non-zero, which means that A is invertible.

Therefore, A is similar to the identity matrix.

This implies that A is diagonalizable.

Thus, A is not similar to matrix 2.

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Answer:

64 square centimeters

Step-by-step explanation:

The surface are of a pyramid is found by finding the sum of the area of the four sides and the base.

Finding the triangular face:

Area of triangle = [tex]\frac{1}{2} b h[/tex] = [tex]\frac{1}{2}*4*6 = 12[/tex]

12 * 4 (4 sides) = 48 square cm

Finding the Base = [tex]w * l = 4 * 4 = 16[/tex]

Finally, we add it together. 48 + 16 = 64

Answer:

tip : (56*18)/100=10,08$

Step-by-step explanation:

The total amount of tip Linda left = $10.08

The question appears to be incomplete.

However, it could be made out from the question that you are asking for the amount of tip Linda left.

The total amount using which Linda had her dinner at restaurant = $56

Percentage of tip she left = 18% of the total amount

To calculate the amount of tip, the total amount can be multiplied to the given percentage divided by 100.

Therefore, the total amount of tip Linda left = 56*18/100

                                                                         = 1008/100

                                                                         = $10.08

Therefore, the total amount of tip Linda left = $10.08

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The integral ∫(rπ/4) tan²(θ) sec²(θ) dθ can be evaluated by applying trigonometric identities and integration techniques.

To evaluate the integral, we can start by using the trigonometric identity tan²(θ) + 1 = sec²(θ). Rearranging this equation gives tan²(θ) = sec²(θ) - 1.

Substituting this identity into the integral, we have ∫(rπ/4) (sec²(θ) - 1) sec²(θ) dθ.

Simplifying further, we get ∫(rπ/4) (sec⁴(θ) - sec²(θ)) dθ.

Now, we can integrate each term separately. The integral of sec⁴(θ) is (1/3)tan(θ)sec²(θ) + (2/3)θ + C, and the integral of sec²(θ) is tan(θ) + C, where C is the constant of integration.

Thus, the final solution to the integral is ((1/3)tan(θ)sec²(θ) + (2/3)θ - tan(θ)) evaluated over the range π/4.

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(a) ker(7) = {0}

(b) nullity(T) = 0

(c) range(T) = {(6s, -3t, 5u) : s, t, u ∈ R}

(d) rank(T) = 3

To find the kernel (ker(7)), nullity (nullity(T)), range (range(T)), and rank (rank(T)) of the linear transformation T(x) = Ax, where A is the given matrix:

A = | 0 -3 5 |

| 6 0 13 |

Let's calculate these values step by step:

(a) ker(7) (If there are an infinite number of solutions, use t as your parameter.)

To find the kernel of the transformation T, we need to find all vectors x such that T(x) = Ax = 0.

We want to find the vectors x such that 7x = 0.

Setting up the equation:

7x = 0

This equation is satisfied when x = 0. So the kernel ker(7) is {0} (the zero vector).

(b) nullity(T)

The nullity of T, denoted as nullity(T), is the dimension of the kernel of T. In this case, the kernel ker(7) contains only the zero vector {0}, so the nullity is 0.

(c) range(T)

The range of T, denoted as range(T), is the set of all possible outputs of T(x) for all vectors x.

We can find the range of T by considering the span of the column vectors of A. Let's denote the column vectors as c₁, c₂, and c₃:

c₁ = | 0 |

| 6 |

c₂ = |-3 |

| 0 |

c₃ = | 5 |

| 13 |

The range of T is the span of these column vectors, which means it's the set of all possible linear combinations of these vectors. So the range(T) is:

range(T) = {(s × c₁) + (t × c₂) + (u × c₃) : s, t, u ∈ R}

Substituting the values of the column vectors:

range(T) = {(s × | 0 |) + (t × |-3 |) + (u × | 5 |) : s, t, u ∈ R}

| 6 | | 0 | | 13 |

Simplifying the expression:

range(T) = {(6s, -3t, 5u) : s, t, u ∈ R}

So the range(T) is {(6s, -3t, 5u) : s, t, u ∈ R}.

(d) rank(T)

The rank of T, denoted as rank(T), is the dimension of the range of T. In this case, the range of T is {(6s, -3t, 5u) : s, t, u ∈ R}, which is a three-dimensional space (R³). So the rank(T) is 3.

In summary:

(a) ker(7) = {0}

(b) nullity(T) = 0

(c) range(T) = {(6s, -3t, 5u) : s, t, u ∈ R}

(d) rank(T) = 3

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(a) The contrapositive of the given statement is: "If n is an odd number, then (n+1)² is an even number." (b) To prove the contrapositive, assume n is an odd number (n = 2k + 1), then (n+1)² = 4(k+1)², which is divisible by 2, proving the contrapositive. (c) The original statement is true because squaring an odd number results in an odd number, and if (n+1)² is odd, it implies n+1 is odd, thus n must be even due to the sum of an even and odd number being odd.

(a) The contrapositive of the given statement is: "If n is an odd number, then (n+1)² is an even number."

(b) To prove the contrapositive statement, we assume that n is an odd number. Therefore, we can write n as 2k + 1, where k is an integer.

Substituting this value of n into the expression (n+1)², we get:

(n+1)² = (2k + 1 + 1)² = (2k + 2)² = 4(k+1)²

Since (k+1) is an integer, we can represent it as m, where m = k + 1. Therefore, the expression becomes:

4(k+1)² = 4m² = 2(2m²)

We can see that the expression 2(2m²) is an even number since it is divisible by 2. Thus, we have proven that if n is an odd number, then (n+1)² is an even number, which is the contrapositive statement.

(c) The original statement is true because it is based on the properties of even and odd numbers. If we square an odd number, the result is always an odd number. On the other hand, if we square an even number, the result is always an even number.

In the given statement, if (n+1)² is an odd number, it means that n+1 is an odd number (since the square of an odd number is odd). If n+1 is odd, then n must be even because adding an odd number to an even number always results in an odd number. Therefore, the original statement holds true based on the properties of even and odd numbers.

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The maximum value of z is 54, and it occurs when x₁ = 11 and x₂ = 14.

To solve the given linear programming problem using the two-stage method, we'll first set up the initial tableau and then perform iterations to find the optimal solution. Let's begin:

Step 1: Set up the initial tableau

We introduce slack variables, s₁, s₂, s₃, and an auxiliary variable, A, to convert the problem into a standard form.

The initial set of equations and inequalities are as follows:

x₁ + 2x₂ + s₁ = 18

x₁ + 3x₂ - s₂ + A = 12

2x₁ + 2x₂ + s₃ = 28

We'll convert these equations into standard form as follows:

x₁ + 2x₂ + s₁ = 18

x₁ + 3x₂ - s₂ + A = 12

2x₁ + 2x₂ + s₃ = 28

Now, we'll construct the initial tableau:

diff

Copy code

---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 | -10|  0 |  0 |  0 |  0 |

---------------------------

|  s₁ |  1 |  2 |  1 |  0 |  0 |  0 |

---------------------------

|  A  |  1 |  3 |  0 | -1 |  1 | 12 |

---------------------------

|  s₃ |  2 |  2 |  0 |  0 |  1 | 28 |

---------------------------

Step 2: Perform iterations

We'll perform iterations to find the optimal solution by applying the simplex method.

Iteration 1:

We choose s₃ as the entering variable and x₂ as the leaving variable. Pivot on the element 2 in the s₃ column.

diff

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---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 |  0 |  0 |  0 | -2 | 56 |

---------------------------

|  s₁ |  1 |  0 |  1 |  0 | -1 | 10 |

---------------------------

|  A  |  1 |  1 |  0 | -1 |  1 | 14 |

---------------------------

|  x₂ |  1 |  1 |  0 |  0 | 1/2| 14 |

---------------------------

Iteration 2:

We choose s₂ as the entering variable and s₃ as the leaving variable. Pivot on the element -1 in the s₂ column.

diff

Copy code

---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 |  0 |  0 |  1 |  0 | 54 |

---------------------------

|  s₁ |  1 |  0 |  1 |  1 |  0 | 11 |

---------------------------

|  A  |  1 |  1 |  0 |  1 |  0 | 13 |

---------------------------

|  x₂ |  1 |  1 |  0 |  0 | 1/2| 14 |

---------------------------

Iteration 3:

No negative coefficients exist in the objective row, indicating the optimal solution has been reached.

Step 3: Read the optimal solution

From the final tableau, we can read the optimal values of the decision variables and the objective function.

x₁ = 11

x₂ = 14

Z = 54

Therefore, the maximum value of z is 54, and it occurs when x₁ = 11 and x₂ = 14.

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The two sets A' n B and O are not equal based on the given Venn diagrams.

In the given problem, we are asked to determine whether the two sets A' n B and O are equal using Venn diagrams. The notation A' represents the complement of set A, and n denotes the intersection of two sets. O represents the null set or the empty set.

To analyze the equality of the sets, we examine the Venn diagrams representing A' n B and O. In a Venn diagram, the intersection of two sets is represented by the overlapping region between them. However, in this case, since O represents the empty set, there is no overlap between the two sets. Thus, we can conclude that A' n B and O are not equal.

The absence of any common elements between A' n B and O indicates that the sets do not have any shared elements. In other words, the intersection of A' n B is empty, which aligns with the definition of the null set O. Therefore, based on the Venn diagrams, we can confidently state that the two sets A' n B and O are not equal.

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a) The convergence of the series Σ[tex]((-1)^i)/(i^2)[/tex] cannot be determined using the ratio test. b) The radius of convergence for the series [tex](i+1)! (2^n + 5)^{(2i)[/tex] is 0.

a) To examine the convergence of the series Σ((-1)^i)/(i^2), we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.

Let's apply the ratio test to the given series:

lim (i→∞)[tex]|((-1)^(i+1))/(i^2+1)| / |((-1)^i)/(i^2)|[/tex]

Taking the absolute values and simplifying, we have:

lim (i→∞) [tex]|(-1)^(i+1)| |i^2| / |i^2+1|[/tex]

Since the absolute value of (-1)^(i+1) is always 1, we can further simplify:

lim (i→∞) [tex]i^2 / (i^2+1)[/tex]

Now, taking the limit as i approaches infinity:

lim (i→∞) i[tex]^2 / (i^2+1) = 1[/tex]

Since the limit is equal to 1, the ratio test is inconclusive. We cannot determine the convergence of the series based on the ratio test.

b) The series [tex](i+1)! (2^n + 5)^(2i) = 7x + 9r^2 + 13x^3 + 21x + ...[/tex] can be rewritten as a power series in the form Σa_nx^n, where a_n is the coefficient of [tex]x^n.[/tex]

To find the radius of convergence for this series, we can use the formula:

R = 1 / lim (n→∞) |a_(n+1)/a_n|

In this case, the coefficient a_n is given by [tex](n+1)! (2^n + 5)^{(2n).[/tex]

Let's calculate the limit:

lim (n→∞) [tex]|(n+2)! (2^(n+1) + 5)^{(2{(n+1)})} / (n+1)! (2^n + 5)^(2n)|[/tex]

Simplifying, we have:

lim (n→∞) [tex]|(n+2)(2^{(n+1)}+ 5)^(2)| / |(2^n + 5)^{(2n)}|[/tex]

Now, taking the limit as n approaches infinity:

lim (n→∞)[tex]|(n+2)(2^{(n+1)} + 5)^(2)| / |(2^n + 5)^{(2n)}|[/tex]= ∞

Since the limit is infinity, the radius of convergence is 0. This means that the series converges only when x = 0, and diverges for all other values of x.

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a) The uniform distribution on [0, 0] has zero probability density for any value of x, the likelihood function becomes:

L(x₁, x₂, ..., xₙ | 0) = 0 × 0 × ... × 0 = 0

b) The maximum value of the samples must be zero for the likelihood function to be maximized.

c) Fe(x) = 0 for 0 ≤ x ≤ 0

Fe(x) = 0 for x > 0

d) The concept of the PDF is not applicable to this random variable.

e) E[o] = 0, which means o is a biased estimator of 0.

f) The small modification to the MLE studied so far does not change the estimator, and o₂ remains the same as o.

a) The likelihood function for the samples x₁, x₂, ..., xₙ from the uniform distribution Unif(0, 0) can be written as:

L(x₁, x₂, ..., xₙ | 0) = f(x₁ | 0) × f(x₂ | 0) × ... × f(xₙ | 0)

Since the uniform distribution on [0, 0] has zero probability density for any value of x, the likelihood function becomes:

L(x₁, x₂, ..., xₙ | 0) = 0 × 0 × ... × 0 = 0

b) The maximum likelihood estimator (MLE) O = max{x₁, x₂, ..., xₙ} is the value that maximizes the likelihood function L(x₁, x₂, ..., xₙ | 0). Since the likelihood function is zero for any non-zero value of O, the likelihood is maximized when O = 0. In other words, the maximum value of the samples must be zero for the likelihood function to be maximized.

We can understand this intuitively by considering the behavior of the likelihood function. The likelihood function assigns a probability to the observed samples given a particular value of the parameter 0. In this case, the likelihood is zero for any non-zero value of O because the samples are drawn from a uniform distribution on [0, 0]. Thus, the likelihood function is maximized when O = 0, as it assigns the highest probability to the observed samples.

c) The cumulative distribution function (CDF) Fe(x) for the random variable Oₙ = max{X₁, X₂, ..., Xₙ} can be computed as follows:

For 0 ≤ x ≤ 0:

Fê(x) = P(Ôₙ ≤ x) = P(X₁ ≤ x, X₂ ≤ x, ..., Xₙ ≤ x)

Since the uniform distribution on [0, 0] has zero probability density for any value of x, the probability of each individual sample Xᵢ being less than or equal to x is also zero. Therefore, for 0 ≤ x ≤ 0, Fê(x) = P(Ôₙ ≤ x) = 0.

For x > 0:

Fê(x) = P(Oₙ ≤ x) = 1 - P(Oₙ > x) = 1 - P(X₁ > x, X₂ > x, ..., Xₙ > x)

Since the samples X₁, X₂, ..., Xₙ are independent and uniformly distributed on [0, 0], the probability of each individual sample being greater than x is given by:

P(Xᵢ > x) = 1 - P(Xᵢ ≤ x) = 1 - F(x) = 1 - 0 = 1

Therefore, for x > 0, Fê(x) = 1 - P(X₁ > x, X₂ > x, ..., Xₙ > x) = 1 - (1 × 1 × ... × 1) = 1 - 1 = 0.

In summary:

Fe(x) = 0 for 0 ≤ x ≤ 0

Fe(x) = 0 for x > 0

d) The probability density function (PDF) fe(x) of Oₙ can be obtained by differentiating the CDF Fe(x) with respect to x. However, in this case, the CDF Fe(x) is discontinuous and does not have a derivative in the traditional sense. Therefore, the concept of the PDF is not applicable to this random variable.

e) Since the random variable Oₙ takes the value 0 with probability 1, its expected value is:

E[Oₙ] = 0 × P(Oₙ = 0) = 0 × 1 = 0

The estimator o, which is the MLE, is not an unbiased estimator of 0 because its expected value E[o] is not equal to 0. The expected value of o can be calculated as:

E[o] = E[max{X₁, X₂, ..., Xₙ}]

However, since the samples X₁, X₂, ..., Xₙ are drawn from a uniform distribution on [0, 0], the maximum value ô will always be 0. Therefore, E[o] = 0, which means o is a biased estimator of 0.

f) To construct an unbiased estimator of 0, we can modify the MLE ô by adding a constant term. Let's define a new estimator o₂ as follows:

o₂ = o + c

where c is a constant. To ensure that o₂ is an unbiased estimator of 0, we need its expected value E[o₂] to be equal to 0. From part e), we know that E[o] = 0. Therefore, we can set:

E[o₂] = E[o + c] = E[o] + E[c] = 0 + c = c

To make o₂ an unbiased estimator of 0, we set c = -E[ô] = 0. Thus, the modified estimator becomes:

o₂ = o - E[ô] = o - 0 = o

Therefore, the small modification to the MLE studied so far does not change the estimator, and o₂ remains the same as o. It is still biased and does not provide an unbiased estimate of 0.

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The notation |{b : O(b)}| means "the number of books b such that O(b) is true". In this case, the number of such books is exactly 2 for library system.

In the design specification of a library system, O(b) denotes the predicates "Book b is overdue". Therefore, the sentence

"There are exactly two books overdue" in symbolic form can be written as follows:2 = |{b : O(b)}|, where | | denotes the cardinality (number of elements) of the set inside the brackets { }.

Symbolic notation is a way of representing mathematical problems, ideas, or concepts in a compact and concise form. The sentence "There are exactly two books overdue" means that the number of books that are overdue is exactly equal to 2. To express this in symbolic form, we can use set notation and cardinality.The set {b : O(b)} consists of all the books b that are overdue. The notation O(b) represents the predicate "Book b is overdue". The symbol ":" means "such that". Therefore, the set {b : O(b)} consists of all the books b such that the predicate O(b) is true.

The cardinality of a set is the number ofelements in that set. To count the number of books that are overdue, we simply count the number of elements in the set {b : O(b)}. If this number is exactly 2, then the sentence "There are exactly two books overdue" is true.The notation 2 = |{b : O(b)}| means that the number of books that are overdue is exactly 2. The symbol "=" means "is equal to", and the vertical bars | | denote cardinality.

Therefore, the notation |{b : O(b)}| means "the number of books b such that O(b) is true". In this case, the number of such books is exactly 2.

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The demand function for a car is given by p= D(x) = 13.2 - 0.2x dollars. The level of production for which the revenue is maximized is 33 units.

To find the level of production for which the revenue is maximized, we need to determine the quantity that maximizes the revenue function. Revenue is calculated by multiplying the quantity sold (x) by the price (p).

The price is given by the demand function: p = 13.2 - 0.2x dollars.

Revenue (R) is given by: R(x) = p × x.

Substituting the demand function into the revenue function, we have:

R(x) = (13.2 - 0.2x) × x

R(x) = 13.2x - 0.2x²

To find the maximum value of R(x), we need to find the critical points by taking the derivative of R(x) with respect to x and setting it equal to zero:

R'(x) = 13.2 - 0.4x

Setting R'(x) = 0:

13.2 - 0.4x = 0

0.4x = 13.2

x = 13.2 / 0.4

x = 33

So, the level of production for which the revenue is maximized is 33 units.

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The derivative of y with respect to x is (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x.

To find the derivative of y with respect to x, we'll apply the rules of differentiation. Let's break down the given expression and differentiate each term separately:

y = 2 ln(3x) cos^5(x) + 45x^2 + 3 / dy/dx - cot(x)

To find the derivative, let's differentiate each term:

1. Differentiate the term 2 ln(3x):

The derivative of ln(u) is du/u, so the derivative of ln(3x) is 1/(3x) multiplied by the derivative of (3x):

d/dx (2 ln(3x)) = 2 * 1/(3x) * 3 = 2/x

2. Differentiate the term cos^5(x):

The derivative of cos(x) is -sin(x), so we'll apply the chain rule here.

d/dx (cos^5(x)) = 5 cos^4(x) * (-sin(x)) = -5 cos^4(x) sin(x)

3. Differentiate the term 45x^2:

The power rule for differentiation states that the derivative of x^n is n * x^(n-1). Applying this to the term 45x^2:

d/dx (45x^2) = 2 * 45x^(2-1) = 90x

4. Differentiate the term 3:

The derivative of a constant is zero, so the derivative of 3 is 0.

5. Differentiate the term dy/dx - cot(x):

Since dy/dx is given as a separate term, its derivative is 0.

Now let's put all the differentiated terms together:

dy/dx = (2/x) * cos^5(x) + (-5 cos^4(x) sin(x)) + 90x + 0

Simplifying the expression, we have:

dy/dx = (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x

Therefore, the derivative of y with respect to x is (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x.

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To find the absolute extrema of the function f(x) = x√(4x²) on the interval [-1, 2], we need to evaluate the function at the critical points and endpoints of the interval.

1. Critical points:

To find the critical points, we need to determine where the derivative of the function is either zero or undefined.

Let's find the derivative of f(x) first:

f'(x) = (d/dx) (x√(4x²))

Using the product rule and chain rule, we have:

f'(x) = √(4x²) + x * (1/2) * (4x²)^(-1/2) * 8x

Simplifying further, we get:

f'(x) = √(4x²) + 4x²/√(4x²)

f'(x) = 2|x| + 4x²/|2x|

To find the critical points, we set the derivative equal to zero and solve for x:

2|x| + 4x²/|2x| = 0

Case 1: x > 0

2x + 4x²/(2x) = 0

2x + 2x = 0

4x = 0

x = 0

Case 2: x < 0

-2x + 4x²/(-2x) = 0

-2x - 2x = 0

-4x = 0

x = 0

So, the critical point is x = 0.

2. Endpoints of the interval:

We also need to evaluate the function at the endpoints of the interval [-1, 2], which are x = -1 and x = 2.

Now, let's find the values of f(x) at the critical points and endpoints:

f(-1) = (-1)√(4(-1)²) = -1√(4) = -1 * 2 = -2

f(0) = (0)√(4(0)²) = 0

f(2) = (2)√(4(2)²) = 2 * 4 = 8

Therefore, the function f(x) = x√(4x²) has absolute extrema on the interval [-1, 2].

The minimum value is -2 and it occurs at x = -1.

The maximum value is 8 and it occurs at x = 2.

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The stone's acceleration is constant at -10 m/s^2. It will reach its maximum height after 4 seconds, at which point it is 240 meters above the ground. The stone's velocity upon impact is -10 m/s.

The equation of motion of the stone is s(t) = -5t² + 30t+200, where s is the directed distance from the ground in meters and t is in seconds. The acceleration of the stone is the derivative of the velocity, which is the derivative of the position.

The derivative of the position is -10t + 30, so the acceleration is -10. The stone will reach its maximum height when the velocity is 0. The velocity is 0 when t = 4, so the stone will reach its maximum height after 4 seconds.

The height of the building is the position of the stone when t = 0, which is 200 meters. The maximum height the stone will reach is the position of the stone when t = 4, which is 240 meters. The velocity of the stone upon impact is the velocity of the stone when t = 8, which is -10 m/s.

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(i) The bipartite graph for A₁ = {1,2,3}, A₂ = {2, 3, 4}, A₃ = {1} has a system of distinct representatives.

(ii) The bipartite graph for A₁ = {1, 2, 3, 4, 5}, A₂ = {1,3}, A₃ = {1,3}, A₄ = {1, 2, 3, 4, 5}, A₅ = {1,3} does not have a system of distinct representatives.

To construct the associated bipartite graph for each family of sets, we can represent the sets as vertices and draw edges between them based on their intersection.

(i) Family of sets: A₁ = {1,2,3}, A₂ = {2, 3, 4}, A₃ = {1}

The bipartite graph for this family of sets would have two sets of vertices, one representing the sets A₁, A₂, A₃ and the other representing the elements 1, 2, 3, 4.

The vertices for sets A₁, A₂, A₃ would be connected to the vertices representing the elements that are contained in those sets. The edges would be drawn based on the intersection of the sets.

To find a system of distinct representatives, we need to find a matching where each vertex on the left side (sets) is connected to a unique vertex on the right side (elements). In this case, a system of distinct representatives exists since every set has at least one unique element.

(ii) Family of sets: A₁ = {1, 2, 3, 4, 5}, A₂ = {1,3}, A₃ = {1,3}, A₄ = {1, 2, 3, 4, 5}, A₅ = {1,3}

The bipartite graph for this family of sets would have two sets of vertices, one representing the sets A₁, A₂, A₃, A₄, A₅ and the other representing the elements 1, 2, 3, 4, 5.

The vertices for sets A₁, A₂, A₃, A₄, A₅ would be connected to the vertices representing the elements that are contained in those sets.

To find a system of distinct representatives, we need to find a matching where each vertex on the left side (sets) is connected to a unique vertex on the right side (elements). In this case, a system of distinct representatives does not exist because element 1 is shared by all the sets A₁, A₂, A₃, A₄, A₅, violating the requirement of distinctness.

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The general solution of the differential equation is [tex]x(t) = Ae^{r_1t} + Be^{r_2t}[/tex]

The given differential equation is a second-order linear homogeneous ordinary differential equation. Let's solve it.

The differential equation is:

D²x/dt² - KmX = 0

To solve this equation, we can assume a solution of the form:

[tex]x(t) = e^{rt}[/tex]

Taking the second derivative of x(t) with respect to t:

[tex]d^x/dt^2 = r^2e^{rt[/tex]

Substituting the assumed solution into the differential equation, we have:

[tex]r^2e^{rt} - Km(e^{rt}) = 0[/tex]

Factoring out [tex]e^{rt[/tex], we get:

[tex]e^{rt}(r^2 - Km) = 0[/tex]

For this equation to hold for all t, the exponential term [tex]e^{rt}[/tex] must be nonzero.

Therefore, we have r² - Km = 0

Solving for r, we find two possible values:

r₁ = √(Km)

r₂ = -√(Km)

Hence, the general solution of the differential equation is a linear combination of these two solutions [tex]x(t) = Ae^{r_1t} + Be^{r_2t}[/tex]

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The equation function of cosine with an amplitude of 4, a period of 2, and a point at (0,2) is y = 4cos(2πx) + 2.

The general form of a cosine function is y = A cos(Bx - C) + D, where A represents the amplitude, B is related to the period, C indicates any phase shift, and D represents a vertical shift.

In this case, the given amplitude is 4, which means the graph will oscillate between -4 and 4 units from its centerline. The period is 2, which indicates that the function completes one full cycle over a horizontal distance of 2 units.

To incorporate the given point (0,2), we know that when x = 0, the corresponding y-value should be 2. Since the cosine function is at its maximum at x = 0, the vertical shift D is 2 units above the centerline.

Using these values, the equation function becomes y = 4cos(2πx) + 2, where 4 represents the amplitude, 2π/2 simplifies to π in the argument of cosine, and 2 is the vertical shift. This equation satisfies the given conditions of the cosine function.

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According to the given information, the equation of the tangent line to f(x) at x = 1 is y = 4 + 5(x - 1). The value of h'(1) is 1.

In order to find h'(1), we need to differentiate the function h(x) = √f(x) with respect to x and then evaluate it at x = 1. Since h(x) is the square root of f(x), we can rewrite it as h(x) = f(x)^(1/2).

Applying the chain rule, the derivative of h(x) with respect to x can be calculated as h'(x) = (1/2) * f(x)^(-1/2) * f'(x).

Since we are interested in finding h'(1), we substitute x = 1 into the derivative expression. Therefore, h'(1) = (1/2) * f(1)^(-1/2) * f'(1).

According to the given information, the equation of the tangent line to f(x) at x = 1 is y = 4 + 5(x - 1). From this equation, we can deduce that f(1) = 4.

Substituting f(1) = 4 into the derivative expression, we have h'(1) = (1/2) * 4^(-1/2) * f'(1). Simplifying further, h'(1) = (1/2) * (1/2) * f'(1) = 1 * f'(1) = f'(1).

Therefore, h'(1) is equal to f'(1), which is given as 1.

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To find the inverse Laplace transform of the given functions, we need to decompose them into partial fractions and then use known Laplace transform formulas. Let's go through each function step by step.

F(s) = (4s + 32)/(s^2 - 16)

First, we need to factor the denominator:

s^2 - 16 = (s + 4)(s - 4)

We can express F(s) as:

F(s) = A/(s + 4) + B/(s - 4)

To find the values of A and B, we multiply both sides by the denominator:

4s + 32 = A(s - 4) + B(s + 4)

Expanding and equating coefficients, we have:

4s + 32 = (A + B)s + (-4A + 4B)

Equating the coefficients of s, we get:

4 = A + B

Equating the constant terms, we get:

32 = -4A + 4B

Solving this system of equations, we find:

A = 6

B = -2

Now, substituting these values back into F(s), we have:

F(s) = 6/(s + 4) - 2/(s - 4)

Taking the inverse Laplace transform, we can find f(t):

f(t) = 6e^(-4t) - 2e^(4t)

F(s) = (2s + 1)/(s^2 - 16)

Again, we need to factor the denominator:

s^2 - 16 = (s + 4)(s - 4)

We can express F(s) as:

F(s) = A/(s + 4) + B/(s - 4)

To find the values of A and B, we multiply both sides by the denominator:

2s + 1 = A(s - 4) + B(s + 4)

Expanding and equating coefficients, we have:

2s + 1 = (A + B)s + (-4A + 4B)

Equating the coefficients of s, we get:

2 = A + B

Equating the constant terms, we get:

1 = -4A + 4B

Solving this system of equations, we find:

A = -1/4

B = 9/4

Now, substituting these values back into F(s), we have:

F(s) = -1/(4(s + 4)) + 9/(4(s - 4))

Taking the inverse Laplace transform, we can find f(t):

f(t) = (-1/4)e^(-4t) + (9/4)e^(4t)

F(s) = (s + a)/(s^2 - s - 2)

We can express F(s) as:

F(s) = A/(s - 1) + B/(s + 2)

To find the values of A and B, we multiply both sides by the denominator:

s + a = A(s + 2) + B(s - 1)

Expanding and equating coefficients, we have:

s + a = (A + B)s + (2A - B)

Equating the coefficients of s, we get:

1 = A + B

Equating the constant terms, we get:

a = 2A - B

Solving this system of equations, we find:

A = (a + 1)/3

B = (2 - a)/3

Now, substituting these values back into F(s), we have:

F(s) = (a + 1)/(3(s - 1)) + (2 - a)/(3(s + 2))

Taking the inverse Laplace transform, we can find f(t):

f(t) = [(a + 1)/3]e^t + [(2 - a)/3]e^(-2t)

F(s) = s/(s^2 + 10s + 2)

We can express F(s) as:

F(s) = A/(s + a) + B/(s + b)

To find the values of A and B, we multiply both sides by the denominator:

s = A(s + b) + B(s + a)

Expanding and equating coefficients, we have:

s = (A + B)s + (aA + bB)

Equating the coefficients of s, we get:

1 = A + B

Equating the constant terms, we get:

0 = aA + bB

Solving this system of equations, we find:

A = -b/(a - b)

B = a/(a - b)

Now, substituting these values back into F(s), we have:

F(s) = -b/(a - b)/(s + a) + a/(a - b)/(s + b)

Taking the inverse Laplace transform, we can find f(t):

f(t) = [-b/(a - b)]e^(-at) + [a/(a - b)]e^(-bt)

These are the inverse Laplace transforms of the given functions.

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If N is a normal subgroup of a finite group G, and the index of subgroup H in G is finite, such that H is a subgroup of N, and the indices [G:H] and [N] are relatively prime, then N is contained in H.

Let's consider the given conditions. N is a normal subgroup of the finite group G, and H is a subgroup of N with finite index [G:H]. Additionally, the indices [G:H] and [N] are relatively prime, which means they have no common factors other than 1.

Since N is a normal subgroup of G, it implies that for any element n in N and any element g in G, the product gng^(-1) is also in N. Since H is a subgroup of N, it follows that for any element h in H and any element n in N, the product hnh^(-1) is also in N.

Now, consider the left cosets of H in G, denoted as gH, where g belongs to G. Since [G:H] is finite, there are only finitely many distinct left cosets. Let's denote the set of left cosets of H in G as {g₁H, g₂H, ..., gₙH}.

Since N is normal, for each left coset gᵢH, the product ngn^(-1) is also in the left coset gᵢH for any element n in N. Therefore, each left coset gᵢH is closed under conjugation by elements of N.

Since the indices [G:H] and [N] are relatively prime, the order of each left coset is relatively prime to the order of N. By applying Lagrange's theorem, the order of N must divide the order of each left coset gᵢH. However, since the order of each left coset is relatively prime to the order of N, it implies that the intersection of N with each left coset is trivial, i.e., N intersects each left coset only at the identity element.

Since N intersects each left coset gᵢH only at the identity element, it means that N is contained in H. Therefore, under the given conditions, N is a subgroup of H.

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To find the scalar equation of the line in two dimensions, we need to determine the slope-intercept form of the equation, which is given by:

y = mx + b

where "m" represents the slope of the line, and "b" represents the y-intercept.

Given the point (-3, 4) and the direction vector (4, -1), we can find the slope "m" using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates of the given point and the direction vector, we have:

m = (-1 - 4) / (4 - (-3))

m = -5 / 7

Now, we have the slope "m" as -5/7. To find the y-intercept "b," we can substitute the coordinates of the given point (-3, 4) into the slope-intercept form equation:

4 = (-5/7)(-3) + b

Simplifying:

4 = 15/7 + b

4 - 15/7 = b

(28 - 15) / 7 = b

13/7 = b

Thus, the y-intercept "b" is 13/7.

Now, we can write the scalar equation of the line in slope-intercept form:

y = (-5/7)x + 13/7

This is the scalar equation of the line passing through the point (-3, 4) and having a direction vector (4, -1) in two dimensions.

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The given integral ∫∫∫ (2ct √(3u^2 + s^2 + u^2)) ds du dt can be evaluated by breaking it down into separate integrals with respect to each variable. The resulting integral involves trigonometric and square root functions, which can be simplified to find the solution.

To evaluate the given integral, we will first integrate with respect to ds, then du, and finally dt. The integration with respect to ds yields s evaluated from 0 to t, the integration with respect to du yields u evaluated from 0 to √3, and the integration with respect to dt yields t evaluated from 0 to 1.

Integrating with respect to ds, we get ∫ (2ct √(3u^2 + s^2 + u^2)) ds = (2ct/2) ∫ √(3u^2 + s^2 + u^2) ds = ct [s√(3u^2 + s^2 + u^2)] evaluated from 0 to t.

Next, integrating with respect to du, we have ∫ ct [s√(3u^2 + s^2 + u^2)] du = cts ∫ √(3u^2 + s^2 + u^2) du = cts [u√(3u^2 + s^2 + u^2)] evaluated from 0 to √3.

Finally, integrating with respect to dt, we obtain ∫ cts [u√(3u^2 + s^2 + u^2)] dt = ct^2s [u√(3u^2 + s^2 + u^2)] evaluated from 0 to 1.

By substituting the limits of integration into the above expression, we can calculate the definite integral and obtain the final result. Please note that the specific values of c and t may affect the final numerical solution.

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a. The 8-bit binary representation for the integer 111 is 01101111.

To convert the decimal number 111 to binary, we divide it by 2 repeatedly until the quotient becomes 0, while keeping track of the remainders. The remainders will give us the binary digits in reverse order.

111 ÷ 2 = 55 (remainder 1)

55 ÷ 2 = 27 (remainder 1)

27 ÷ 2 = 13 (remainder 1)

13 ÷ 2 = 6 (remainder 0)

6 ÷ 2 = 3 (remainder 0)

3 ÷ 2 = 1 (remainder 1)

1 ÷ 2 = 0 (remainder 1)

Reading the remainders from bottom to top, we get 01101111 as the 8-bit binary representation for the integer 111.

b. The 8-bit binary representation for the integer 98 is 01100010.

Explanation:

Following the same process as above, we divide 98 by 2 repeatedly:

98 ÷ 2 = 49 (remainder 0)

49 ÷ 2 = 24 (remainder 1)

24 ÷ 2 = 12 (remainder 0)

12 ÷ 2 = 6 (remainder 0)

6 ÷ 2 = 3 (remainder 0)

3 ÷ 2 = 1 (remainder 1)

1 ÷ 2 = 0 (remainder 1)

Reading the remainders from bottom to top, we obtain 01100010 as the 8-bit binary representation for the integer 98.

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In summary, for the given parametric equations, the point (x, y) on the curve where the tangent line is horizontal is (4, 16). This is found by differentiating y(t), setting the derivative equal to zero, and solving for t. Substituting the value of t back into the parametric equations gives the coordinates (x, y) of the point on the curve.

The point (x, y) on the curve where the tangent line is horizontal can be found by determining the value of t that makes the derivative of y(t) with respect to t equal to zero. For the parametric equations 4(sin(t) + cos(t)) and y = 28 — 12 cos²(t) — 24 sin(t), we can differentiate y(t) with respect to t, set the derivative equal to zero, and solve for t. Once t is known, the corresponding values of x and y can be found.

To elaborate, we differentiate y(t) = 28 — 12 cos²(t) — 24 sin(t) with respect to t, yielding dy/dt = 48 sin(t) - 24 cos(t). Setting dy/dt equal to zero, we have 48 sin(t) - 24 cos(t) = 0. Simplifying the equation, we obtain 2 sin(t) - cos(t) = 0. Solving for t, we find t = π/6. Substituting t = π/6 into the parametric equations, we get x = 4 and y = 16. Therefore, the point (x, y) where the tangent line is horizontal is (4, 16).

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a) Definition of Limit: Let f(x) be defined on an open interval containing c, except possibly at c itself.

We say that the limit of f(x) as x approaches c is L and write: 

[tex]limx→cf(x)=L[/tex]

if for every number ε>0 there exists a corresponding number δ>0 such that |f(x)-L|<ε whenever 0<|x-c|<δ.

Let's prove that f(x) = 2x³ - 1 is continuous at the point x = 2; that is, [tex]lim-2 2x³ - 1[/tex]= 15.

Let [tex]limx→2(2x³-1)[/tex]= L than for ε > 0, there exists δ > 0 such that0 < |x - 2| < δ implies

|(2x³ - 1) - 15| < ε

|2x³ - 16| < ε

|2(x³ - 8)| < ε

|x - 2||x² + 2x + 4| < ε

(|x - 2|)(x² + 2x + 4) < ε

It can be proved that δ can be made equal to the minimum of 1 and ε/13.

Then for

0 < |x - 2| < δ

|x² + 2x + 4| < 13

|x - 2| < ε

Thus, [tex]limx→2(2x³-1)[/tex]= 15.

b) (i) Definition of Contractions: Let f: [a, b] → [a, b] be a function.

We say f is a contraction if there exists a constant 0 ≤ k < 1 such that for any x, y ∈ [a, b],

|f(x) - f(y)| ≤ k |x - y| and |k|< 1.

(ii) We need to prove that h(x) = cos(sin x) is continuous at every point x = x0; that is, [tex]limx→x0[/tex] | cos(sin x)| = | cos(sin(x0)).

First, we prove that cos(x) is a contraction function on the interval [0, π].

Let f(x) = cos(x) be defined on the interval [0, π].

Since cos(x) is continuous and differentiable on the interval, its derivative -sin(x) is continuous on the interval.

Using the Mean Value Theorem, for all x, y ∈ [0, π], we have cos (x) - cos(y) = -sin(c) (x - y),

where c is between x and y.

Then,

|cos(x) - cos(y)| = |sin(c)|

|x - y| ≤ 1 |x - y|.

Therefore, cos(x) is a contraction on the interval [0, π].

Now, we need to show that h(x) = cos(sin x) is also a contraction function.

Since sin x takes values between -1 and 1, we have -1 ≤ sin(x) ≤ 1.

On the interval [-1, 1], cos(x) is a contraction, with a contraction constant of k = 1.

Therefore, h(x) = cos(sin x) is also a contraction function on the interval [0, π].

Hence, by the Contraction Mapping Theorem, h(x) = cos(sin x) is continuous at every point x = x0; that is,

[tex]limx→x0 | cos(sin x)| = | cos(sin(x0)).[/tex]

(c) (i) Given a common deviation bound of 0.00025 for both x - 2 and y - 2, we need to prove that x + y is accurate to +2 by 3 decimal places.

Let x - 2 = δ and y - 2 = ε.

Then,

x + y - 4 = δ + ε.

So,

|x + y - 4| ≤ |δ| + |ε|

≤ 0.00025 + 0.00025

= 0.0005.

Therefore, x + y is accurate to +2 by 3 decimal places.(ii) The mapping diagram is shown below:

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The expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12) is f(x) = (7/2)x - 3/2.

To find an expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12), we can use the point-slope form of a linear equation.

The point-slope form is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line.

Given the points (1, -2) and (5, 12), we can calculate the slope:

m = (y₂ - y₁) / (x₂ - x₁) = (12 - (-2)) / (5 - 1) = 14 / 4 = 7/2

Now, let's choose one of the points, say (1, -2), and plug it into the point-slope form:

y - (-2) = (7/2)(x - 1)

Simplifying the equation, we get:

y + 2 = (7/2)(x - 1)

Next, we can rewrite the equation in slope-intercept form (y = mx + b):

y = (7/2)x - 7/2 + 2

y = (7/2)x - 7/2 + 4/2

y = (7/2)x - 3/2

Therefore, the expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12) is f(x) = (7/2)x - 3/2.

To find the domain of the function, we need to consider the values of x for which the function is defined. Since f(x) is a linear function, it is defined for all real numbers. Therefore, the domain of the function f(x) = (7/2)x - 3/2 is (-∞, +∞) or (-∞, ∞) in interval notation, indicating that it is defined for all values of x.

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