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Find the exact values of the following, giving your answers as fractions
a) 4¹
b) 2³
c) 3

Answer:

Option 2 is better (pls give brainliest lol!)

Step-by-step explanation:

To determine which is a better deal, we can compare the cost per mango for each option.

Option 1: $5.00 for 4 mangoes

Cost per mango = $5.00/4 = $1.25

Option 2: $6.00 for 5 mangoes

Cost per mango = $6.00/5 = $1.20

Based on the calculations, we can see that Option 2 has a lower cost per mango, making it the better deal. Therefore, buying 5 mangoes for $6.00 is a better deal than buying 4 mangoes for $5.00.

Answer:22

Step-by-step explanation:

First you put

40/100

and that makes

11/22

log₂(7) + log₂(8) is equal to log₂(56).

Logarithmic is a mathematical concept that is used to describe the relationship between a number and its exponent. In particular, a logarithm is the power to which a base must be raised to produce a given number. For example, if we have a base of 2 and a number of 8, the logarithm (base 2) of 8 is 3, since 2 raised to the power of 3 equals 8.

Logarithmic functions are commonly used in mathematics, science, and engineering to describe exponential growth and decay, as well as to solve various types of equations. They are particularly useful in dealing with large numbers, as logarithms allow us to express very large or very small numbers in a more manageable way.

The logarithmic function is typically denoted as log(base a) x, where a is the base and x is the number whose logarithm is being taken. There are several different bases that are commonly used, including base 10 (common logarithm), base e (natural logarithm), and base 2 (binary logarithm). The properties of logarithmic functions, including rules for combining and simplifying logarithmic expressions, are well-defined and widely used in mathematics and other fields.

We can use the logarithmic rule that states that the sum of the logarithms of two numbers is equal to the logarithm of the product of the two numbers. That is,

log₂(7) + log₂(8) = log₂(7 × 8)

Now we can simplify the product of 7 and 8 to get:

log₂(7) + log₂(8) = log₂(56)

Therefore, log₂(7) + log₂(8) is equal to log₂(56).

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There are [tex](1/2) * (2^{100} - 2)[/tex] partitions of {1, 2, ..., 100} into two parts.

We can use the following formula:

Number of partitions = (n choose k)/2, where n is the total number of elements, and k is the number of elements in one of the two parts.

In this case, we want to divide the set {1, 2, ..., 100} into two parts, each with k elements.

Since we are not distinguishing between the two parts, we divide the total number of partitions by 2.

The number of ways to choose k elements from a set of n elements is given by the binomial coefficient (n choose k).

So the number of partitions of {1, 2, ..., 100} into two parts is:

(100 choose k)/2

where k is any integer between 1 and 99 (inclusive).

To find the total number of partitions, we need to sum this expression for all values of k between 1 and 99:

Number of partitions = (100 choose 1)/2 + (100 choose 2)/2 + ... + (100 choose 99)/2

This is equivalent to:

Number of partitions = (1/2) * ([tex]2^{100}[/tex] - 2)

Therefore, there are (1/2) * ([tex]2^{100][/tex] - 2) partitions of {1, 2, ..., 100} into two parts.

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Answer:

perimeter= 96feet

area= 470 feet ^2

Step-by-step explanation:

to find the perimeter u add all the sides together

top missing side= 10feet

right missing side= 19feet

perimeter= 28+20+10+9+10+19

perimeter= 96 feet

area= 20×19=380

9×10=90

area=380+90

area=470 feet^2

Answer: Top box : 10 Ft. , Side box: 21 Ft. , Area: 510 Ft^2, Perimeter: 98 Ft.

Step-by-step explanation:

- Think about it as two shapes. A smaller rectangle that the sheep is in and a larger one with the rest of the pen. Doing this visually will help.

Top box:

20-10 = 10

- We minus 10 feet from 200 because we are dealing with the 'smaller' shape first, to find the length of its missing side we must subtract the known lengths; we removed the excess.

Side box:

28-9=21

- We do this because 28 Ft was a whole length from end to end when we only need the bigger shape, hence we remove the excess which is 9 Ft.

Area:

-Now we know all our lengths, deal with the two self-allocated 'shapes' as you would normally.

10 x 9 = 90. (Smaller shape.)

20 x 21 = 420. (Larger shape.)

- Then we add them to find the area of the WHOLE shape combined.

90 + 420 = 410 FT²

Perimeter:

- Once again, we know all our lengths and simply add them all together.

10 + 28 + 20 + 21 + 10 + 9 = 98 FT

If this helped, consider dropping a thanks ! Have a good day !

The missing number is 7/13.

We have the Sequence,

2/3, 4/7, x, 11/21, 16/31

As, the sequence in Numerator are +2, +3, +4, +5,

and, the sequence of denominator are 4, 6, 8 and 10.

Then, the numerator of missing fraction is

= 4 +3 = 7

and, denominator = 7 + 6 =13

Thus, the required number is 7/13.

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Answer:

1,837

Step-by-step explanation:

From the given information, we know that {an} is a sequence of positive terms, so all of its terms are greater than 0. We also know that sn = m∑ ak, which means that sn is a sum of a finite number of positive terms.

Now, let's look at the given limit: lim an+1 = 0 as n approaches infinity. This tells us that the terms of {an} must approach 0 as n approaches infinity since the limit of an+1 is dependent on the limit of an. Therefore, we can conclude that {an} is a decreasing sequence of positive terms. Using this information, we can determine the following:- ak must converge: Since {an} is decreasing and positive, we know that the terms of {ak} are also decreasing and positive. Therefore, {ak} must converge by the Monotone Convergence Theorem. - ak must converge to 0: Since {an} approaches 0 as n approaches infinity, we know that the terms of {ak} must also approach 0. Therefore, {ak} must converge to 0.
- {sn} must be bounded: Since {ak} converges to 0, we know that there exists some N such that ak < 1 for all n > N. Therefore, sn < m(N-1) + m for all n > N. This shows that {sn} is bounded above by some constant.
- {sn} is monotonic: Since {an} is decreasing and positive, we know that {ak} is also decreasing and positive. Therefore, sn+1 = sn + ak+1 < sn, which shows that {sn} is a decreasing sequence. - limn→∞ sn does not exist: Since {an} approaches 0 as n approaches infinity, we know that {sn} approaches a finite limit if and only if {ak} approaches a nonzero limit. However, we know that {ak} approaches 0, so {sn} does not approach a finite
Therefore, the correct answers
- ak must converge
- ak must converge to 0
- {sn} must be bounded
- {sn} is monotonic
- limn→∞ sn does not exist

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The average slope of f(x) on the interval [-6,4] is equal to f'(3.5) = -12(3.5) = -42.

The Mean Value Theorem (MVT) for a function f(x) on the interval [a,b] states that there exists a point c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).

In this problem, we are asked to find the average slope of the function f(x) = 3 - 6x² on the interval [-6,4]. The average slope is:

(f(4) - f(-6))/(4 - (-6)) = (3 - 6(4)² - (3 - 6(-6)²))/(4 + 6) = -42

So, we need to find a point c in (-6,4) such that f'(c) = -42. The derivative of f(x) is:

f'(x) = -12x

Setting f'(c) = -42, we get:

-12c = -42

c = 3.5

Therefore, the point c = 3.5 satisfies the conditions of the Mean Value Theorem, and the average slope of f(x) on the interval [-6,4] is equal to f'(3.5) = -12(3.5) = -42.

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The decay rate of f is approximately 4.0822% per unit of x.

To change [tex]f(x) = 40(0.96)^x[/tex] to an exponential function with base e, we can use the fact that:

[tex]e^{ln(a)} = a[/tex], where a is a positive real number.

First, we can rewrite 0.96 as[tex]e^{ln(0.96)}[/tex]:

[tex]f(x) = 40(e^{ln(0.96)})^x[/tex]

Then, we can use the property of exponents to simplify this expression:

[tex]f(x) = 40e^{(x*ln(0.96))}[/tex]

This is an exponential function with base e.

To approximate the decay rate of f, we can look at the exponent x*ln(0.96).

The coefficient of x represents the rate of decay. In this case, the coefficient is ln(0.96).

Using a calculator, we can approximate ln(0.96) as -0.040822. This means that the decay rate of f is approximately 4.0822% per unit of x.

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The n(l ∪ m) = 950. This can also be said as the size of the union of sets l and m is 950.

In the question, we have

n(l) = 750, n(m) = 230 and n(l ∩ m) = 30,

To find n(l ∪ m), we need to add the number of elements in both sets, but since they have some overlap n(l ∩ m), we need to subtract that overlap to avoid counting those elements twice.

n(l ∪ m) = n(l) + n(m) - n(l ∩ m)

Substituting the given values, we get:

n(l ∪ m) = 750 + 230 - 30
n(l ∪ m) = 950

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The P-value for the given test statistic, z0 = -2.33, in a one-tailed hypothesis test with H0: µ = 11 and H1: µ < 11 is approximately 0.01.


1. Identify the null hypothesis (H0) and alternative hypothesis (H1). In this case, H0: µ = 11 and H1: µ < 11.

2. Determine the test statistic. Here, z0 = -2.33.

3. Since H1: µ < 11, we are performing a one-tailed test (left-tailed).

4. Look up the corresponding P-value for z0 = -2.33 using a standard normal (Z) table or an online calculator.

5. In a standard normal table, find the row and column corresponding to -2.3 and 0.03, respectively. The intersection gives the value 0.0099, which is approximately 0.01.

6. The P-value is about 0.01, which represents the probability of observing a test statistic as extreme or more extreme than z0 = -2.33 under the null hypothesis.

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For Problem 9, the characteristic equation is r² + 3 = 0, which has roots r = +/- i*sqrt(3).

Since this is a nonhomogeneous equation with a constant on the right-hand side, we guess a particular solution of the form y_p = A, where A is a constant. Plugging this into the differential equation, we get A = -3, so our particular solution is y_p = -3.

For Problem 10, the characteristic equation is r² + 2r - 1 = 0, which has roots r = (-2 +/- sqrt(8))/2 = -1 +/- sqrt(2).

Again, this is a nonhomogeneous equation with a constant on the right-hand side, so we guess a particular solution of the form y_p = B, where B is a constant. Plugging this into the differential equation, we get B = 10/3, so our particular solution is y_p = 10/3.

For Problem 11, the characteristic equation is r^2 + 1 = 0, which has roots r = +/- i.

This is a nonhomogeneous equation with a constant on the right-hand side, so we guess a particular solution of the form y_p = C, where C is a constant. Plugging this into the differential equation, we get C = 2, so our particular solution is y_p = 2.

For Problem 12, this is a first-order differential equation, so we can use the method of integrating factors.

The integrating factor is e^int(1/2, dx) = e^(x^2/4), so we multiply both sides of the equation by e^(x^2/4) to get (e^(x^2/4) x)' = 312 e^(x^2/4). Integrating both sides with respect to x, we get e^(x^2/4) x = 312/2 int(e^(x^2/4), dx) = 156 e^(x^2/4) + C, where C is a constant of integration. Solving for x, we get x = 156 e^(-x^2/4) + Ce^(-x^2/4). This is our particular solution.

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The partial derivatives of u with respect to x, y, and t are, [tex]\dfrac{\partial u}{\partial x}[/tex] = 22, [tex]\dfrac{\partial u}{\partial y}[/tex] = 18 and [tex]\dfrac{\partial u}{\partial t}[/tex] = 40.

We can use the chain rule to find the partial derivatives of u with respect to x, y, and t.

First, we will find the partial derivative of u with respect to r and s:

u = r² + s²

[tex]\dfrac{\partial u}{\partial r}[/tex] = 2r

[tex]\dfrac{\partial u}{\partial s}[/tex] = 2s

Next, we will find the partial derivatives of r with respect to x, y, and t:

r = y + xcos(t)

[tex]\dfrac{\partial r}{\partial x}[/tex] = cos(t)

[tex]\dfrac{\partial r}{\partial y}[/tex] = 1

[tex]\dfrac{\partial r}{\partial t}[/tex] = -xsin(t)

Similarly, we will find the partial derivatives of s with respect to x, y, and t:

s = x + ysin(t)

[tex]\dfrac{\partial s}{\partial x}[/tex] = 1

[tex]\dfrac{\partial s}{\partial y}[/tex] = sin(t)

[tex]\dfrac{\partial s}{\partial t}[/tex] = ycos(t)

Now, we can use the chain rule to find the partial derivatives of u with respect to x, y, and t:

[tex]\dfrac{\partial u}{\partial x} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial x} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial x}[/tex]

[tex]\dfrac{\partial u}{\partial x}[/tex] = 2r * cos(t) + 2s * 1

At x = 4, y = 5, t = 0, we have:

r = 5 + 4cos(0) = 9

s = 4 + 5sin(0) = 4

Substituting these values into the partial derivative formula, we get:

[tex]\dfrac{\partial u}{\partial x}[/tex] = 2(9)(1) + 2(4)(1) = 22

Similarly, we can find the partial derivatives with respect to y and t:

[tex]\dfrac{\partial u}{\partial y} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial y} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial y}[/tex]

[tex]\dfrac{\partial u}{\partial y}[/tex] = 2r * 1 + 2s * sin(t)

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2(9)(1) + 2(4)(0) = 18

[tex]\dfrac{\partial u}{\partial t} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial t} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial t}[/tex]

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2r * (-xsin(t)) + 2s * (ycos(t))

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2(9)(-4sin(0)) + 2(4)(5cos(0)) = 40

Therefore, the partial derivatives of u with respect to x, y, and t are:

[tex]\dfrac{\partial u}{\partial x}[/tex] = 22

[tex]\dfrac{\partial u}{\partial y}[/tex] = 18

[tex]\dfrac{\partial u}{\partial t}[/tex] = 40

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The number of years it will  take the schools to have the same enrollment is 4 years.

We are given that;

The enrollment at high school R has been increasing by 20 students per year.

Currently high school R has 200 students attending.

High school T currently has 400 students, but it’s enrollment is decreasing in size by an average of 30 students per year.

Let x be the number of years from now, and y be the enrollment of the schools. Then we have:

y=200+20x

for high school R, and

y=400−30x

for high school T. To find when the schools have the same enrollment, we set the two equations equal to each other and solve for x:

200+20x=400−30x

Adding 30x to both sides, we get:

50x=200

Dividing both sides by 50, we get:

x=4

At that time, they will both have y = 200 + 20(4) = 280 students.

Therefore, by the linear equation the answer will be 4 years.

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Okay, let's think this through step-by-step:

* A to D is 1 km

* B to C is 2 km

* B to D is 3 km

* A to B is 4 km

* C to D is 5 km

So we have:

A -> B = 4 km

B -> C = 2 km

C -> D = 5 km

We want to find A -> C.

A -> B is 4 km

B -> C is 2 km

So A -> C = 4 + 2 = 6 km

Therefore, the distance between stops A and C is 6 km.

The length of line AX is 3p/4q.

The length of side AY is  9p²/4q + 3p/4.

The length of line AX is calculated as follows;

From the given figure, we can apply the principle of congruent sides of the parallellogram.

AD/DC = CX/AX

8q/6p = 1/AX

AX = 6p/8q

AX = 3p/4q

The length of side AY is calculated by applying the following formula as shown below.

Apply similar principle of congruent sides;

AX/CX = AY/CY

3p/4q / 1 = AY/(3p + q)

AY = 3p/4q(3p + q)

AY = 9p²/4q + 3p/4

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The inverse Laplace transform of f(s) is: f(t) = -2t*u(t)[tex]e^{-t}[/tex] - 4u(t)[tex]e^{-t}[/tex]+ 4u(t)

What is convolution theorem?

The convolution theorem is a fundamental result in mathematics and signal processing that relates the convolution operation in the time domain to multiplication in the frequency domain.

To find the inverse Laplace transform of the given function, we will use the convolution theorem, which states that the inverse Laplace transform of the product of two functions is the convolution of their inverse Laplace transforms.

We can rewrite the given function as:

f(s) = 1/(s+1)² * (s² + 4)

Taking the inverse Laplace transform of both sides, we get:

[tex]L^{-1}[/tex]{f(s)} = [tex]L^{-1}[/tex]{1/(s+1)²} *[tex]L^{-1}[/tex]{s² + 4}

We can use partial fraction decomposition to find the inverse Laplace transform of 1/(s+1)²:[tex]e^{-t}[/tex]

1/(s+1)² = d/ds(-1/(s+1))

Thus, [tex]L^{-1}[/tex]{1/(s+1)²} = -t*[tex]e^{-t}[/tex]

To find the inverse Laplace transform of s²+4, we can use the table of Laplace transforms and the property of linearity of the Laplace transform:

L{[tex]t^{n}[/tex]} = n!/[tex]s^{(n+1)}[/tex]

L{4} = 4/[tex]s^{0}[/tex]

[tex]L^{-1}[/tex]{s² + 4} = L^-1{s²} + [tex]L^{-1}[/tex]{4} = 2*d²/dt²δ(t) + 4δ(t)

Now, we can use the convolution theorem to find the inverse Laplace transform of f(s):

[tex]L^{-1}[/tex]{f(s)} = [tex]L^{-1}[/tex]{1/(s+1)²} * [tex]L^{-1}[/tex]{s² + 4} = (-te^(-t)) * (2d²/dt²δ(t) + 4δ(t))

Simplifying this expression, we get:

[tex]L^{-1}[/tex]{f(s)} = -2[tex]te^{-t}[/tex]δ''(t) - 4[tex]te^{-t}[/tex]δ'(t) + 4[tex]e^{-t}[/tex]δ(t)

Therefore, the inverse Laplace transform of f(s) is:

f(t) = -2t*u(t)[tex]e^{-t}[/tex] - 4u(t)[tex]e^{-t}[/tex]+ 4u(t).

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Answer:

10,11

Step-by-step explanation:

Givne set: {7, 8 , 9 , 10 , 11}

To solve the inequality, isolate 'h'.

        2h - 3 > 15

Add 3 to both sides,

     2h - 3 + 3 > 15 + 3

               2h  > 18

Divide both sides by 2,

                [tex]\sf \dfrac{2h}{2} > \dfrac{18}{2}[/tex]

                 h > 9

h = {10 , 11}

The integral along the actual process path will not accurately represent the maximum possible entropy change for the system.

To determine the entropy change for an irreversible process between states 1 and 2, the integral ∫1 2 δq/t should be performed along an imaginary reversible path. This is because entropy is a state function and is independent of the path taken to reach a particular state. Therefore, the entropy change between two states will be the same regardless of whether the process is reversible or irreversible.

However, performing the integral along an imaginary reversible path will give a more accurate measure of the entropy change as it represents the maximum possible work that could have been obtained from the system. In contrast, an irreversible process will always result in a lower amount of work being obtained due to losses from friction, heat transfer to the surroundings, and other factors.

Therefore, performing the integral along the actual process path will not accurately represent the maximum possible entropy change for the system.

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the volume percent of ethylene glycol in the solution is 30.25%.

Why is it?

To calculate the volume percent of ethylene glycol in the solution, we need to know the volume of ethylene glycol and the total volume of the solution.

Given:

Volume of ethylene glycol = 357 ml

Total volume of solution = 1.18 × 10²3 ml

The volume percent of ethylene glycol is calculated as:

Volume percent = (volume of ethylene glycol / total volume of solution) x 100%

Volume percent = (357 ml / 1.18 × 10²3 ml) x 100%

Volume percent = 30.25%

Therefore, the volume percent of ethylene glycol in the solution is 30.25%.

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There will be 2,600 prize-winning opportunities in a year for all 200 members combined.

Assuming that the 200 Club follows the rules and conducts all the draws specified, there will be a total of 13 prize-winning opportunities in a year for each member.

The breakdown of the prize-winning opportunities is as follows:

Monthly draws: There are 12 monthly draws in a year, and each draw has 3 prizes - a first prize of £15, a second prize of £5, and a main prize of £20. Therefore, there are 36 prize-winning opportunities in total for the monthly draws.

Annual Grand draw: There is one annual grand draw, which has 2 prizes - a first prize of £50 and a second prize of £30.

So, for each member, there will be 13 prize-winning opportunities in a year - 12 monthly draws and 1 annual grand draw. However, it is important to note that each member can only win one prize per monthly draw, and their card remains valid for the entire year even if they have won a prize already.

Therefore, in total, there will be 2,600 prize-winning opportunities in a year for all 200 members combined (13 prize-winning opportunities per member multiplied by 200 members).

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The derivatives are:

[tex]y' = (2x + 4y) / (4x - 6y)[/tex]

[tex]y'' = [(4x - 6y)(2 + 4((2x + 4y) / (4x - 6y))) - (2x + 4y)(4 - 6((2x + 4y) / (4x - 6y)))] / (4x - 6y)^2[/tex]

To find y' and y'' for the given equation x^2 + 4xy - 3y^2 = 8, follow these steps:

Step 1: Differentiate both sides of the equation with respect to x.
For the left side, use the product rule for 4xy and the chain rule for -3y^2.
[tex]d(x^2)/dx + d(4xy)/dx - d(3y^2)/dx = d(8)/dx[/tex]

Step 2: Calculate the derivatives.
[tex]2x + 4(dy/dx * x + y) - 6y(dy/dx) = 0[/tex]

Step 3: Solve for y'.
Rearrange the equation to isolate dy/dx (y'):
[tex]y' = (2x + 4y) / (4x - 6y)[/tex]

Step 4: Differentiate y' with respect to x to find y''.
Use the quotient rule: [tex](v * du/dx - u * dv/dx) / v^2[/tex],

where u = (2x + 4y) and v = (4x - 6y).
[tex]y'' = [(4x - 6y)(2 + 4(dy/dx)) - (2x + 4y)(4 - 6(dy/dx))] / (4x - 6y)^2[/tex]

Step 5: Substitute y' back into the equation for y''.
[tex]y'' = [(4x - 6y)(2 + 4((2x + 4y) / (4x - 6y))) - (2x + 4y)(4 - 6((2x + 4y) / (4x - 6y)))] / (4x - 6y)^2[/tex]

This is the expression for y'' in terms of x and y.

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The area of the other octagon is 75 square inches.

To find the area of the other octagon, we can use the concept of scale factors. The scale factor of 4:5 tells us that corresponding lengths of the two similar octagons are in a ratio of 4:5.

Since the scale factor applies to the lengths, it will also apply to the areas of the two octagons. The area of a shape is proportional to the square of its corresponding side length.

Let's assume the area of the other octagon (with the scale factor of 4:5) is A.

The ratio of the areas of the two octagons can be expressed as:

(Area of the given octagon) : A = (Side length of the given octagon)^2 : (Side length of the other octagon)^2

48 : A = (4/5)^2

48 : A = 16/25

Cross-multiplying:

25 * 48 = 16A

1200 = 16A

Dividing both sides by 16:

75 = A

Therefore, the area of the other octagon is 75 square inches.

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Answer:

Set your calculator to degree mode.

[tex] { \sin }^{ - 1} \frac{18}{20} = 64 [/tex]

So theta measures approximately 64 degrees.

The derivative of the function f(x) = (9[tex]x^{6}[/tex] + 8x³)³ is f'(x) = 4(9[tex]x^{6}[/tex] + 8x³)³(54x³ + 24x²).

To find the derivative of the function f(x) = (9x² + 8x³)³, you need to apply the Chain Rule. The Chain Rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, let u = 9x + 8x.

First, find the derivative of the outer function with respect to u: d( u³ )/du = 4u³.
Next, find the derivative of the inner function with respect to x: d(9x² + 8x³)/dx = 54x³ + 24x².

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In a binary tree with leaves l1, l2, ..., lM at depths d1, d2, ..., dM respectively, the sum of [tex]2^-^d^_i[/tex] for all leaves is always less than or equal to 1: Σ  [tex]2^-^d^_i[/tex] <= 1.

In a binary tree, each leaf node is reached by following a unique path from the root. Since it is a binary tree, each internal node has two child nodes.

Consider a full binary tree, where all leaves have the maximum number of nodes at each depth. For a full binary tree, the total number of leaves is  [tex]2^d[/tex] , where d is the depth.

Each leaf node contributes [tex]2^-^d[/tex] to the sum. Thus, the sum for a full binary tree is Σ  [tex]2^-^d[/tex] = (2⁰ + 2⁰ + ... + 2⁰) = [tex]2^d[/tex] * [tex]2^-^d[/tex]  = 1. Now, if we remove any node from the full binary tree, the sum can only decrease, as we are reducing the number of terms in the sum. Hence, for any binary tree, the sum Σ [tex]2^-^d^_i[/tex]  will always be less than or equal to 1.

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The length of the curve L is [tex]\int\limits^3_{-2} {\sqrt{1+(15x^4+15)^2} } \, dx[/tex] where a = -2, b= 3 and f'(x) =  [tex]15x^4[/tex] + 15.

To find the length of the curve defined by y = [tex]3x^5[/tex] + 15x from the point (-2, -126) to the point (3, 774), you'd actually need to compute the arc length using the formula:
L = [tex]\int\limits^b_a {\sqrt{(1+(f'(x)^2)} } \, dx[/tex]
First, find the derivative of the function, f'(x):
f'(x) = d([tex]3x^5[/tex] + 15x)/dx = [tex]15x^4[/tex] + 15
Now, substitute f'(x) into the arc length formula:
L = [tex]\int\limits^b_a {\sqrt{(1+(15x^4+15)^2)} } \, dx[/tex]
Here, the points given are (-2, -126) and (3, 774). Therefore, the limits of integration are:
a = -2
b = 3
So the final integral to compute the length of the curve is:
L = [tex]\int\limits^3_{-2} {\sqrt{1+(15x^4+15)^2} } \, dx[/tex]

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The length of the curve L is [tex]\int\limits^3_{-2} {\sqrt{1+(15x^4+15)^2} } \, dx[/tex] where a = -2, b= 3 and f'(x) =  [tex]15x^4[/tex] + 15.

To find the length of the curve defined by y = [tex]3x^5[/tex] + 15x from the point (-2, -126) to the point (3, 774), you'd actually need to compute the arc length using the formula:
L = [tex]\int\limits^b_a {\sqrt{(1+(f'(x)^2)} } \, dx[/tex]
First, find the derivative of the function, f'(x):
f'(x) = d([tex]3x^5[/tex] + 15x)/dx = [tex]15x^4[/tex] + 15
Now, substitute f'(x) into the arc length formula:
L = [tex]\int\limits^b_a {\sqrt{(1+(15x^4+15)^2)} } \, dx[/tex]
Here, the points given are (-2, -126) and (3, 774). Therefore, the limits of integration are:
a = -2
b = 3
So the final integral to compute the length of the curve is:
L = [tex]\int\limits^3_{-2} {\sqrt{1+(15x^4+15)^2} } \, dx[/tex]

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The functions $y =[tex]e^{-4x}[/tex]$ and $y = [tex]e^{2x}[/tex] $ are solutions to the differential equation $y'' + 2y' - 8y = 0$.

To see if the function $y = e^{rx}$ is a solution to the differential equation $y'' + 2y' - 8y = 0$, we substitute it in place of $y$ and its derivatives:

y=[tex]e^{rx}[/tex]

y' = [tex]re^{rx}[/tex]

y" = [tex]r^{2} e^{rx}[/tex]

Substituting these expressions into the differential equation, we get:

[tex]r^{2} e^{rx} + 2re^{rx} - 8e^{rx} = 0[/tex]

Dividing both sides by $ [tex]$e^{rx}$[/tex] $, we get:

[tex]r^{2} + 2r - 8 = 0[/tex]

This is a quadratic equation in $r$. Solving for $r$, we get:

r = -4,2

Therefore, the functions $y =[tex]e^{-4x}[/tex]$ and $y = [tex]e^{2x}[/tex] $ are solutions to the differential equation $y'' + 2y' - 8y = 0$.

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