A natural cubic spline g on [0, 2] is defined by (91(1), g(x) = 0≤x≤1 (92(1), 1≤ ≤2. where g₁(z) = 1 + 2x-r³ and ga(z) = a + b(x-1) + c(x − 1)² + d(x - 1)³. (a) What conditions should gi(z) and 92 (7) satisfy at z = 1? (b) What conditions must 9₁(x) satisfy at x = 0? (c) What conditions must 92(x) satisfy at x = = 2? (d) Apply the conditions in (a) to (c) to find a, b, c and d.

The height of the object will be 180 feet, we need to solve the equation 16t² + 175t + 12 = 180.The units for the time will be in seconds since the given equation represents time in seconds. The units for height will be in feet since the equation represents height in feet.

To find when the height of the object will be 180 feet, we set the equation 16t² + 175t + 12 equal to 180 and solve for t. This gives us:

16t² + 175t + 12 = 180.

Rearranging the equation, we have:

16t² + 175t - 168 = 0.

We can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of t when the height is 180 feet.

To find when the object reaches the ground, we set the equation 16t² + 175t + 12 equal to 0. This represents the height of the object being 0, which occurs when the object hits the ground. Solving this equation will give us the time at which the object reaches the ground.

The units for the time will be in seconds since the given equation represents time in seconds. The units for height will be in feet since the equation represents height in feet.

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Therefore, you would need to mix approximately 167 gallons of the 90% antifreeze solution with 100 gallons of the 25% antifreeze solution to obtain a mixture that is 80% antifreeze.

Using the six-step method, we can solve the problem as follows:

Step 1: Assign variables to the unknown quantities. Let's call the number of gallons of the 90% antifreeze solution needed as "x."

Step 2: Translate the problem into equations. We are given that the strength (concentration) of the 90% antifreeze solution is 90% and that of the 25% antifreeze solution is 25%. We need to find the number of gallons of the 90% antifreeze solution required to obtain a mixture with a strength of 80%.

Step 3: Write the equation for the total amount of antifreeze in the mixture. The amount of antifreeze in the 90% solution is 90% of x gallons, and the amount of antifreeze in the 25% solution is 25% of 100 gallons. The total amount of antifreeze in the mixture is the sum of these two amounts.

0.90x + 0.25(100) = 0.80(x + 100)

Step 4: Solve the equation. Distribute the terms and combine like terms:

0.90x + 25 = 0.80x + 80

Step 5: Solve for x. Subtract 0.80x from both sides and subtract 25 from both sides:

0.10x = 55

x = 55 / 0.10

x = 550

Step 6: Round the answer. Since we are dealing with gallons, round the answer to the nearest whole number:

x ≈ 550

Therefore, you would need to mix approximately 167 gallons of the 90% antifreeze solution with 100 gallons of the 25% antifreeze solution to obtain a mixture that is 80% antifreeze.

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The (LCM) least common multiple of these two expressions is 56yu.

To find the least common multiple (LCM) of two expressions 14yu and 8x,

we need to find the prime factorization of each expression.

The prime factorization of 14yu is: 2 × 7 × y × u

The prime factorization of 8x is: 2³ × x

LCM is the product of all unique prime factors of each expression raised to their highest powers.

So, LCM of 14yu and 8x = 2³ × 7 × y × u = 56yu

The LCM of the given expressions is 56yu.

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Answer:

$18

Step-by-step explanation:

If she starts with $50 and has $32 left when she's done then. 50-32= 18

So she spent $18 on clothing.

The approximate value of sin(2x³) - x², using its Maclaurin expansion and x = 1/3, is approximately -0.0800.

The Maclaurin series expansion of sin(x) is given by the equation sin(x) = x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + ... . To find the value of sin(2x³) - x², we substitute 2x³ in place of x in the Maclaurin series expansion of sin(x). Thus, we have sin(2x³) = (2x³) - ((2x³)³/3!) + ((2x³)⁵/5!) - ((2x³)⁷/7!) + ... .

Now, we substitute x = 1/3 into the expression. We have sin(2(1/3)³) = (2(1/3)³) - ((2(1/3)³)³/3!) + ((2(1/3)³)⁵/5!) - ((2(1/3)³)⁷/7!) + ... .

Simplifying this expression, we get sin(2(1/3)³) = (2/27) - ((2/27)³/3!) + ((2/27)⁵/5!) - ((2/27)⁷/7!) + ... .

To approximate the value within 0.0001, we can stop the calculation after a few terms. Evaluating the expression, we find that sin(2(1/3)³) ≈ 0.0741 - 0.0001 - 0.0043 + 0.0002 = -0.0800.

Therefore, the approximate value of sin(2x³) - x², using its Maclaurin expansion and x = 1/3, is approximately -0.0800, satisfying the given accuracy requirement.

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The graph of the set of real numbers {-4 < x ≤ 2} is drawn.

Here is the graph of the set of real numbers {-4 < x ≤ 2}.

The closed dot at -4 represents the boundary point where x is greater than -4, and the closed dot at 2 represents the boundary point where x is less than or equal to 2. The line segment between -4 and 2 indicates the set of numbers between -4 and 2, including -4 and excluding 2.

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10 1 2 {=}} 0 2 {}} -8 -5 -8 is not a linearly independent set.

Let us first arrange the given vectors horizontally:[tex]$$\begin{bmatrix}10 & 1 & 2 & 0 & 2 & -8 & -5 & -8\end{bmatrix}$$[/tex]

Now let us row reduce the matrix:[tex]$$\begin{bmatrix}10 & 1 & 2 & 0 & 2 & -8 & -5 & -8 \\ 0 & -9/5 & -6/5 & 0 & -6/5 & 12/5 & 7/5 & 4/5 \\ 0 & 0 & 2/3 & 0 & 2/3 & -4/3 & -1/3 & -1/3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}$$[/tex]

Since there are no pivots in the last row of the row-reduced matrix, we can conclude that the set of vectors is linearly dependent.

This is because the corresponding homogeneous system, whose coefficient matrix is the above row-reduced matrix, has infinitely many solutions.

Hence, 10 1 2 {=}} 0 2 {}} -8 -5 -8 is not a linearly independent set.

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Based on the normal distribution table, the probability corresponding to the z score is 0.8577

Since the heights of men are normally distributed, we will apply the formula for normal distribution which is expressed as

z = (x - u)/s

Where x is the height of men

u = mean height

s = standard deviation

From the information we have;

u = 1.75 cm

s = 7.1 cm

We need to find the probability that the mean height of 1.83 cm is less than 7.1 inches.

Thus It is expressed as

P(x < 7.1 )

For x = 7.1

z = (7.1 - 1.75 )/1.83 = 1.07

Based on the normal distribution table, the probability corresponding to the z score is 0.8577

P(x < 7.1 ) = 0.8577

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To determine whether the sets (A ∩ B)' and B are equal, we can use Venn diagrams. The Venn diagram representations of the two sets will help us visualize their elements and determine if they have the same elements or not.

The set (A ∩ B)' represents the complement of the intersection of sets A and B, while B represents set B itself. By using Venn diagrams, we can compare the two sets and see if they have the same elements or not.

If the two sets are equal, it means that they have the same elements. In terms of Venn diagrams, this would mean that the regions representing (A ∩ B)' and B would overlap completely, indicating that every element in one set is also in the other.

If the two sets are not equal, it means that they have different elements. In terms of Venn diagrams, this would mean that the regions representing (A ∩ B)' and B do not overlap completely, indicating that there are elements in one set that are not in the other.

To determine the equality of the sets (A ∩ B)' and B, we can draw the Venn diagrams for A and B, shade the region representing (A ∩ B)', and compare it to the region representing B. If the shaded region and the region representing B overlap completely, then the two sets are equal. If there is any part of the region representing B that is not covered by the shaded region, then the two sets are not equal.

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The critical points of the function f(x, y) = x * e^(-r^2) - y^3 are (3, 3).

To find the critical points of a function, we need to find the values of x and y where the partial derivatives with respect to x and y are equal to zero or do not exist. In this case, we have the function f(x, y) = x * e^(-r^2) - y^3, where r is the distance from the origin given by r^2 = x^2 + y^2.

Taking the partial derivatives, we have:

∂f/∂x = e^(-r^2) - 2x^2 * e^(-r^2)

∂f/∂y = -3y^2

Setting these partial derivatives equal to zero and solving the equations, we find that x = 3 and y = 3. Therefore, the critical point is (3, 3).

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The equation of the tangent line to the curve y = ln(xe²³) at (1, 1) is y = 24x - 23. The slope is determined by evaluating the derivative at the given point.

The equation of the tangent line to the curve y = ln(xe²³) at the point (1, 1) can be found by taking the derivative of the equation and substituting the x-coordinate of the given point.

First, we find the derivative of y = ln(xe²³) using the chain rule. The derivative is given by dy/dx = 1/x + 23.

Next, we substitute x = 1 into the derivative to find the slope of the tangent line at (1, 1). Thus, the slope is 1/1 + 23 = 24.

Finally, using the point-slope form of a line, we can write the equation of the tangent line as y - 1 = 24(x - 1), which simplifies to y = 24x - 23.

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The equation of the blue graph, g(x) is (d) g(x) = 1/5x²

From the question, we have the following parameters that can be used in our computation:

The functions f(x) and g(x)

In the graph, we can see that

However, the blue graph is 5 times wider than f(x)

This means that

g(x) = 1/5 * f(x)

Recall that

f(x) = x²

This means that

g(x) = 1/5x²

This means that the equation of the blue graph is g(x) = 1/5x²

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the maximum value between 5.39 and 0 is 5.39, therefore the maximum value of the Payoff will be 5.39. Thus, the payoff can be rewritten as:

Payoff = max(So.25 - 20e 0.5, 5,0).

The below is the solution to the given problem.

As per the problem, U(t) = So * e^rt

From this formula, the value of U(0.75) can be calculated as follows:

U(0.75) = So * e^(0.06 × 0.75)U(0.75) = So * e^0.045U(0.75)

= 20 * e^0.045U(0.75)

= 21.1592

Hence, we have U(0.75) = 21.1592.

Now, we can easily determine U(0.25) as follows:

U(0.25) = Ster (0.75 - 0.25)U(0.25)

= Ster 0.5U(0.25) = 2.23607

Now, we can find the value of the option at the maturity of 9 months as follows:

Payoff = max(U(0.25) - 20, 0)

= max(2.23607 - 20, 0) = 0

Now, we can rewrite the formula for the payoff as:

Payoff = max(So × e^0.5 - 20, 0)

= max(20 × e^0.5 - 20, 0)

= 5.39

Since the maximum value between 5.39 and 0 is 5.39, therefore the maximum value of the Payoff will be 5.39. Thus, the payoff can be rewritten as:

Payoff = max(So.25 - 20e 0.5, 5,0).

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Given that the function f(x) has the following values known:

I-102f(x) 9 4 12

(a) To find the Lagrange polynomial through the 3 points.

The Lagrange interpolation is given as:

P(x) = f(x0) L0(x) + f(x1) L1(x) + f(x2) L2(x)

where L0(x) = (x - x1) (x - x2) / (x0 - x1) (x0 - x2)

L1(x) = (x - x0) (x - x2) / (x1 - x0) (x1 - x2) and

L2(x) = (x - x0) (x - x1) / (x2 - x0) (x2 - x1)

Substituting the given values in the above equations, we get:

L0(x) = (x - 4) (x - 12) / (9 - 4) (9 - 12)

= (x2 - 16x + 48) / 15L1(x)

= (x - 9) (x - 12) / (4 - 9) (4 - 12)

= -(x2 - 21x + 108) / 20L2(x)

= (x - 9) (x - 4) / (12 - 9) (12 - 4)

= (x2 - 13x + 36) / 15

Thus, the Lagrange polynomial through the 3 points is:

P(x) = 9[(x2 - 16x + 48) / 15] - 102[(x2 - 21x + 108) / 20] + 4[(x2 - 13x + 36) / 15]

= (4/15)x2 - (41/15)x + 60(b)

To find the Newton's interpolating polynomial through the 3 points.

Newton's Interpolation formula is given as:

f(x) = f(x0) + (x - x0) f[x0, x1] + (x - x0) (x - x1) f[x0, x1, x2]

where f[xi, xi+1, ...] is the divided difference of order i.

We find the divided difference of order 1:

f[x0, x1] = (f(x1) - f(x0)) / (x1 - x0)

= (4 - 9) / (4 - 9)

= -1

f[x1, x2] = (f(x2) - f(x1)) / (x2 - x1)

= (12 - 4) / (12 - 4)

= 8

Thus,

f(x) = 9 - (x - 4) + 8(x - 4)(x - 9)

= - (x2 - 13x + 36) / 15 + 9

(c) To approximate f(1) using the Lagrange polynomial or the Newton's interpolating polynomial.

f(1) using the Lagrange polynomial:

P(1) = (4/15)(1)2 - (41/15)(1) + 60

= 45.6f(1)

using the Newton's interpolating polynomial:

f(1) = -[tex](1^2 - 13(1) + 36) / 15 + 9[/tex]

= 19/15

(d) To approximate f'(1) using the Newton's interpolating polynomial.

f'(x) = - (2x - 13) / 15

Thus,

f'(1) = - (2(1) - 13) / 15

= - 11/15

(e) To approximate the integral of f(z) dz using the Newton's interpolating polynomial.

The integral of f(z) dz can be obtained as follows:

∫f(z) dz = F(x) + C

where F(x) is the antiderivative of f(x).

Thus,

∫f(z) dz = [tex]∫[- (z^2 - 13z + 36) / 15 + 9] dz[/tex]

= [tex](-z^3 / 45 + (13/30) z^2 - 12z) / 15 + C[/tex]

Substituting the limits of integration from 0 to z, we get:

∫f(z) dz =[tex][(-z^3 / 45 + (13/30) z^2 - 12z) / 15][/tex]

z=[tex]1 - [(-1^3 / 45 + (13/30) 1^2 - 12(1)) / 15][/tex]

= 19/3

Therefore, the value of the integral of f(z) dz is approximately 19/3.

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Area:

Approximately 211.24 square feet

Explanation:

Im going to assume you are asking for the surface area of a sphere with a diameter of 8.2ft. The equation to find this is: [tex]A = 4\pi r^2[/tex]

Firstly, we need to convert the diameter to the radius. The diameter is always twice the length of the radius, so the radius must be 4.1ft

Plug this value in:

[tex]A = 4\pi (4.1)^2\\A = 4\pi(16.81)\\A = 211.24[/tex]

Using the given graph figure, we can say that:

Equation for position a is: y = x² + 3

Equation for position B is: y = x² - 5

To shift a function left by b units we will add inside the domain of the function's argument to get: f(x + b) shifts f(x) b units to the left.

Shifting to the right works the same way, f(x - b) shifts f(x) by b units to the right.

To translate the function up and down, you simply add or subtract numbers from the whole function.

If you add a positive number (or subtract a negative number), you translate the function up.

If you subtract a positive number (or add a negative number), you translate the function down.

Looking at the given graph, we can say that:

Equation for position a is: y = x² + 3

Equation for position B is: y = x² - 5

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the equation that models the baby's weight is:
y = 1.5x + 7To model the baby's weight as it grows, we can use the equation y = mx + b, where y represents the weight in pounds and x represents the age of the baby in months.

Given that the baby weighs 13 pounds at 4 months and is expected to gain 1.5 pounds each month, we can determine the equation:

y = 1.5x + b

To find the value of b, we substitute the given information that the baby weighs 13 pounds at 4 months:

13 = 1.5(4) + b

Simplifying the equation:

13 = 6 + b
b = 13 - 6
b = 7

, the equation that models the baby's weight is:
y = 1.5x + 7

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The line equation which models the data plotted on the graph is y = -16.67X + 1100

The equation for the line of best fit is expressed by the relation :

b = slope ; c = intercept

The slope , b = (change in Y/change in X)

Using the points : (28, 850) , (40, 650)

slope = (850 - 650) / (28 - 40)

slope = -16.67

The intercept is the point where the best fit line crosses the y-axis

Hence, intercept is 1100

Line of best fit equation :

Therefore , the equation of the line is y = -16.67X + 1100

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The derivative of the function f(x) = (x - 4)(4x + 4) can be found using the Product Rule. The correct option is OC i.e., the derivative is 8x - 12.

To find the derivative of a product of two functions, we can use the Product Rule, which states that the derivative of the product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).

Applying the Product Rule to the given function f(x) = (x - 4)(4x + 4), we differentiate the first function (x - 4) and keep the second function (4x + 4) unchanged, then add the product of the first function and the derivative of the second function.

a. Using the Product Rule, the derivative of f(x) is:

f'(x) = (x - 4)(4) + (1)(4x + 4)

Simplifying this expression, we have:

f'(x) = 4x - 16 + 4x + 4

Combining like terms, we get:

f'(x) = 8x - 12

Therefore, the correct answer is OC. The derivative is 8x - 12.

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The statement (a U b) = a U b is true for all symbols a and b, regardless of the set to which they belong, which implies that the statement is true for every alphabet and every symbol that belongs to it. Consequently, the given statement is true for each alphabet and each symbol a that belongs to it.

Yes, the given statement is true for each alphabet and each symbol a that belongs to it.

Let's see why this is true:

A set is a collection of unique elements that is denoted by capital letters such as A, B, C, etc. Elements are enclosed in braces, e.g., {1, 2, 3}.

The union of two sets is a set of elements that belong to either of the two sets. A∪B reads as A union B. A union B is the combination of all the elements from sets A and B.

(A∪B) means the union of sets A and B. It consists of all the elements in set A and all the elements in set B. The elements of set A and set B are combined without any repetition.

A set is said to be a subset of another set if all its elements are present in the other set. It is denoted by ⊆. If A is a subset of B, then B ⊇ A.

The union of a set A with a set B is denoted by A U B, and it contains all elements that are in A or B, or in both. Let a be a symbol belonging to the alphabet set. The given statement is (a U b) = a U b which means the set containing a and b is the same as the set containing b and a, where a and b are elements of an arbitrary set.

Suppose a is an element of set A and b is an element of set B. Then (a U b) means the union of A and B and it contains all the elements of A and all the elements of B. The order of the elements in a set does not matter, so (a U b) = (b U a) = A U B.

The statement (a U b) = a U b is true for all symbols a and b, regardless of the set to which they belong, which implies that the statement is true for every alphabet and every symbol that belongs to it. Consequently, the given statement is true for each alphabet and each symbol a that belongs to it.

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The chosen differential equation is: dv/dt = A / (v + B) - (v - 20) / B

The differential equation that models the velocity of the Prius in this scenario can be chosen as:

dv/dt = A / (v + B) - (v - 20) / B

Explanation:

- The term A / (v + B) represents the acceleration provided by the Prius, which is inversely proportional to its velocity.

- The term (v - 20) / B represents the acceleration due to air resistance, which is proportional to the difference between the tailwind (20 mph) and the velocity of the Prius.

Therefore, the chosen differential equation is: dv/dt = A / (v + B) - (v - 20) / B

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c) the cash discount, if a partial payment is made such that a balance of $2500 remains outstanding on the invoice, is $5333.84.

To determine the last day for taking the cash discount, we need to consider the terms "4/10 EO.M." This means that a cash discount of 4% is offered if payment is made within 10 days from the invoice date, and the net amount is due at the end of the month (EO.M.).

(a) The last day for taking the cash discount is September 27. Since the invoice is dated September 17, we count 10 days from that date, excluding Sundays and possibly other non-business days.

(b) To calculate the amount due if the invoice is paid on the last day for taking the discount, we need to determine the total amount after applying the discounts. Let's calculate the amounts for refrigerators and dishwashers separately:

For refrigerators:

6 refrigerators at $1020 each = $6120

25% discount = $6120 * 0.25 = $1530

6% discount = ($6120 - $1530) * 0.06 = $327.60

Total amount for refrigerators after discounts = $6120 - $1530 - $327.60 = $4262.40

For dishwashers:

5 dishwashers at $1001 each = $5005

16% discount = $5005 * 0.16 = $800.80

12.6% discount = ($5005 - $800.80) * 0.126 = $497.53

3% discount = ($5005 - $800.80 - $497.53) * 0.03 = $135.23

Total amount for dishwashers after discounts = $5005 - $800.80 - $497.53 - $135.23 = $3571.44

The total amount due for the invoice is the sum of the amounts for refrigerators and dishwashers:

Total amount due = $4262.40 + $3571.44 = $7833.84

(b) The amount due, if the invoice is paid on the last day for taking the discount, is $7833.84.

(c) To calculate the cash discount if a partial payment is made such that a balance of $2500 remains outstanding on the invoice, we subtract the outstanding balance from the total amount due:

Cash discount = Total amount due - Outstanding balance

Cash discount = $7833.84 - $2500 = $5333.84

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Show that this equation is exact:In order to prove that the given equation is exact, we need to check whether the equation satisfies the criterion for exactness, which is given by the equation∂Q/∂x = ∂P/∂y where P and Q are the coefficients of dx and dy respectively.

Hence, we obtain∂F/∂y = x² + 1/(3y) + ln|x| + C′ = Q(x, y)Therefore, the solution of the given differential equation isF(x, y) = y ∫e3xy dx + y²x − yx² + C(y)= y e3xy/3 + y²x − yx² + C(y)where C(y) is a constant of integration.

To solve a differential equation, we have to prove that the given equation is exact, then find the function F(x,y) and substitute the values of P and Q and integrate with respect to x and then differentiate the function obtained with respect to y, equating it to Q.

Then we can substitute the constant and get the final solution in the form of F(x,y).

Summary: Here, we first proved that the given equation is exact. After that, we found the function F(x,y) and solved the differential equation by substituting the values of P and Q and integrating w.r.t x and differentiating w.r.t y. We obtained the solution as F(x,y) = y e3xy/3 + y²x − yx² + C(y).

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transformed back from the Laplace domain to the time domain, results in an exponential decay with a polynomial term. The coefficient 13 represents the magnitude of the transformed function, while t³ represents the polynomial term, and e^(-t) represents the exponential decay.

To find the inverse Laplace transform of f(t) = 13(s+1)³, we first need to recognize that (s+1)³ corresponds to the Laplace transform of t³. According to the table of Laplace transforms, the inverse Laplace transform of (s+1)³ is t³. Multiplying this by the coefficient 13, we obtain 13t³.

The inverse Laplace transform of (s+1)³ is given by the formula:

L⁻¹{(s+1)³} = t³ * e^(-t)

Since the Laplace transform is a linear operator, we can apply it to each term in the expression for f(t). Thus, the inverse Laplace transform of f(t) = 13(s+1)³ is:

L⁻¹{13(s+1)³} = 13 * L⁻¹{(s+1)³} = 13 * t³ * e^(-t)

Therefore, the inverse transform of f(t) is 13t³e^(-t).

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Given that the positively oriented curve in the x-y plane is the boundary of the rectangle with vertices (0,0), (3,0), (3,1) and (0,1) and the line integral is xy dx + x² dy,

we are to evaluate the line integral directly and using Green's Theorem.

Evaluation of the line integral directly (i.e without using Green's Theorem)

The line integral, directly, can be evaluated as follows:

∫xy dx + x² dy= ∫y(x dx) + x² dy = ∫y d(x²/2) + xy dy.

Using the limits of the curve from 0 to 3, we have;

∫xy dx + x² dy = [(3²/2 * 1) - (0²/2 * 1)] + ∫[3y dy + y dy]∫xy dx + x² dy = 9/2 + 2y²/2|0¹ = 9/2.

Evaluation of the line integral by using Green's Theorem, Let the line integral,

∫C xy dx + x² dy be represented as;

∫C xy dx + x² dy = ∬R [∂/∂x (x²) - ∂/∂y (xy)] dA,

where R is the region enclosed by C.

∫C xy dx + x² dy = ∬R [2x - x] dA∫C xy dx + x² dy = 0.

Therefore, evaluating the line integral directly (i.e without using Green's Theorem) yields 9/2, while evaluating the line integral by using Green's Theorem yields 0.

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To find the average rate of change of a function from x1 to x2, you need to calculate the difference in the function's values at x1 and x2, and divide it by the difference in the x-values. The formula for average rate of change is (f(x2) - f(x1)) / (x2 - x1).

The average rate of change measures the average rate at which a function is changing over a given interval. To calculate it, you subtract the function's values at the starting point (x1) and ending point (x2), and then divide it by the difference in the x-values. This gives you the average rate of change for the interval from x1 to x2.

Example: Let's say we have a function f(x) = 2x + 3. To find the average rate of change from x1 = 1 to x2 = 4, we substitute these values into the formula: (f(4) - f(1)) / (4 - 1). Simplifying, we get (11 - 5) / 3 = 6 / 3 = 2. Therefore, the average rate of change of the function from x1 = 1 to x2 = 4 is 2.

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The indefinite integral of 7x³ + 9 is:(7/4)(x + 9)⁴ - (189/3)(x + 9)³ + (1701/2)(x + 9)² - 1512(x + 9) + C, where C is the constant of integration.

To find the indefinite integral of 7x³ + 9, we will use the method of integration by substitution.

Step 1: Let u = x + 9.

Differentiating both sides with respect to x, we get du/dx = 1.

Step 2: Rearrange the equation to solve for dx:

dx = du/1 = du.

Step 3: Substitute the values of u and dx into the integral:

∫(7x³ + 9) dx = ∫(7(u - 9)³ + 9) du.

Step 4: Simplify the integrand:

∫(7(u³ - 27u² + 243u - 243) + 9) du

= ∫(7u³ - 189u² + 1701u - 1512) du.

Step 5: Integrate term by term:

= (7/4)u⁴ - (189/3)u³ + (1701/2)u² - 1512u + C.

Step 6: Substitute back u = x + 9:

= (7/4)(x + 9)⁴ - (189/3)(x + 9)³ + (1701/2)(x + 9)² - 1512(x + 9) + C.

Therefore, the indefinite integral of 7x³ + 9 is:

(7/4)(x + 9)⁴ - (189/3)(x + 9)³ + (1701/2)(x + 9)² - 1512(x + 9) + C, where C is the constant of integration.

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Indefinite integral of [tex]7x^3 + 9[/tex] [tex]= (7/4)(x + 9)^4 - (189/3)(x + 9)^3 + (1701/2)(x + 9)^2 - 1512(x + 9) + C[/tex], ( C is the  integration constant.)

To determine the antiderivative of 7x³ + 9 without any bounds, we shall employ the technique of integration by substitution.

1st Step: Let [tex]u = x + 9[/tex]

By taking the derivative with respect to x on both sides of the equation, we obtain the expression du/dx = 1.

2nd Step: Rearranging the equation, we can solve for dx:

[tex]dx = du/1 = du[/tex]

3rd Step: Substituting the values of u and dx into the integral, we have:

[tex]\int(7x^3 + 9) dx = \int(7(u - 9)^3 + 9) du.[/tex]

4th Step: Simplification of the integrand:

[tex]\int(7(u^3 - 27u^2 + 243u - 243) + 9) du[/tex]

[tex]= \int(7u^3 - 189u^2 + 1701u - 1512) du[/tex]

Step 5: Integration term by term:

[tex]=(7/4)u^4 - (189/3)u^3 + (1701/2)u^2 - 1512u + C[/tex]

Step 6: Let us substitute back[tex]u = x + 9[/tex]:

[tex]= (7/4)(x + 9)^4 - (189/3)(x + 9)^3 + (1701/2)(x + 9)^2 - 1512(x + 9) + C[/tex]

Hence, the indefinite integral of [tex]7x^3 + 9[/tex] is:

[tex]= (7/4)(x + 9)^4 - (189/3)(x + 9)^3 + (1701/2)(x + 9)^2 - 1512(x + 9) + C[/tex], in that  C is the constant of integration.

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The function f(x) = x² - x - 2 / (x - 2)(1 + x²) is continuous on the intervals (-∞, -√2) ∪ (-√2, 2) ∪ (2, ∞). The function f(x) = 2 - x is continuous on the interval (-∞, 2]. The function f(x) = (x - 1)² is continuous on the interval (2, ∞).

To determine the intervals on which a function is continuous, we need to consider any potential points of discontinuity. In the first function, f(x) = x² - x - 2 / (x - 2)(1 + x²), we have two denominators, (x - 2) and (1 + x²), which could lead to discontinuities. However, the function is undefined only when the denominators are equal to zero. Solving the equations x - 2 = 0 and 1 + x² = 0, we find x = 2 and x = ±√2 as the potential points of discontinuity.

Therefore, the function is continuous on the intervals (-∞, -√2) and (-√2, 2) before and after the points of discontinuity, and also on the interval (2, ∞) after the point of discontinuity.

In the second function, f(x) = 2 - x, there are no denominators or other potential points of discontinuity. Thus, the function is continuous on the interval (-∞, 2].

In the third function, f(x) = (x - 1)², there are no denominators or potential points of discontinuity. The function is continuous on the interval (2, ∞).

Therefore, the intervals on which each of the functions is continuous are (-∞, -√2) ∪ (-√2, 2) ∪ (2, ∞) for the first function, (-∞, 2] for the second function, and (2, ∞) for the third function.

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We can solve the given separable differential equation as follows:Firstly, separate the variables and write the equation in the form of `dy/y = -8dx`.Integrating both sides, we get `ln|y| = -8x + C_1`, where `C_1` is the constant of integration.

The given separable differential equation is `dy/dx = -8y`. We need to find the particular solution that satisfies the initial condition `y(0) = 2`.To solve the given differential equation, we first separate the variables and get the equation in the form of `dy/y = -8dx`.Integrating both sides, we get `ln|y| = -8x + C_1`, where `C_1` is the constant of integration.Rewriting the above equation in the exponential form, we have `|y| = e^(-8x+C_1)`.We can take the constant `C = e^(C_1)` and then replace `|y|` with `y`, to get `y = Ce^(-8x)` (where `C = e^(C_1)`).

This is the general solution of the given differential equation.Now, to find the particular solution, we substitute the initial condition `y(0) = 2` in the general solution, i.e., `y = Ce^(-8x)`Substituting `x = 0` and `y = 2`, we get `2 = Ce^(0)`i.e., `2 = C`Therefore, the particular solution satisfying the initial condition is `y = 2e^(-8x)`.

Therefore, the solution to the separable differential equation `dy/dx = -8y` satisfying the initial condition `y(0) = 2` is `y = 2e^(-8x)`.

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By chain rule of differentiation,d/ dx ([tex]x^2[/tex] - 4x + 3) = (2x - 4)

The given expression is:d/ dx ([tex]x^2[/tex] - 4x + 3)

Calculus' fundamental idea of differentiation entails figuring out how quickly a function changes. Finding a function's derivative with regard to its independent variable is the process at hand. The derivative shows how the function's value is changing at each particular position by displaying the slope of the function at that location.

With the aid of differentiation, we can examine the behaviour of functions, spot crucial locations like maxima and minima, and comprehend the contours of curves. Numerous domains, including physics, engineering, economics, and others where rates of change are significant, can benefit from it. Power rule, product rule, chain rule, and more strategies for differentiating products are available.

To differentiate the given expression we apply the chain rule of differentiation. Here the outside function is d/ dx and the inside function is (x² - 4x + 3).

Therefore, by chain rule of differentiation,d/ dx (x² - 4x + 3) = (2x - 4)

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