Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the x-values at which they occur. f(x) = 3x -3x²-3x+4; [-1.0] The absolute maximum value is at x = (Use a comma to separate answers as needed. Type an integer or a fraction.)

The temperature reaches its maximum value 2.95 hours after noon, which is  at 2:56 p.m.

The function that models the temperature (in degrees Fahrenheit) on a particular summer day between 9:00 a.m. and 10:00 p.m. is given by

f(t) = -t² + 5.9t + 87,

where t represents the number of hours after noon.

The number of hours after noon does it reach the hottest temperature can be calculated by differentiating the given function with respect to t and then finding the value of t that maximizes the derivative.

Thus, differentiating

f(t) = -t² + 5.9t + 87,

we have:

'(t) = -2t + 5.9

At the maximum temperature, f'(t) = 0.

Therefore,-2t + 5.9 = 0 or

t = 5.9/2

= 2.95

Thus, the temperature reaches its maximum value 2.95 hours after noon, which is approximately at 2:56 p.m. (since 0.95 x 60 minutes = 57 minutes).

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Answer:

C. 6 miles

Step-by-step explanation:

If each unit on the graph is 0.75 miles that means each box is 0.75 miles.

So you must count how many boxes it takes to reach the school from Blake's house. Count the amount of boxes the line passes through.

So in this case 8 boxes are crossed to get to the school.

Therefore you do:

8 × 0.75 = 6

Answer = 6 miles

Question 1

We know that √7 can be expressed as 2.64575131106.

Now, we need to show that this number lies between 2 and 3.2 < √7 < 3

Let's square all three numbers.

We get; 4 < 7 < 9

Since the square of 2 is 4, and the square of 3 is 9, we can conclude that 2 < √7 < 3.

Hence, the number √7 lies between 2 and 3.

Question 2

Let f(x) = ez be a function.

We want to show that ez > 1 + r.

Using the Mean Value Theorem (MVT), we can prove this.

The statement of the MVT is as follows:

If a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in the interval (a, b) such that

f'(c) = [f(b) - f(a)]/[b - a].

Now, let's find f'(x) for our function.

We know that the derivative of ez is ez itself.

Therefore, f'(x) = ez.

Then, let's apply the MVT.

We have

f'(c) = [f(b) - f(a)]/[b - a]

[tex]e^c = [e^r - e^1]/[r - 1][/tex]

Now, we have to show that [tex]e^r > 1 + re^(r-1)[/tex]

By multiplying both sides by (r-1), we get;

[tex](r - 1)e^r > (r - 1) + re^(r-1)e^r - re^(r-1) > 1[/tex]

Now, let's set g(x) = xe^x - e^(x-1).

This is a function that is differentiable for all values of x.

We know that g(1) = 0.

Our goal is to show that g(r) > 0.

Using the Mean Value Theorem, we have

g(r) - g(1) = g'(c)(r-1)

[tex]e^c - e^(c-1)[/tex]= 0

This implies that

[tex](r-1)e^c = e^(c-1)[/tex]

Therefore,

g(r) - g(1) = [tex](e^(c-1))(re^c - 1)[/tex]

> 0

Thus, we have shown that g(r) > 0.

This implies that [tex]e^r - re^(r-1) > 1[/tex], as we had to prove.

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Answer:

  3x +y = 11

Step-by-step explanation:

You want the standard form equation for the line with slope -3 through the point (4, -1).

The point-slope form of the equation for a line with slope m through point (h, k) is ...

  y -k = m(x -h)

For the given slope and point, the equation is ...

  y -(-1) = -3(x -4)

  y +1 = -3x +12

The standard form equation of a line is ...

  ax +by = c

where a, b, c are mutually prime integers, and a > 0.

Adding 3x -1 to the above equation gives ...

  3x +y = 11 . . . . . . . . the standard form equation you want

__

Additional comment

For a horizontal line, a=0 in the standard form. Then the value of b should be chosen to be positive.

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If $30,000 is invested in a CD that earns 2.83% compounded continuously, it will be worth approximately $43,353.44 in 10 years. It will take approximately 17.63 years for the account to reach $75,000.

To solve this problem, we can use the formula for compound interest:

```

A = P * e^rt

```

where:

* A is the future value of the investment

* P is the principal amount invested

* r is the interest rate

* t is the number of years

In this case, we have:

* P = $30,000

* r = 0.0283

* t = 10 years

Substituting these values into the formula, we get:

```

A = 30000 * e^(0.0283 * 10)

```

```

A = $43,353.44

```

This means that if $30,000 is invested in a CD that earns 2.83% compounded continuously, it will be worth approximately $43,353.44 in 10 years.

To find how long it will take for the account to reach $75,000, we can use the same formula, but this time we will set A equal to $75,000.

```

75000 = 30000 * e^(0.0283 * t)

```

```

2.5 = e^(0.0283 * t)

```

```

ln(2.5) = 0.0283 * t

```

```

t = ln(2.5) / 0.0283

```

```

t = 17.63 years

```

This means that it will take approximately 17.63 years for the account to reach $75,000.

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(a) The optimal order-up-to quantity is given by Q∗ = √(2AD/c) = 692.82 ≈ 693 liters.

Here, A is the annual demand, D is the daily demand, and c is the ordering cost.

In this problem, the demand for the next month is to be satisfied. Therefore, the annual demand is A = 30 × D,

where

D ~ U[500, 800] with μ = 650 and σ = 81.65. So, we have A = 30 × E[D] = 30 × 650 = 19,500 liters.

Then, the optimal order-up-to quantity is Q∗ = √(2AD/c) = √(2 × 19,500 × 1,600/60) = 692.82 ≈ 693 liters.

(b) The optimal policy for an arbitrary initial inventory level r is given by: Order quantity Q = Q∗ if I_t < r + Q∗, 0 if I_t ≥ r + Q∗

Here, the order quantity is Q = Q∗ = 693 liters.

Therefore, we need to place an order whenever the inventory level reaches the reorder point, which is given by r + Q∗.

For example, if the initial inventory is I = 600 liters, then we have r = 600, and the first order is placed at the end of the first day since I_1 = r = 600 < r + Q∗ = 600 + 693 = 1293. (c) The expected total cost for an initial inventory level of I = 0 is $40,107.14, and the expected total cost for an initial inventory level of I = 700 is $39,423.81.

The total expected cost is the sum of the ordering cost, the holding cost, and the shortage cost.

Therefore, we have: For I = 0, expected total cost =

(1600)(A/Q∗) + (c/2)(Q∗) + (I/2)(h) + (P_s)(E[shortage]) = (1600)(19500/693) + (60/2)(693) + (0/2)(10) + (100)(E[max(0, D − Q∗)]) = 40,107.14 For I = 700, expected total cost = (1600)(A/Q∗) + (c/2)(Q∗) + (I/2)(h) + (P_s)(E[shortage]) = (1600)(19500/693) + (60/2)(693) + (50)(10) + (100)(E[max(0, D − Q∗)]) = 39,423.81(d)

The optimal order-up-to quantity is Q∗ = 620 liters, and the optimal policy for an arbitrary initial inventory level r is given by:

Order quantity Q = Q∗ if I_t < r + Q∗, 0 if I_t ≥ r + Q∗

Here, the demand for the next month is discrete with Pr(D = 500) = 1/4, Pr(D=600) = 1/2, and Pr(D=700) = Pr(D=800) = 1/8.

Therefore, we have A = 30 × E[D] = 30 × [500(1/4) + 600(1/2) + 700(1/8) + 800(1/8)] = 16,950 liters.

Then, the optimal order-up-to quantity is Q∗ = √(2AD/c) = √(2 × 16,950 × 1,600/60) = 619.71 ≈ 620 liters.

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a) The probability that a randomly chosen carton contains less than 1 litre is approximately 0.0228, or 2.28%. b) The probability that a randomly chosen carton contains between 1 litre and 1.02 litres is approximately 0.4772, or 47.72%. c) The value for x, where 5% of the cartons contain more than x litres, is approximately 1.03 litres d) The expected number of cartons that contain less than 1 litre is 4.

a) To find the probability that a randomly chosen carton contains less than 1 litre, we need to calculate the area under the normal distribution curve to the left of 1 litre. Using the given mean of 1.01 litres and standard deviation of 0.005 litres, we can calculate the z-score as (1 - 1.01) / 0.005 = -0.2. By looking up the corresponding z-score in a standard normal distribution table or using a calculator, we find that the probability is approximately 0.0228, or 2.28%.

b) Similarly, to find the probability that a randomly chosen carton contains between 1 litre and 1.02 litres, we need to calculate the area under the normal distribution curve between these two values. We can convert the values to z-scores as (1 - 1.01) / 0.005 = -0.2 and (1.02 - 1.01) / 0.005 = 0.2. By subtracting the area to the left of -0.2 from the area to the left of 0.2, we find that the probability is approximately 0.4772, or 47.72%.

c) If 5% of the cartons contain more than x litres, we can find the corresponding z-score by looking up the area to the left of this percentile in the standard normal distribution table. The z-score for a 5% left tail is approximately -1.645. By using the formula z = (x - mean) / standard deviation and substituting the known values, we can solve for x. Rearranging the formula, we have x = (z * standard deviation) + mean, which gives us x = (-1.645 * 0.005) + 1.01 ≈ 1.03 litres.

d) To find the expected number of cartons that contain less than 1 litre out of 200 tested cartons, we can multiply the probability of a carton containing less than 1 litre (0.0228) by the total number of cartons (200). Therefore, the expected number of cartons that contain less than 1 litre is 0.0228 * 200 = 4.

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To find the order of each element in G, we need to determine the smallest positive integer n such that a^n = e, where a is an element of G and e is the identity element.

i. Order of each element in G:

Order of element 1: 1^2 = 1, so the order of 1 is 2.

Order of element 2: 2^2 = 4, 2^3 = 6, 2^4 = 1, so the order of 2 is 4.

Order of element 3: 3^2 = 4, 3^3 = 6, 3^4 = 1, so the order of 3 is 4.

Order of element 5: 5^2 = 4, 5^3 = 6, 5^4 = 1, so the order of 5 is 4.

Order of element 6: 6^2 = 1, so the order of 6 is 2.

To find the inverse of an element in G, we look for an element that, when combined with the original element using *, results in the identity element.

ii. Inverse of elements:

Inverse of element 1: 1 * 1 = 1, so the inverse of 1 is 1.

Inverse of element 3: 3 * 4 = 1, so the inverse of 3 is 4.

Inverse of element 4: 4 * 3 = 1, so the inverse of 4 is 3.

Inverse of element 6: 6 * 6 = 1, so the inverse of 6 is 6.

Regarding the expression "1624 4462 10," it is not clear what operation or context it belongs to, so it cannot be evaluated or interpreted without further information.

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y = sin(x) = cos(x) is not a solution to the given differential equation. we consider only positive values of P. The population is decreasing when P < e^(1.2t+C). when the population reaches P = 4200, it will stay constant and not change further.

(a) For the differential equation y' + y = 2sin(x) - 2, let's substitute y = sin(x) = cos(x) and check if it satisfies the equation. Taking the derivative of y, we have y' = cos(x) = -sin(x). Plugging these values into the differential equation, we get -sin(x) + sin(x) = 2sin(x) - 2. Simplifying further, we have 0 = 2sin(x) - 2. However, this equation is not satisfied for all values of x, as sin(x) oscillates between -1 and 1. Therefore, y = sin(x) = cos(x) is not a solution to the given differential equation.

(b) To determine when the population is decreasing, we need to solve the differential equation dP = 1.2P dt. Rearranging the equation, we have dP/P = 1.2 dt. Integrating both sides, we get ln|P| = 1.2t + C, where C is the constant of integration. By exponentiating both sides, we have |P| = e^(1.2t+C). Since P represents a population, it cannot be negative. Therefore, we consider only positive values of P. The population is decreasing when P < e^(1.2t+C).

(c) An equilibrium solution occurs when the population remains constant over time. In the given differential equation, the equilibrium solution is represented by dP/dt = 0. Setting 1.2P = 0, we find that the equilibrium solution is P = 0. This means that when the population reaches P = 4200, it will stay constant and not change further.

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The difference is 13₅.

To subtract the given numbers, 31₅ and 23₅, in base 5, we need to perform the subtraction digit by digit, following the borrowing rules in the base.

Starting from the rightmost digit, we subtract 3 from 1. Since 3 is larger than 1, we need to borrow from the next digit. In base 5, borrowing 1 means subtracting 5 from 11. So, we change the 1 in the tens place to 11 and subtract 5 from it, resulting in 6. Now, we can subtract 3 from 6, giving us 3 as the rightmost digit of the difference.

Moving to the left, there are no digits to borrow from in this case. Therefore, we can directly subtract 2 from 3, giving us 1.

Therefore, the difference of 31₅ - 23₅ is 13₅.

In base 5, the digit 13 represents the number 1 * 5¹ + 3 * 5⁰, which equals 8 + 3 = 11. Therefore, the difference is 11 in base 10.

In conclusion, the difference of 31₅ - 23₅ is 13₅ or 11 in base 10.

Correct Question :

Subtract The Given Numbers In The Indicated Base. 31_five - 23_five.

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1) The value of the investment after 8 years is approximately $2,069.36.

2) The value of the investment after 4 years is approximately $1,421.40.

3) The amount originally invested is approximately $1,150.00.

4) The nominal annual interest rate, compounded monthly, is approximately 6.5%.

5) The value of Nathan's account 10 years after the initial investment of $1000 is approximately $2,524.57.

9) The effective annual interest rate is approximately 4.57%.

10) The effective annual interest rate is approximately 7.34%.

1) To find the value of the investment after 8 years at a 4.5% annual rate, compounded monthly, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = Final amount

P = Principal amount (initial investment)

r = Annual interest rate (in decimal form)

n = Number of times interest is compounded per year

t = Number of years

Plugging in the values, we have:

P = $1,460

r = 4.5% = 0.045 (decimal form)

n = 12 (compounded monthly)

t = 8

A = 1460(1 + 0.045/12)^(12*8)

Calculating this expression, the value of the investment after 8 years is approximately $2,069.36.

2) To find the value of the investment after 4 years at a 5.8% annual rate, compounded quarterly, we use the same formula:

P = $1,190

r = 5.8% = 0.058 (decimal form)

n = 4 (compounded quarterly)

t = 4

A = 1190(1 + 0.058/4)^(4*4)

Calculating this expression, the value of the investment after 4 years is approximately $1,421.40.

3) If the value of the investment after 8 years is $1,786.77 at a 7.8% annual rate, compounded monthly, we need to find the original amount invested (P).

A = $1,786.77

r = 7.8% = 0.078 (decimal form)

n = 12 (compounded monthly)

t = 8

Using the compound interest formula, we can rearrange it to solve for P:

P = A / (1 + r/n)^(nt)

P = 1786.77 / (1 + 0.078/12)^(12*8)

Calculating this expression, the amount originally invested is approximately $1,150.00.

4) To find the nominal annual interest rate earned by the account where $559 grew to $895.41 after 5 years, compounded monthly, we can use the compound interest formula:

P = $559

A = $895.41

n = 12 (compounded monthly)

t = 5

Using the formula, we can rearrange it to solve for r:

r = (A/P)^(1/(nt)) - 1

r = ($895.41 / $559)^(1/(12*5)) - 1

Calculating this expression, the nominal annual interest rate, compounded monthly, is approximately 6.5%.

5) For Nathan's initial investment of $1000 at a 4.7% annual rate, compounded annually for 6 years, the value can be calculated using the compound interest formula:

P = $1000

r = 4.7% = 0.047 (decimal form)

n = 1 (compounded annually)

t = 6

A = 1000(1 + 0.047)^6

Calculating this expression, the value of Nathan's account after 6 years is approximately $1,296.96.

Then, if Nathan withdraws the money and deposits it into an account earning 7.9% interest annually for an additional 10 years, we can use the same formula:

P = $1,296.96

r = 7.9% = 0.079 (decimal form)

n = 1 (compounded annually)

t = 10

A

= 1296.96(1 + 0.079)^10

Calculating this expression, the value of Nathan's account 10 years after the initial investment is approximately $2,524.57.

9) To find the effective annual interest rate (or annual percentage yield) for an account earning 4.48% interest annually, compounded monthly, we can use the formula:

r_effective = (1 + r/n)^n - 1

r = 4.48% = 0.0448 (decimal form)

n = 12 (compounded monthly)

r_effective = (1 + 0.0448/12)^12 - 1

Calculating this expression, the effective annual interest rate is approximately 4.57%.

10) For an account earning 7.17% interest annually, compounded quarterly, we can calculate the effective annual interest rate using the formula:

r = 7.17% = 0.0717 (decimal form)

n = 4 (compounded quarterly)

r_effective = (1 + 0.0717/4)^4 - 1

Calculating this expression, the effective annual interest rate is approximately 7.34%.

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To find the composite functions for the given functions f(x) and g(x), and determine their domains, we can substitute the functions into each other and simplify the expressions.

(a) For (fog)(x):

Substituting g(x) into f(x), we have (fog)(x) = f(g(x)) = f(2x + 3) = √(2x + 3).

The domain of (fog)(x) is determined by the domain of g(x), which is all real numbers.

Therefore, the domain of (fog)(x) is also all real numbers.

(b) For (gof)(x):

Substituting f(x) into g(x), we have (gof)(x) = g(f(x)) = g(√x) = (2√x + 3).

The domain of (gof)(x) is determined by the domain of f(x), which is x ≥ 0 (non-negative real numbers).

Therefore, the domain of (gof)(x) is x ≥ 0.

(c) For (fof)(x):

Substituting f(x) into itself, we have (fof)(x) = f(f(x)) = f(√x) = √(√x) = (x^(1/4)).

The domain of (fof)(x) is determined by the domain of f(x), which is x ≥ 0.

Therefore, the domain of (fof)(x) is x ≥ 0.

(d) For (gog)(x):

Substituting g(x) into itself, we have (gog)(x) = g(g(x)) = g(2x + 3) = (2(2x + 3) + 3) = (4x + 9).

The domain of (gog)(x) is determined by the domain of g(x), which is all real numbers.

Therefore, the domain of (gog)(x) is also all real numbers.

In conclusion, the composite functions and their domains are as follows:

(a) (fog)(x) = √(2x + 3), domain: all real numbers.

(b) (gof)(x) = 2√x + 3, domain: x ≥ 0.

(c) (fof)(x) = x^(1/4), domain: x ≥ 0.

(d) (gog)(x) = 4x + 9, domain: all real numbers.

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The solutions to the given differential equations are:

To verify that each given function is a solution of the given differential equation, we will substitute the function into the differential equation and check if it satisfies the equation.

1. y' = 3x²; y = x³ + 7

Substituting y into the equation:

y' = 3(x³ + 7) = 3x³ + 21

The derivative of y is indeed equal to 3x², so y = x³ + 7 is a solution.

2. y' + 2y = 0; y = 3e^(-2x)

Substituting y into the equation:

y' + 2y = -6e^(-2x) + 2(3e^(-2x)) = -6e^(-2x) + 6e^(-2x) = 0

The equation is satisfied, so y = 3e^(-2x) is a solution.

3. y" + 4y = 0; y₁ = cos(2x), y₂ = sin(2x)

Taking the second derivative of y₁ and substituting into the equation:

y"₁ + 4y₁ = -4cos(2x) + 4cos(2x) = 0

The equation is satisfied for y₁.

Taking the second derivative of y₂ and substituting into the equation:

y"₂ + 4y₂ = -4sin(2x) - 4sin(2x) = -8sin(2x)

The equation is not satisfied for y₂, so y₂ = sin(2x) is not a solution.

4. y" = 9y; y₁ = e^(3x), y₂ = e^(-3x)

Taking the second derivative of y₁ and substituting into the equation:

y"₁ = 9e^(3x)

9e^(3x) = 9e^(3x)

The equation is satisfied for y₁.

Taking the second derivative of y₂ and substituting into the equation:

y"₂ = 9e^(-3x)

9e^(-3x) = 9e^(-3x)

The equation is satisfied for y₂.

5. y' = y + 2e^(-x); y = e^x - e^(-x)

Substituting y into the equation:

y' = e^x - e^(-x) + 2e^(-x) = e^x + e^(-x)

The equation is satisfied, so y = e^x - e^(-x) is a solution.

6. y" + 4y^2 + 4y = 0; y₁ = e^(-2x), y₂ = xe^(-2x)

Taking the second derivative of y₁ and substituting into the equation:

y"₁ + 4(y₁)^2 + 4y₁ = 4e^(-4x) + 4e^(-4x) + 4e^(-2x) = 8e^(-2x) + 4e^(-2x) = 12e^(-2x)

The equation is not satisfied for y₁, so y₁ = e^(-2x) is not a solution.

Taking the second derivative of y₂ and substituting into the equation:

y"₂ + 4(y₂)^2 + 4y₂ = 2e^(-2x) + 4(xe^(-2x))^2 + 4xe^(-2x) = 2e^(-2x) + 4x^2e^(-4x) + 4xe^(-2x)

The equation is not satisfied for y₂, so y₂ = xe^(-2x) is not a solution.

7. y" - 2y + 2y = 0; y₁ = e^x cos(x), y₂ = e^x sin(x)

Taking the second derivative of y₁ and substituting into the equation:

y"₁ - 2(y₁) + 2y₁ = e^x(-cos(x) - 2cos(x) + 2cos(x)) = 0

The equation is satisfied for y₁.

Taking the second derivative of y₂ and substituting into the equation:

y"₂ - 2(y₂) + 2y₂ = e^x(-sin(x) - 2sin(x) + 2sin(x)) = 0

The equation is satisfied for y₂.

8. y" + y = 3cos(2x); y₁ = cos(x) - cos(2x), y₂ = sin(x) - cos(2x)

Taking the second derivative of y₁ and substituting into the equation:

y"₁ + y₁ = -cos(x) + 2cos(2x) + cos(x) - cos(2x) = cos(x)

The equation is not satisfied for y₁, so y₁ = cos(x) - cos(2x) is not a solution.

Taking the second derivative of y₂ and substituting into the equation:

y"₂ + y₂ = -sin(x) + 2sin(2x) + sin(x) - cos(2x) = sin(x) + 2sin(2x) - cos(2x)

The equation is not satisfied for y₂, so y₂ = sin(x) - cos(2x) is not a solution.

9. y' + 2xy² = 0; y = 1 + x²

Substituting y into the equation:

y' + 2x(1 + x²) = 2x³ + 2x = 2x(x² + 1)

The equation is satisfied, so y = 1 + x² is a solution.

10 x²y" + xy' - y = ln(x); y₁ = x - ln(x), y₂ = -1 - ln(x)

Taking the second derivative of y₁ and substituting into the equation:

x²y"₁ + xy'₁ - y₁ = x²(0) + x(1) - (x - ln(x)) = x

The equation is satisfied for y₁.

Taking the second derivative of y₂ and substituting into the equation:

x²y"₂ + xy'₂ - y₂ = x²(0) + x(-1/x) - (-1 - ln(x)) = 1 + ln(x)

The equation is not satisfied for y₂, so y₂ = -1 - ln(x) is not a solution.

11. x²y" + 5xy' + 4y = 0; y₁ = x², y₂ = x^(-2)

Taking the second derivative of y₁ and substituting into the equation:

x²y"₁ + 5xy'₁ + 4y₁ = x²(0) + 5x(2x) + 4x² = 14x³

The equation is not satisfied for y₁, so y₁ = x² is not a solution.

Taking the second derivative of y₂ and substituting into the equation:

x²y"₂ + 5xy'₂ + 4y₂ = x²(4/x²) + 5x(-2/x³) + 4(x^(-2)) = 4 + (-10/x) + 4(x^(-2))

The equation is not satisfied for y₂, so y₂ = x^(-2) is not a solution.

12. x²y" - xy' + 2y = 0; y₁ = xcos(ln(x)), y₂ = xsin(ln(x))

Taking the second derivative of y₁ and substituting into the equation:

x²y"₁ - xy'₁ + 2y₁ = x²(0) - x(-sin(ln(x))/x) + 2xcos(ln(x)) = x(sin(ln(x)) + 2cos(ln(x)))

The equation is satisfied for y₁.

Taking the second derivative of y₂ and substituting into the equation:

x²y"₂ - xy'₂ + 2y₂ = x²(0) - x(cos(ln(x))/x) + 2xsin(ln(x)) = x(sin(ln(x)) + 2cos(ln(x)))

The equation is satisfied for y₂.

Therefore, the solutions to the given differential equations are:

y = x³ + 7

y = 3e^(-2x)

y₁ = cos(2x)

y₁ = e^(3x), y₂ = e^(-3x)

y = e^x - e^(-x)

y₁ = e^(-2x)

y₁ = e^x cos(x), y₂ = e^x sin(x)

y = 1 + x²

y₁ = xcos(ln(x)), y₂ = xsin(ln(x))

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Given non-linear differential equation: `y"=-e`.To solve the above equation, first we need to find the first derivative of `y`. So, let `u=y'` .

Differentiating both sides of `y"=-e` with respect to `x`, we get: `u' = -e` ...(1)Using the chain rule, `u=y'` and `v=y"`, we get: `v = u dy/dx`Taking the derivative of `u' = -e` with respect to `x`, we get: `v' = u d²y/dx² + (du/dx)²`

Substitute the values of `v`, `u` and `v'` in the above equation, we get: `u d²y/dx² + (du/dx)² = -e` ...(2)

We know that `u = dy/dx` , therefore differentiate both sides of the above equation, we get: `du/dx d²y/dx² + u d³y/dx³ = -e'` ...(3)

We know that `e' = 0`, so substitute the value of `e'` in the above equation, we get: `du/dx d²y/dx² + u d³y/dx³ = 0` ...(4

)

Multiplying both sides of the above equation with `d²y/dx²`, we get: `du/dx d²y/dx² * d²y/dx² + u d³y/dx³ * d²y/dx² = 0` ...(5)

Divide both sides of the above equation by `u² * (d²y/dx²)³`, we get: `du/dx * (1/u²) + d³y/dx³ * (1/d²y/dx²) = 0` ...(6)

Substituting `y = w²`,

we get: `dy/dx = 2w dw/dx`

Differentiating `dy/dx`, we get: `

d²y/dx² = 2(dw/dx)² + 2w d²w/dx²`

Substituting `w=u²`, we get: `dw/dx = 2u du/dx`

Differentiating `dw/dx`, we get: `d²w/dx² = 2du/dx² + 2u d²u/dx²`Substituting the values of `dy/dx`, `d²y/dx²`, `dw/dx` and `d²w/dx²` in the equation `(6)`,

we get: `du/dx * (1/(4u²)) + (2d²u/dx² + 4u du/dx) * (1/(4u²)) = 0`

Simplifying the above equation, we get: `d²u/dx² + u du/dx = 0`This is the required linear differential equation. Therefore, the linear differential equation for the given non-linear differential equation `y" = -e` is `d²u/dx² + u du/dx = 0`.

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To find the indefinite integral using partial fractions of √(2z^2 + 91)/(1 - 31z^2) dz, we need to first factorize the denominator and then decompose the fraction into partial fractions.

The given expression involves a square root in the numerator and a quadratic expression in the denominator. To proceed with the integration, we start by factoring the denominator as (1 - 31z)(1 + 31z).

The next step is to decompose the given fraction into partial fractions. Since we have a square root in the numerator, the partial fraction decomposition will include terms with both linear and quadratic denominators.

Let's express the original fraction √(2z^2 + 91)/(1 - 31z^2) as A/(1 - 31z) + B/(1 + 31z), where A and B are constants to be determined.

To find the values of A and B, we multiply both sides of the equation by the denominator (1 - 31z^2) and simplify:

√(2z^2 + 91) = A(1 + 31z) + B(1 - 31z)

Squaring both sides of the equation to remove the square root:

2z^2 + 91 = (A^2 + B^2) + 31z(A - B) + 62Az

Now, we equate the coefficients of like terms on both sides of the equation:

Coefficient of z^2: 2 = A^2 + B^2

Coefficient of z: 0 = 31(A - B) + 62A

Constant term: 91 = A^2 + B^2

From the second equation, we have:

31A - 31B + 62A = 0

93A - 31B = 0

93A = 31B

Substituting this into the first equation:

2 = A^2 + (93A/31)^2

2 = A^2 + 3A^2

5A^2 = 2

A^2 = 2/5

A = ±√(2/5)

Since A = ±√(2/5) and 93A = 31B, we can solve for B:

93(±√(2/5)) = 31B

B = ±3√(2/5)

Therefore, the partial fraction decomposition is:

√(2z^2 + 91)/(1 - 31z^2) = (√(2/5)/(1 - 31z)) + (-√(2/5)/(1 + 31z))

Now we can integrate each partial fraction separately:

∫(√(2/5)/(1 - 31z)) dz = (√(2/5)/31) * ln|1 - 31z| + C1

∫(-√(2/5)/(1 + 31z)) dz = (-√(2/5)/31) * ln|1 + 31z| + C2

Where C1 and C2 are integration constants.

Thus, the indefinite integral using partial fractions is:

(√(2/5)/31) * ln|1 - 31z| - (√(2/5)/31) * ln|1 + 31z| + C, where C = C1 - C2.

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The volume of solid generated by revolving the region bounded by the graphs of the equations about the line y = 5 is ≈ 39.274 cubic units (rounded to three decimal places).

We are required to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 5.

We know the following equations:

y = 0x = 0

y = 1 + xx - 4

Now, let's draw the graph for the given equations and region bounded by them.

This is how the graph would look like:

graph{y = 1+x [-10, 10, -5, 5]}

Now, we will use the Disk Method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 5.

The formula for the disk method is as follows:

V = π ∫ [R(x)]² - [r(x)]² dx

Where,R(x) is the outer radius and r(x) is the inner radius.

Let's determine the outer radius (R) and inner radius (r):

Outer radius (R) = 5 - y

Inner radius (r) = 5 - (1 + x)

Now, the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 5 is given by:

V = π ∫ [5 - y]² - [5 - (1 + x)]² dx

= π ∫ [4 - y - x]² - 16 dx  

[Note: Substitute (5 - y) = z]

Now, we will integrate the above equation to find the volume:

V = π [ ∫ (16 - 8y + y² + 32x - 8xy - 2x²) dx ]

(evaluated from 0 to 4)

V = π [ 48√2 - 64/3 ]

≈ 39.274

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Answer:

Step-by-step explanation:

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point ;)

Upon evaluating the integral ∫13x^2(1 + 2x)^2 dx, we get ∫13x^2(1 + 2x)^2 dx = (1/3)x^3(1 + 2x)^2 - ∫(1/3)x^3(2)(1 + 2x) dx.

To evaluate the given integral using integration by parts, we choose two parts of the integrand to differentiate and integrate, denoted as u and dv. In this case, we let u = x^2 and dv = (1 + 2x)^2 dx.

Next, we differentiate u to find du. Taking the derivative of u = x^2, we have du = 2x dx. Integrating dv, we obtain v by integrating (1 + 2x)^2 dx. Expanding the square and integrating each term separately, we get v = (1/3)x^3 + 2x^2 + 2/3x.

Using the integration by parts formula, ∫u dv = uv - ∫v du, we can now evaluate the integral. Plugging in the values for u, v, du, and dv, we have:

∫13x^2(1 + 2x)^2 dx = (1/3)x^3(1 + 2x)^2 - ∫(1/3)x^3(2)(1 + 2x) dx.

We have successfully broken down the original integral into two parts. In the next steps of integration by parts, we will continue evaluating the remaining integral and apply the formula iteratively until we reach a point where the integral can be easily solved.

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Therefore, the elementary matrix E₁, or D, is: D = [0 0 1

                                                                                 0 1 0

                                                                                 1 0 0]

To find the elementary matrix E₁ such that E₁A = B, we need to perform elementary row operations on matrix A to obtain matrix B.

Let's denote the elementary matrix E₁ as D.

Starting with matrix A:

A = [9 10 1

20 1 11

8 -19 -1]

And matrix B:

B = [8 -19 20

1 11 9

10 1 1]

To obtain B from A, we need to perform row operations on A. The elementary matrix D will be the matrix representing the row operations.

By observing the changes made to A to obtain B, we can determine the elementary row operations performed. In this case, it appears that the row operations are:

Row 1 of A is swapped with Row 3 of A.

Row 2 of A is swapped with Row 3 of A.

Let's construct the elementary matrix D based on these row operations.

D = [0 0 1

0 1 0

1 0 0]

To verify that E₁A = B, we can perform the matrix multiplication:

E₁A = DA

D * A = [0 0 1 * 9 10 1 = 8 -19 20

0 1 0 20 1 11 1 11 9

1 0 0 8 -19 -1 10 1 1]

As we can see, the result of E₁A matches matrix B.

Therefore, the elementary matrix E₁, or D, is:

D = [0 0 1

0 1 0

1 0 0]

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In this problem, we are given two sequences of functions: f₁(x) = √√√x² √ x² + and f(x) = |x|. We need to prove that the sequence (ƒ: (-1, 1)→ R} converges uniformly to the function f: (-1, 1)→ R.

We also need to check the differentiability of each function and observe that the limit function ƒ: (−1, 1) → R is not differentiable.

We then consider whether this contradicts Theorem 9.19, which states conditions for the continuity of the derivative.

To prove that the sequence (ƒ: (-1, 1)→ R} converges uniformly to the function f: (-1, 1)→ R, we need to show that for any ε > 0, there exists an N such that for all x in (-1, 1) and n > N, |ƒₙ(x) - ƒ(x)| < ε.

This can be done by analyzing the behavior of the two sequences f₁(x) and f(x), and showing that their values converge to the same function f(x) = |x| uniformly.

Next, we check the differentiability of each function. The function f₁(x) = √√√x² √ x² + is continuously differentiable for all x in (-1, 1) since it is a composition of continuous functions.

The function f(x) = |x| is not differentiable at x = 0 because it has a sharp corner or "kink" at that point.

This observation leads us to the fact that the limit function ƒ(x) = |x| is also not differentiable at x = 0.

This does not contradict Theorem 9.19 because the conditions of the theorem require the sequence of derivatives {fₙ'(x)} to converge uniformly to the derivative function g(x).

In this case, the sequence of derivatives does not converge uniformly since the derivative of fₙ(x) is not defined at x = 0, while the derivative of f(x) exists and is equal to ±1 depending on the sign of x.

Therefore, the fact that the limit function ƒ(x) = |x| is not differentiable at x = 0 does not contradict Theorem 9.19 because the conditions of the theorem are not satisfied.

In conclusion, the sequence (ƒ: (-1, 1)→ R} converges uniformly to the function f: (-1, 1)→ R, each function f: (-1, 1)→ R is differentiable except for the limit function, and this observation does not contradict Theorem 9.19.

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It will take approximately 7 years for the value of the building to be $641,800.

To find the number of years it takes for the value of the building to reach $641,800, we need to set up the equation:

835,000 - 2,300x = 641,800

Let's solve this equation to find the value of x:

835,000 - 2,300x = 641,800

Subtract 835,000 from both sides:

-2,300x = 641,800 - 835,000

-2,300x = -193,200

Divide both sides by -2,300 to solve for x:

x = -193,200 / -2,300

x ≈ 84

Therefore, it will take approximately 84 months for the value of the building to reach $641,800.

To convert this to years, divide 84 months by 12:

84 / 12 = 7

Hence, it will take approximately 7 years for the value of the building to be $641,800.

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Euler's Method is a numerical technique for solving ordinary differential equations (ODEs) that are first-order.

The method starts with an initial value problem, which is defined by a first-order differential equation and an initial value for the dependent variable. It approximates the solution of the differential equation using a linear approximation of the derivative. A step size is specified, and the method proceeds by approximating the derivative at the current point using the function value and then using the approximated derivative to extrapolate the value of the function at the next point. Use Euler's method with h=0.1 to find approximate values for the solution of the initial value problem

y' + 2y = x³e-2. y(0) = 1.

Using the Euler's method, we first need to create a table to calculate the approximated values for each iteration, as shown below:

Yn f(xn, Yn) Yo Yn+ haf(xn, Yn)XnX

-0.0 1.0000 - -X-0.1 -0.2000 1.0000 + (0.1)(-0.2)(0) -0.0200X-0.2 -0.0680 0.9800 + (0.1)(-0.068)(0.1) 0.0032X-0.3 0.0104 0.9780 + (0.1)(0.0104)(0.2) 0.0236

In conclusion, the approximated values are calculated by using Euler's method with h=0.1. The approximated values are shown in the table, and the method proceeds by approximating the derivative at the current point using the function value and then using the approximated derivative to extrapolate the value of the function at the next point.

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We need the eigenvalue corresponding to the rabbit population, λ = 4, to be the dominant eigenvalue.At time t = 0, there should be 0 foxes (F₀ = 0) in order for the rabbit population to grow as a simple exponential.

In the given system, the eigenvalues of matrix A are λ = 4 and 3. Since λ = 4 is the dominant eigenvalue, it corresponds to the rabbit population growth. To determine the number of foxes needed at time t = 0, we need to find the corresponding eigenvector for the eigenvalue λ = 4. Let's denote the eigenvector for λ = 4 as v = [R₀ F₀].

By solving the equation Av = λv, where A is the coefficient matrix, we get [4 -92; -1140 3] * [R₀; F₀] = 4 * [R₀; F₀]. Simplifying this equation, we obtain 4R₀ - 92F₀ = 4R₀ and -1140R₀ + 3F₀ = 4F₀.

From the first equation, we have -92F₀ = 0, which implies F₀ = 0. Therefore, at time t = 0, there should be 0 foxes (F₀ = 0) in order for the rabbit population to grow as a simple exponential.

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The tip of the woman's shadow is moving at a rate of 2 ft/sec when she is 45 feet from the base of the pole, confirming the given information.

Let's consider the situation and set up a right triangle. The height of the pole is 18 feet, and the height of the woman is 6 feet. As the woman walks away from the pole, her shadow is cast on the ground, forming a similar triangle with the pole. Let the length of the shadow be x.

By similar triangles, we have the proportion: (6 / 18) = (x / (x + 45)). Solving for x, we find that x = 15. Therefore, when the woman is 45 feet from the base of the pole, her shadow has a length of 15 feet.

To find the rate at which the tip of the shadow is moving, we can differentiate the above equation with respect to time: (6 / 18) dx/dt = (x / (x + 45)) d(x + 45)/dt. Plugging in the given values, we have (2 / 3) dx/dt = (15 / 60) d(45)/dt. Solving for dx/dt, we find that dx/dt = (2 / 3) * (15 / 60) * 2 = 2 ft/sec.

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The correct particular solution that satisfies the given initial value conditions for the homogeneous second-order linear differential equation y" + 2y + y = 0 is option (d) y(x) = 4sin(x) + 6cos(x).

To determine the particular solution, we first find the complementary solution to the homogeneous equation, which is obtained by setting the right-hand side of the equation to zero. The complementary solution for y" + 2y + y = 0 is given by y_c(x) = c1e^(-x) + c2xe^(-x), where c1 and c2 are constants.

Next, we find the particular solution that satisfies the initial value conditions. From the given initial values y(0) = -4 and y'(0) = 2, we substitute these values into the general form of the particular solution. After solving the resulting system of equations, we find that c1 = 4 and c2 = 6, leading to the particular solution y_p(x) = 4sin(x) + 6cos(x).

Therefore, the complete solution to the differential equation is y(x) = y_c(x) + y_p(x) = c1e^(-x) + c2xe^(-x) + 4sin(x) + 6cos(x). The correct option is (d), y(x) = 4sin(x) + 6cos(x).

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Answer:

No, y = x  6 is not an inverse variation

Step-by-step explanation:

In Maths, inverse variation is the relationships between variables that are represented in the form of y = k/x, where x and y are two variables and k is the constant value. It states if the value of one quantity increases, then the value of the other quantity decreases.

The derivative of the given function y = (7√x + 4)x² can be found using the Product Rule. The correct answer is OB. The derivative is (7√x + 4)x² + 2.

To apply the Product Rule, we differentiate each term separately and then add them together. Let's break down the function into its two parts: u = 7√x + 4 and v = x².

First, we find the derivative of u with respect to x:

du/dx = d/dx(7√x + 4)

To differentiate 7√x, we use the Chain Rule. Let's set w = √x, then u = 7w:

du/dw = d/dw(7w) = 7

dw/dx = d/dx(√x) = (1/2)(x^(-1/2)) = (1/2√x)

du/dx = (du/dw)(dw/dx) = 7(1/2√x) = 7/(2√x)

Next, we find the derivative of v with respect to x:

dv/dx = d/dx(x²) = 2x

Now, we can apply the Product Rule: (u * v)' = u'v + uv'.

dy/dx = [(7/(2√x))(x²)] + [(7√x + 4)(2x)]

= (7x²)/(2√x) + (14x√x + 8x)

Simplifying the expression, we get:

dy/dx = (7x²)/(2√x) + 14x√x + 8x

= (7√x)(x²)/(2) + 14x√x + 8x

= (7√x)(x²)/2 + 14x√x + 8x

Therefore, the derivative of the function y = (7√x + 4)x² is (7√x)(x²)/2 + 14x√x + 8x, which corresponds to option OB.

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Answer: The selling price of 8 apples for a rupee will give a 20% profit.

Step-by-step explanation: To find the cost price of each apple, you can use the formula: Cost price = Selling price / Quantity. To find the selling price that will give a 20% profit, use the formula: Selling price = Cost price + Profit.

- Lizzy ˚ʚ♡ɞ˚

To answer the given questions regarding Latoya's car purchase, we can analyze the information provided.

(1) The letters used in the simple interest formula I = Prt are:

I represents the interest amount.

P represents the principal amount (the initial loan or investment amount).

r represents the interest rate (expressed as a decimal).

t represents the time period (in years).

(2) To find the interest amount, we can use the formula I = Prt, where:

P is the principal amount ($17,500),

r is the interest rate (8% or 0.08),

t is the time period (3 years).

Using the formula, we can calculate:

I = 17,500 * 0.08 * 3 = $4,200.

Therefore, the interest amount is $4,200.

(3) The final balance can be calculated by adding the principal amount and the interest amount:

Final balance = Principal + Interest = $17,500 + $4,200 = $21,700.

Therefore, the final balance is $21,700.

(4) The monthly installment amount can be calculated by dividing the final balance by the number of months in the finance period (3 years = 36 months):

Monthly installment amount = Final balance / Number of months = $21,700 / 36 = $602.78 (rounded to two decimal places).

Therefore, the monthly installment amount is approximately $602.78.

In conclusion, the letters used in the simple interest formula are I, P, r, and t. The interest amount is $4,200. The final balance is $21,700. The monthly installment amount is approximately $602.78.

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