express dw / dt for w=x^2 -y , x=cos(t) , y=sin(t)

The shape and size of the figure are unchanged. and other soltuions are below

The rule for translating a point 4 units left and 8 units up is (x, y) → (x - 4, y + 8).

Using the above rule

After the translation, the new coordinates of A are (3, 13).

The rule for reflecting a point over the y-axis is (x, y) → (-x, y).

Using the above rule

After the reflection, the coordinates of A become (-7, 5).

The final figure is congruent to the original figure because translation and reflection are both rigid transformations, which preserve distance and angles between points.

Therefore, the shape and size of the figure are unchanged.

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If the curve x2/3+y2/3=4 and Let L be the tangent line to this curve at the point (1,3√3), and let A and B be the x- and y-intercepts of L, then the length of line segment AB is √38 (option b).

Explanation:

To find the length of line segment AB, follow these steps:

Step 1: We need to first find the equation of tangent line L at the given point (1, 3√3) on the curve x^(2/3) + y^(2/3) = 4.

Step 1. Find the derivative dy/dx using implicit differentiation:
(2/3)x^(-1/3) + (2/3)y^(-1/3)(dy/dx) = 0

Step 2. Solve for dy/dx (the slope of tangent line L) at point (1, 3√3):
(2/3)(1)^(-1/3) + (2/3)(3√3)^(-1/3)(dy/dx) = 0
dy/dx = -1/2

Step 3. Use the point-slope form to find the equation of tangent line L:
y - 3√3 = -1/2(x - 1)

Step 4. Find the x- and y-intercepts (A and B) of line L:
x-intercept (A): y = 0
0 - 3√3 = -1/2(x - 1)
x = 2 + 6√3

y-intercept (B): x = 0
y - 3√3 = -1/2(0 - 1)
y = 3√3 + 1/2

Step 5. Calculate the length of the line segment AB using the distance formula:
AB = √[(2 + 6√3 - 0)^2 + (3√3 + 1/2 - 0)^2] = √38

The length of line segment AB is √38 (option b).

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The singular, affirmative and formal command for each of the following are as follows: 1.tener: tenga, 2.conocer: conozca 3.buscar:busque 4.ir:vaya 5.ser:sea

What Spanish Affirmative and Negative commands?

The indicative, subjunctive, and imperative verb moods are the three primary categories of verb moods in Spanish.

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The height of the pillar is 14 feet.

We can use the ratio of the building's height to the length of its shadow to solve for the pillar's height and apply it to the pillar as well. This is because similar-shaped objects will produce shadows that are proportional to their size.

Let h represent the pillar's height, and let's establish a proportion:

height of building / length of building's shadow = height of pillar / length of pillar's shadow

Substituting the given values:

70 / 50 = h / 10

We can simplify this proportion by cross-multiplying:

70 x 10 = 50h

700 = 50h

And solving for h:

h = 700 / 50 = 14

Therefore, the height of the pillar is 14 feet.

In conclusion, we can solve for the pillar's height by establishing a ratio between the building's height and the length of its shadow and applying that ratio to the pillar.

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Using De Moivre's theorem, we have: z = r^(1/10) * (cos(θ/10) + i*sin(θ/10)). As we are looking for 10 roots, we need to find each root by varying k from 0 to 9: z_k = (√910)^(1/10) * (cos(2πk/10) + i*sin(2πk/10))Substitute k from 0 to 9 to obtain all the real and complex roots of the equation z^10 = 910.

To find all real and complex roots of the equation z^10 = 910, we can use the polar form of complex numbers. First, we can write 910 in polar form: 910 = 910(cos(0) + i sin(0)) Next, we can express z in polar form as well: z = r(cos(θ) + i sin(θ)) Substituting these expressions into the equation z^10 = 910 and using De Moivre's Theorem, we get: r^10(cos(10θ) + i sin(10θ)) = 910(cos(0) + i sin(0)) Equating the real and imaginary parts, we get: r^10 cos(10θ) = 910 cos(0) = 910 r^10 sin(10θ) = 910 sin(0) = 0

The second equation gives us two possible values of θ: θ = 0 and θ = π (since sin(π) = 0). For θ = 0, the first equation gives us: r^10 cos(0) = 910 r^10 = 910 r = (910)^(1/10) So one possible solution is z = (910)^(1/10). For θ = π, the first equation gives us: r^10 cos(10π) = 910 r^10 (-1) = 910 r^10 = -910 Since r must be real, this equation has no real solutions.

However, we can find 5 complex solutions by using the 5th roots of -910: (-910)^(1/5) = 2(cos(π/5) + i sin(π/5)) (-910)^(1/5) = 2(cos(3π/5) + i sin(3π/5)) (-910)^(1/5) = 2(cos(5π/5) + i sin(5π/5)) = -2 (-910)^(1/5) = 2(cos(7π/5) + i sin(7π/5)) (-910)^(1/5) = 2(cos(9π/5) + i sin(9π/5)) Using these values of r and θ, we can write the 6 solutions to z^10 = 910 as: z = (910)^(1/10) z = 2(cos(π/5) + i sin(π/5)) z = 2(cos(3π/5) + i sin(3π/5)) z = -2 z = 2(cos(7π/5) + i sin(7π/5)) z = 2(cos(9π/5) + i sin(9π/5))

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let's firstly get the EQUATion, of the graph before we get the inequality.

so we have a quadratic with two zeros, at -6 and 8, hmmm and we also know that it passes through (-2 , 10)

[tex]\begin{cases} x = -6 &\implies x +6=0\\ x = 8 &\implies x -8=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +6 )( x -8 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that } \begin{cases} x=-2\\ y=10 \end{cases}[/tex]

[tex]a ( -2 +6 )( -2 -8 ) = 10\implies a(4)(-10)=10\implies -40a=10 \\\\\\ a=\cfrac{10}{-40}\implies a=-\cfrac{1}{4} \\\\[-0.35em] ~\dotfill\\\\ -\cfrac{1}{4}(x+6)(x-8)=y\implies -\cfrac{1}{4}(x^2-2x-48)=y \\\\\\ ~\hfill {\Large \begin{array}{llll} -\cfrac{x^2}{4}+\cfrac{x}{2}+12=y \end{array}}~\hfill[/tex]

now, hmmm let's notice something, the line of the graph is a solid line, that means the borderline is included in the inequality, so we'll have either ⩾ or ⩽.

so hmmm we could do a true/false region check by choosing a point and shade accordingly, or we can just settle with that, since the bottom is shaded, we're looking at "less than or equal" type, or namely ⩽, so that's our inequality

[tex]{\Large \begin{array}{llll} -\cfrac{x^2}{4}+\cfrac{x}{2}+12\geqslant y \end{array}}[/tex]

the critical numbers of g(t) are approximately -0.690 and 0.690

To find the critical numbers of the function g(t) = t⁵ - t, we need to first find the derivative of the function.

g'(t) = 5t⁴ - 1

Then we set the derivative equal to zero and solve for t:

5t⁴  - 1 = 0
5t⁴ = 1
t⁴ = 1/5
t = [tex]±(1/5)^{(1/4)}[/tex]

However, we need to check if these values are in the domain of the function, which is t < 4.

[tex](1/5)^{(1/4)}[/tex]≈ 0.690, which is less than 4, so it is a valid critical number.
-[tex](1/5)^{(1/4)}[/tex] ≈ -0.690, which is also less than 4, so it is also a valid critical number.

Therefore, the critical numbers of g(t) are approximately -0.690 and 0.690, and we can write them as a comma-separated list:

-0.690, 0.690

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The length of the curve is approximately 4.7 units when rounded to one decimal place.

Explanation:

To find the length of the curve, follow these steps:

Step 1: To find the length of the curve, we need to use the formula:

length = ∫(a to b) √(dx/dt)^2 + (dy/dt)^2 dt

Step 2: In this case, we have x = cos(2t) and y = sin(3t) for 0 ≤ t ≤ 2, First, find the derivatives dx/dt and dy/dt so we can find dx/dt and dy/dt as:

dx/dt = -2sin(2t)
dy/dt = 3cos(3t)

Step 3: Substituting these into the formula, we get:

length = ∫ (0 to 2) √((-2sin(2t))^2 + (3cos(3t))^2) dt

length = ∫ (0 to 2) √(4sin^2(2t) + 9cos^2(3t)) dt
This integral must be evaluated numerically.

Step 4: Using a calculator or software to evaluate the integral numerically, we get:

length ≈ 4.7

Therefore, the length of the curve is approximately 4.7 units when rounded to one decimal place.

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Answer:

The statements that are true are:

6 + 3 = 9 and 4 · 4 = 16 (both are true statements)

5 · 3 = 15 or 7 + 5 = 20 (at least one of these statements is true, since the word "or" means that only one of the two statements needs to be true for the entire statement to be true)

The other two statements are false:

If 6 > 10, then 8 · 3 = 24 (this statement is false, because the premise "6 > 10" is false, and a false premise can never imply a true conclusion)

If 6 · 3 = 18, then 4 + 8 = 20 (this statement is false, because the conclusion "4 + 8 = 20" does not follow logically from the premise "6 · 3 = 18")

Answer: 87

Step-by-step explanation: 3 x 29=87 because you are multiplying the amount of people in the survey by 3

No, W is not a subspace of R3 because it is not closed under vector addition and scalar multiplication, even though it contains the zero vector.

A set must meet three requirements to be a subspace of a vector space: (1) it must include the zero vector, (2) it must be closed under vector addition, and (3) it must be closed under scalar multiplication.

While W includes the zero vector (0, 0, 0), vector addition does not close it. For example, the triples (3, 4, 5) and (5, 12, 13) are both Pythagorean, but their addition (8, 16, 18) is not. As a result, W does not meet the second requirement and is not a subspace of R3.

Under scalar multiplication, W is likewise not closed. When we multiply the Pythagorean triple (3, 4, 5) by -1, we obtain (-3, -4, -5), which is not a Pythagorean triple. Therefore, W does not satisfy the third condition and is not a subspace of R.

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Answer:5/6

Step-by-step explanation: Theres 6 people that you are splitting 5 oranges among. so you do the equation 5÷6 or 5/6 which is the answer your looking for

Answer:

5/6

Step-by-step explanation:

=0.8333.......................

(x - (a+48))² + (y - (b+19))² = 24.5² This is the equation for all possible positions (x,y) on the boundary of the cell tower's service coverage.

To write an equation for all possible positions (x,y) on the boundary of the cell tower's service coverage, we first need to determine the coordinates of the center of the coverage.

We know that the cell tower is located 48 miles east and 19 miles north of the center of the small town, so we can add these distances to the coordinates of the town's center. If we let (a,b) be the coordinates of the town's center, then the coordinates of the cell tower would be (a+48,b+19).

Next, we know that the cell tower services everything within a radius of 24.5 miles from it. This means that any point (x,y) on the boundary of the coverage circle would be exactly 24.5 miles away from the cell tower. We can use the distance formula to write an equation for this:

√[(x - (a+48))² + (y - (b+19))²] = 24.5

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The text is asking for a step-by-step sequence to solve quadratic problems using the quadratic formula.

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one squared term. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and x is the variable. The quadratic formula is used to find the solution(s) of a quadratic equation. The formula is x = (-b ± sqrt(b² - 4ac)) / 2a.

Answer:

Step-by-step explanation:

put your equation into

ax²+bx+c

determine a, b, and c

plug into formula

simplify numbers under the square root first (b²-4ac)

then simplify the root.  ex.  √12 can be simplified to 2√3

then reduce if the bottom can be reduced with both of the terms on top

Answer:

[tex] \frac{y}{28} = \frac{7}{25} [/tex]

[tex]25y = 196[/tex]

[tex]y = 7.84[/tex]

To find the value of probability p that maximizes the variance of a Bernoulli random variable, we need to take the derivative of the variance formula with respect to p and set it equal to 0: d/dp [p(1-p)] = 1-2p = 0.

The value of probability p that maximizes the variance of a Bernoulli random variable is 1/2.The variance of a Bernoulli random variable is given by the formula Var(X) = p(1-p), where p is the probability of success. To find the value of p that maximizes the variance, you can take the derivative of the variance formula with respect to p and set it to zero.d(Var(X))/dp = d(p(1-p))/dp = 1 - 2pSetting the derivative equal to zero:1 - 2p = 0Solving for p:p = 1/2So, the value of probability p that maximizes the variance of a Bernoulli random variable is 0.5 or 1/2.

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The corresponding z-score for a random variable x that is normally distributed with a mean of 395 and a standard deviation of 23, when  x = 35 is:  -15.65

To find the z-score, you can use the following formula:

z = (x - μ) / σ

where z is the z-score, x is the value of the random variable, μ is the mean, and σ is the standard deviation.

Step 1: Identify the values.
x = 35, μ = 395, and σ = 23.

Step 2: Substitute the values into the formula.
z = (35 - 395) / 23

Step 3: Calculate the z-score.
z = (-360) / 23 = -15.65

So, the corresponding z-score for x = 35 is approximately -15.65.

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The resultant matrix represents the vertices of the translated triangle, with A' at (-3, 7), B' at (-2, 0), and C' at (1, 7).

A transformation matrix is a mathematical matrix that represents a geometric transformation of space.

To translate the triangle by (-2, 4), we need to add -2 to the x-coordinates and add 4 to the y-coordinates of each vertex. The transformation matrix for this translation is:

[tex]\left[\begin{array}{ccc}1&0&-2\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}0&1&4\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}0&0&1\end{array}\right][/tex]

To apply this transformation to the vertices of the triangle, we can represent each vertex as a column vector with a third coordinate of 1:

[tex]A=\left[\begin{array}{ccc}-1&3&1\end{array}\right][/tex]

[tex]B=\left[\begin{array}{ccc}0&-4&1\end{array}\right][/tex]

[tex]C=\left[\begin{array}{ccc}3&3&1\end{array}\right][/tex]

To apply the translation, we multiply each vertex by the transformation matrix:

[tex]A'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}-1&3&1\end{array}\right] \left[\begin{array}{ccc}-3&7&1\end{array}\right][/tex]

      [0 1  4] × [ 0 -4  1] = [-2  0  1]

      [0 0  1]   [ 1  3  1]   [-1  7  1]

[tex]B'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}0&-4&1\end{array}\right] \left[\begin{array}{ccc}-2&-4&1\end{array}\right][/tex]

      [0 1  4] × [ 0 -4  1] = [-2   0  1]

      [0 0  1]   [ 0  3  1]   [-2   7  1]

[tex]C'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}3&3&1\end{array}\right] \left[\begin{array}{ccc}1&7&1\end{array}\right][/tex]

      [0 1  4] × [ 3  3  1] = [ 1  7  1]

      [0 0  1]   [ 3  3  1]   [ 1  7  1]

The resultant matrix represents the vertices of the translated triangle, with A' at (-3, 7), B' at (-2, 0), and C' at (1, 7).

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The surface given by the function f(x, y) = y² is a cylinder with rulings parallel to the x-axis. The generating curve is described by z = y². The domain of the function is the entire xy-plane, and the range is z ≥ 0.

The function f(x, y) = y² describes a surface in three-dimensional space. Since the variable x is missing, the surface will not depend on x, and the rulings (lines) of the surface will be parallel to the x-axis. This makes the surface a cylinder with rulings parallel to the x-axis.

The generating curve of the surface is given by z = y², which means that the z-coordinate of any point on the surface is equal to the square of the y-coordinate. This generates a parabolic shape along the y-axis, extending infinitely in the positive and negative y-directions.

The domain of the function is the entire xy-plane, which means that the function is defined for all values of x and y. There are no restrictions on the values of x and y in the domain.

The range of the function is z ≥ 0, which means that the z-coordinate of any point on the surface will always be greater than or equal to zero. This is because the function f(x, y) = y² always produces non-negative values for z, since any real number squared is always non-negative.

Therefore, the surface described by the function f(x, y) = y² is a cylinder with rulings parallel to the x-axis, the generating curve is given by z = y², the domain is the entire xy-plane, and the range is z ≥ 0.

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Answer:

B) 11

Step-by-step explanation:

PEMDAS=parenthesis, exponents, multiplication, division, addition, and subtraction.

First start with the parenthesis:

20 x 1/4=5

9 + (5) - 6 ÷ 2

We don't have exponents or multiplication, so go onto division:

-6 ÷ 2 = -3

9+5-3

Finally addition and subtraction:

9+5-3=11

Hope this helps!

Using the limit comparison test, we determined that the series (A) diverges, (B) converges, (C) diverges, and (D) converges.

We can use the limit comparison test with the series 1/n to determine whether the series converges or diverges:

lim n→∞ (n7 + n2) / n = lim n→∞ (n7/n + n2/n) = ∞

Since this limit diverges to infinity, we cannot use the limit comparison test with the series 1/n. We can try another convergence test.

We can use the limit comparison test with the series 1/n3 to determine whether the series converges or diverges:

lim n→∞ (3n3 − 2n) / (6n5 + 2n + 1) = lim n→∞ (n2 − 2/n2) / (2n5 + 1/n + 1/n5) = 1/2

Since this limit is a positive finite number, the series converges if and only if the series ∑ 1/n^3 converges. Since the p-series with p = 3 converges, the series ∑ (3n^3 - 2n) / (6n^5 + 2n + 1) also converges.

We can use the limit comparison test with the series 1/n to determine whether the series converges or diverges:

lim n→∞ 2n / (4n − n2) = lim n→∞ 2/n(4 − n) = 0

Since this limit is a finite number, the series converges if and only if the series ∑ 1/n converges. Since the harmonic series diverges, the series ∑ 2n / (4n - n^2) also diverges.

We can use the limit comparison test with the series 1/n^2 to determine whether the series converges or diverges:

lim n→∞ sin(1/n) / (1/n^2) = lim n→∞ sin(1/n) * n^2 = 1

Since this limit is a positive finite number, the series converges if and only if the series ∑ 1/n^2 converges. Since the p-series with p = 2 converges, the series ∑ sin(1/n) / n also converges.

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The statement "consider a wave form s(t)=5 sin 10 π t 2 sin 12 π t. the signal s(t) is sampled at 10 hz. do you expect to see aliasing" is true because aliasing is expected.

When sampling a signal s(t) = 5 sin(10πt) * 2 sin(12πt) at 10 Hz, you can expect to see aliasing. The Nyquist sampling theorem states that a signal should be sampled at least twice the highest frequency present in the signal to avoid aliasing.

The two sinusoids in s(t) have frequencies of 5 Hz (10πt) and 6 Hz (12πt). The highest frequency is 6 Hz, so according to the Nyquist theorem, the signal should be sampled at least at 12 Hz (2 times the highest frequency) to avoid aliasing. Since the signal is sampled at 10 Hz, which is lower than the required 12 Hz, aliasing will occur.

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The final image is real since it is located to the right of the second lens. It is also upright since the magnification is positive.

(a) Using the thin lens formula, 1/f = 1/p + 1/q, where f is the focal length, p is the object distance, and q is the image distance, we have:

1/f = 1/p - 1/q1

Substituting the given values, we get:

1/11.6 = 1/16.2 - 1/q1

Solving for q1, we get:

q1 = 6.97 cm

Therefore, the image formed by the first lens is located 6.97 cm to the right of the lens.

(b) The image formed by the first lens acts as an object for the second lens. Using the thin lens formula again, we have:

1/f = 1/p2 + 1/q1

Substituting the given values, we get:

1/5 = 1/p2 + 1/6.97

Solving for p2, we get:

p2 = 3.32 cm

The distance from the second lens to the image of the first lens is then:

q2 = p2 + q1 = 3.32 + 6.97 = 10.29 cm

Therefore, the image of the first lens is located 10.29 cm to the right of the second lens.

(c) The object distance for the second lens is simply the image distance of the first lens:

p2 = q1 = 6.97 cm

(d) Using the thin lens formula again, we have:

1/f = 1/p2 + 1/q2

Substituting the given values, we get:

1/5 = 1/6.97 + 1/q2

Solving for q2, we get:

q2 = 13.95 cm

Therefore, the final image is located 13.95 cm to the right of the second lens.

(e) The magnification of the first lens is given by:

m1 = -q1/p = -6.97/16.2 ≈ -0.43

(f) The magnification of the second lens is given by:

m2 = -q2/p2 = -13.95/3.32 ≈ -4.20

(g) The total magnification of the system is given by the product of the magnifications of the two lenses:

m = m1 × m2 ≈ 1.81

(h) The final image is real since it is located to the right of the second lens. It is also upright since the magnification is positive.

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The overall order of the reaction is 2nd order.

Option B is the correct answer.

We have,

The overall order of a chemical reaction is the sum of the orders of the reactants in the rate law.

In this case,

The rate law is given as:

Rate = k[X][Y]

The order with respect to X is 1, and the order with respect to Y is 1.

Therefore, the overall order of the reaction is:

1 + 1 = 2

Thus,

The overall order of the reaction is 2nd order.

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Answer:

Step-by-step explanation:

To find the average value of the function f(x) = x^2(x^3 + 30) on the interval [-3, 3], we use the formula:

fave = (1/(b-a)) * ∫[a,b] f(x) dx

where a = -3 and b = 3.

So, we have:

fave = (1/(3-(-3))) * ∫[-3,3] x^2(x^3 + 30) dx

fave = (1/6) * [∫[-3,3] x^5 dx + 30∫[-3,3] x^2 dx]

fave = (1/6) * [0 + 30(2*3^3)]

fave = 2430

Therefore, the average value of the function f on the given interval is 2430.

To find the average value of the function h(u) = In(u) on the interval [1, 5], we use the formula:

have = (1/(b-a)) * ∫[a,b] h(u) du

where a = 1 and b = 5.

So, we have:

have = (1/(5-1)) * ∫[1,5] ln(u) du

have = (1/4) * [u ln(u) - u] from 1 to 5

have = (1/4) * [(5 ln(5) - 5) - (ln(1) - 1)]

have = (1/4) * (5 ln(5) - 4)

have = 0.962

Therefore, the average value of the function h on the given interval is approximately 0.962.

To find all numbers b such that the average value of f(x) = 6 + 10x - 9x^2 on the interval [0, b] is equal to 7, we use the formula:

fave = (1/(b-a)) * ∫[a,b] f(x) dx

where a = 0 and b = b.

So, we have:

7 = (1/b) * ∫[0,b] (6 + 10x - 9x^2) dx

7b = [6x + 5x^2 - 3x^3/3] from 0 to b

7b = 2b^2 - 3b^3/3 + 6

21b = 6b^2 - b^3 + 18

b^3 - 6b^2 + 21b - 18 = 0

Using synthetic division, we find that b = 2 is a root of this polynomial equation. Dividing by (b-2), we get:

(b-2)(b^2 - 4b + 9) = 0

The quadratic factor has no real roots, so the only solution is b = 2.

Therefore, the only number b such that the average value of f(x) on the interval [0, b] is equal to 7 is 2.

To estimate the average world population to the nearest million during the time period of 1950 to 1980, we need to find:

ave = (1/(1980-1950)) * ∫[1950,1980] P(t) dt

ave = (1/30) * ∫[1950,1980] 2560e^(0.017185t) dt

Using the formula for integrating exponential functions, we get:

ave = (1/30) * [2560

Based on the analysis, there is no 3x3 matrix A such that A^2 = -I_3. To understand this analysis let's consider whether there exists a 3x3 matrix A such that A^2 = -I_3, where I_3 is the 3x3 identity matrix.

Step:1. Start by assuming that there is a 3x3 matrix A such that A^2 = -I_3.
Step:2. Recall that the determinant of a matrix squared (det(A^2)) is equal to the determinant of the matrix (det(A)) squared: det(A^2) = det(A)^2.
Step:3. Compute the determinant of both sides of the equation A^2 = -I_3: det(A^2) = det(-I_3).
Step:4. For the 3x3 identity matrix I_3, its determinant is 1. Therefore, the determinant of -I_3 is (-1)^3 = -1.
Step:5. From step 2, we know that det(A^2) = det(A)^2. Since det(A^2) = det(-I_3) = -1, we have det(A)^2 = -1.
Step:6. However, no real number squared can equal -1, which means det(A)^2 cannot equal -1.
Based on the analysis, there is no 3x3 matrix A such that A^2 = -I_3.

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The arc length of the polar curve r = 2 from θ = 0 to θ = 2 is 4.

Explanation:

To find the arc length of the polar curve r = 2 from θ = 0 to θ = 2, Follow these steps:

Step 1: To find the arc length of the polar curve r = 2 from θ = 0 to θ = 2, we can use the arc length formula for polar coordinates:

Arc length (L) = ∫√(r^2 + (dr/dθ)^2) dθ, from θ = 0 to θ = 2

Given r = 2, dr/dθ = 0 (since r is a constant)

Step 2: Now substitute r and dr/dθ into the formula:

L = ∫√(2^2 + 0^2) dθ, from θ = 0 to θ = 2

L = ∫√(4) dθ, from θ = 0 to θ = 2

L = ∫2 dθ, from θ = 0 to θ = 2

Step 3: Integrate with respect to θ:

L = 2θ | from θ = 0 to θ = 2

Step 4: Evaluate the definite integral:

L = 2(2) - 2(0) = 4

So the arc length of the polar curve r = 2 from θ = 0 to θ = 2 is 4.

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The inverse of g' (-1/5) is -15625 / 4^{5/4}.

To find g' (-1/5), we first need to find g(x) which is the inverse of f(x). To do this, we start by setting y = f(x) and solving for x.
y = f(x) = {x^7} / {x^4 + 4}

Multiplying both sides by x^4 + 4, we get:
y(x^4 + 4) = x^7

Expanding the left side, we get:
y(x^4) + 4y = x^7

Substituting u = x^4, we get:
yu + 4y = u^(7/4)

Rearranging and solving for y, we get:
y = (u^(7/4)) / (u + 4)

Substituting back u = x^4, we get:
y = (x^7) / (x^4 + 4)

Thus, g(x) = (x^7) / (x^4 + 4).

Now, to find g'(-1/5), we need to take the derivative of g(x) and evaluate it at x = -1/5.

Using the quotient rule, we get:
g'(x) = [7x^6(x^4+4) - x^7(4x^3)] / (x^4+4)^2

Substituting x = -1/5, we get:
g'(-1/5) = [7(-1/5)^6((-1/5)^4+4) - (-1/5)^7(4(-1/5)^3)] / ((-1/5)^4+4)^2

Simplifying and expressing in exact form, we get:
g'(-1/5) = [-(4/3125)^(3/4)] / (4/3125)^2 = -1 / (4/3125)^{5/4} = -15625 / 4^{5/4}

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