Find, from first principle the deriva- tive of 1/(x²+1)
​

The absolute potential (V) at a point due to a point charge (Q) is given by the formula: V = kQ / r

where k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), Q is the charge in Coulombs, and r is the distance between the point and the charge.

First, we'll find the absolute potential at point A (3, 2, 3) m due to the charge Q = 0.4 nC at the origin.

1. Convert Q to Coulombs: Q = 0.4 nC = 0.4 x 10^(-9) C
2. Find the distance r between the origin and point A using the Pythagorean theorem: r = √(3^2 + 2^2 + 3^2) = √(9 + 4 + 9) = √22
3. Calculate V at point A: V_A = (8.99 x 10^9)(0.4 x 10^(-9)) / √22 ≈ 25.67 V

Now, we'll calculate the absolute potential at the new position (5, 3, 3) m due to the charge Q relocated to point B (5, 3, 3) m.

1. Find the distance r between point B and the new position: r = √((5-5)^2 + (3-3)^2 + (3-3)^2) = 0 (same point)
2. Since r = 0, the absolute potential at the new position is undefined (potential would go to infinity if r approaches 0).

So, the absolute potential at point A is approximately 25.67 V, and the absolute potential at the new position is undefined.

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Answer:

1.2

Step-by-step explanation:

based on your description the top sides are corresponding pairs, the bottom sides are corresponding sides, the left sides are corresponding sides, and the right sides are corresponding sides between the 2 polygons.

the dilation (scaling) is happening for all points on the polygon with the same scaling factor.

so, we only need to find the scaling factor f between one of these corresponding pairs.

8×f = 9.6

f = 9.6/8 = 1.2

Answer: The answer is 1.2

Step-by-step explanation:

Just trust me.

The congruency statement that describes the figures is:
ΔDEF ≅ ΔRSU

To answer your question, let's first find the image of triangle DEF after reflecting over the y-axis and then translating down 4 units and right 3 units.

1. Reflect ΔDEF over the y-axis:
D'(−6, 4), E'(−5, 8), F'(−1, 2)

2. Translate ΔD'E'F' down 4 units and right 3 units:
D''(−3, 0), E''(−2, 4), F''(2, −2)

Now, we have ΔD''E''F'' with points (−3, 0), (−2, 4), and (2, −2). Comparing this to ΔRSU with points (−2, 4), (−3, 0), and (2, −2), we can see that:

ΔD''E''F'' ≅ ΔRSU
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Answer:

ΔDEF ≅ ΔRSU

Step-by-step explanation:

The length of the straight line between the school and the fire station is 4.6 miles.

We can form a right-angled triangle with the school at the right-angle.

The distance between the school and the fire station is the hypotenuse of this triangle.

Using the Pythagorean theorem, we can calculate the length of the hypotenuse:

h^2 = 4.3^2 + 1.7^2

h^2 = 21.38

h ≈ 4.62

Rounding to the nearest tenth, the length of the straight line between the school and the fire station is approximately 4.6 miles.

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Step-by-step explanation:

Use Pythagorean theorem for right triangles

c^2 = a^2 + b^2      where c = hypotenuse and   a   and b are the legs

8.6^2 = 5^2 + r^2

8.6^2 - 5^2 = r^2

r = ~ 7  units

The measure of the angle x in the circle is 65 degrees

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

The circle

On the circle, we have the angle at the vertex of the triangle to be

Angle = 100/2

Angle = 50

The sum of angles in a triangle is 180

So, we have

x + x + 50 = 180

Evaluate the like terms,

2x = 130

So, we have

x = 65

Hence, the angle is 65 degrees

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Answer:

Step-by-step explanation:

The value of 2s in 43,290 is 2,000, while the value of 2s in 32,865 is 20.

B. 200 is 1/20 the value of 2,000.

This statement is correct, as 200 is 1/10 of 2,000, and there are two 0s in the value of 2s in 43,290 compared to one 0 in the value of 2s in 32,865.

There exists at least one c in the open interval (a, b) such that f'(c) = 0.

There are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.

To prove that the given equation has exactly 3 distinct real solutions, let's follow the steps mentioned in the question.

First, consider the polynomial p(x) = (x-a)(x-b)(x-c)(x-d). Since a, b, c, and d are distinct real numbers, p(x) has 4 distinct real roots, namely a, b, c, and d.

Now, let's find the derivative p'(x) using logarithmic differentiation. Taking the natural logarithm of both sides, we have:

[tex]ln(p(x)) = ln((x-a)(x-b)(x-c)(x-d))[/tex]

Differentiating both sides with respect to x, we get:

[tex]p'(x)/p(x) = 1/(x-a) + 1/(x-b) + 1/(x-c) + 1/(x-d)[/tex]

Multiplying both sides by p(x) and simplifying, we have:

[tex]p'(x) = (x-b)(x-c)(x-d) + (x-a)(x-c)(x-d) + (x-a)(x-b)(x-d) + (x-a)(x-b)(x-c)[/tex]

Now, we apply Rolle's Theorem, which states that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.

Since p(x) has 4 distinct real roots, there must be 3 intervals between these roots where the function p(x) satisfies the conditions of Rolle's Theorem. Therefore, there are 3 distinct points where p'(x) = 0, which corresponds to the 3 distinct real solutions of the given equation.

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The critical point(s) of the function f(x) = x^3 + x - 3 + 2 are determined to find the x-coordinate(s) of the local minimum or local maximum.

To find the critical point(s) of the given function, we need to first find the derivative of the function and then solve for the value(s) of x that make the derivative equal to zero.

Given function: f(x) = x^3 + x - 3 + 2

Find the derivative of the function f(x) with respect to x.

f'(x) = 3x^2 + 1

Set the derivative f'(x) equal to zero and solve for x.

3x^2 + 1 = 0

Subtract 1 from both sides of the equation.

3x^2 = -1

Divide both sides of the equation by 3.

x^2 = -1/3

Take the square root of both sides of the equation.

x = ±√(-1/3)

Since the square root of a negative number is not a real number, the function f(x) does not have any real critical points. Therefore, the critical point(s) for the function f(x) = x^3 + x - 3 + 2 is DNE (Does Not Exist) in terms of real numbers.

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Answer:

Step-by-step explanation:

Rounded percent of change = 500.0% Therefore, the percent of change is an increase of 500.0%.

I think sorry if I’m wrong

No, there is not convincing evidence that p, the true proportion of 17-year-olds at this high school with driver's licenses, is not 0.75.

This is because the P-value of 0.9315 is very large, and the 95% confidence interval contains 0.75 (0.6300, 0.8811). This means that there is not enough evidence to reject the null hypothesis that the true proportion of 17-year olds with driver's licenses is 0.75. The 95% confidence interval also supports this, as it includes 0.75. Therefore, there is no convincing evidence to suggest that the student's claim is incorrect.

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In ΔMNO, possible values of ∠N are 64.6° and 115.4°.

To find the possible values of ∠N, follow these steps:

1. Since the sum of angles in a triangle is 180°, we first find ∠O by subtracting ∠M from 180°: 180° - 38° = 142°.

2. Next, we use the Law of Sines to find the sine of ∠N: sin(∠N) = (n * sin(∠O)) / m = (88 * sin(142°)) / 60.

3. Solve for sin(∠N), which gives us two possible values: sin(∠N) ≈ 0.8988 and sin(∠N) ≈ -0.8988.

4. Find the inverse sine (arcsin) of both values to get the possible angles for ∠N: arcsin(0.8988) ≈ 64.6° and arcsin(-0.8988) ≈ 115.4° (adding 180° to the negative result).

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Answer:

Step-by-step explanation:

If we let M be a number, then 5 times as big as M would be 5M. Not that hard :/

Answer:

The largest volume that such a box can have is (3/4)²(3/2)²/4 = 1.6875 m³

Step-by-step explanation:

4. Let the sides of the rectangle be 'l' and 'w', where lw = A, the fixed area. The perimeter P is given by P = 2l + 2w. To minimize P, we need to find the values of 'l' and 'w' that make P as small as possible. Solving the equation for 'w' in terms of 'l' from lw = A, we get w = A/l. Substituting this into the equation for P, we get P = 2l + 2(A/l). Taking the derivative of P with respect to 'l' and setting it to zero, we get 2 - 2A/l² = 0, which implies l = √A. Substituting this value into lw = A, we get w = √A. Therefore, a square with sides of length √A has the minimum perimeter among all rectangles with a fixed area of A.

Let the side length of the square base be 'x' and the height of the box be 'h'. Then the volume of the box is V = x²h = 400. We need to minimize the surface area S of the box, which is given by S = x² + 4xh. Solving the equation for 'h' in terms of 'x' from V = x²h, we get h = 400/x². Substituting this into the equation for S, we get S = x²+ 4x(400/x²) = x² + 1600/x. Taking the derivative of S with respect to 'x' and setting it to zero, we get 2x - 1600/x² = 0, which implies x = 10 cm. Therefore, the dimensions of the box that minimizes the amount of material used are 10 cm x 10 cm x 4 cm.

Let the side length of the square cut out from each corner be 'x', and the height of the box be 'h'. Then the volume of the box is V = x²h. The length and width of the base of the box are (3-2x) and (3-2x) respectively. We need to maximize the volume V of the box subject to the constraint that the length and width of the base are positive. Taking the derivative of V with respect to 'x' and setting it to zero, we get h = (3-2x)²/4. Substituting this into the equation for V, we get V = x²(3-2x)²/4. Taking the derivative of V with respect to 'x' and setting it to zero, we get x = 3/4 m. Therefore, the largest volume that such a box can have is (3/4)²(3/2)²/4 = 1.6875 m³

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Using the exponential function to estimate the average price of a gallon of orange juice in 2020, the average price in 2020 is $6.39

To estimate the average price of a gallon of orange juice in 2020 using the given exponential function f(x) = 1.07(1.03^x) + 3.79, first, determine the number of years after 1990, which is r:

r = 2020 - 1990 = 30

Next, substitute r with 30 into the function:

f(30) = 1.07(1.03^30) + 3.79

Calculate the value of f(30):

f(30) ≈ 1.07(2.427) + 3.79 ≈ 2.599 + 3.79 ≈ 6.389

Round your answer to the nearest cent:

Average price in 2020 ≈ $6.39

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The volume of the sphere is 4 π × 3 × h. Option C

To determine the expression, we need to know the formula for volume of a sphere.

The formula that is used for calculating the volume of a sphere is expressed as;

V = 1/3 πr²h

Given that the parameters of the formula are;

Now, substitute the values, we have;

Volume, V= 4/3 × π × 3² × h

Multiply the values, we get;

Volume =4 π × 3² × h/3

Divide the values

Volume =4 π × 3 × h

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Answer:

Answer:17

Explanation: 5-0.75=4.25 4.25÷0.25=17

Hope this helps

Answer:

Step-by-step explanation:

The cost of 1 pound of peanuts can be found by dividing the total cost of 8 pounds by 8:

Cost of 1 pound of peanuts = $24 / 8 pounds = $3 per pound

The cost of 1 pound of walnuts can be found by dividing the total cost of 6 pounds by 6 and then multiplying by 2 (since the cost of 6 pounds is half that of the peanuts for the same weight):

Cost of 1 pound of walnuts = ($24 / 6 pounds) x 2 = $8 per pound

Therefore, we see that walnuts are more expensive than peanuts by $5 per pound ($8 - $3).

In other words, 1 pound of walnuts costs $5 more than 1 pound of peanuts.

Where the above graph and conditions are given,  the value of k that satisfies g(x) = f(x+k) is k = -2.

We can determine the value of k by using the given relationship between g(x) and f(x+k).

If g(x) = f(x + k), then we can substitute the given values of x in g(x) to get:

g(-1) = f(-1 + k) --> 5 = f(-1 + k)

g(0) = f(0 + k) --> 4 = f(k)

g(1) = f(1 + k) --> 3 = f(1 + k)

g(2) = f(2 + k) --> 2 = f(2 + k)

g(3) = f(3 + k) --> 3 = f(3 + k)

We know that f(x) is a v-shaped graph with a vertex at (0,2) and points at (-1,3) and (1,3). Therefore, we can conclude that f(k) = 4, which means that k is the x-coordinate of the vertex of f(x) shifted to the left or right.

Since the vertex of f(x) is at (0,2), and the x-coordinate of the vertex of f(x+k) is at k, we have:

k = 0 --> vertex of f(x+k) is at (0,2)

k = -1 --> vertex of f(x+k) is at (-1,2)

k = 1 --> vertex of f(x+k) is at (1,2)

k = 2 --> vertex of f(x+k) is at (2,2)

Therefore, the value of k that satisfies g(x) = f(x+k) is k = -2.

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Answer:

Step-by-step explanation:

Rules:

Take out the GCF (greatest common factor)

a)  [tex]9m^{4} -9m^{3}[/tex]             >take out GCF, what both terms can be divided by

  =9m³(m-1)                >when taking out GCF, divide both terms by GCF

b)  [tex]25x^{9}y^{10}-35x^{7}y^{5}[/tex]    >GCF is [tex]=5x^{7}y^{5}[/tex]

[tex]=5x^{7}y^{5}(5x^{2} y^{5}-7)[/tex]

c)   (x-1)(x-1)-3(x-1)​        >GCF is (x-1)

  =(x-1) [(x-1) - 3]          >within the bracket you can combine like terms

  =(x-1) (x-4)

Step-by-step explanation:

A) 9m^4 - 9m^3 = 9m^3 (m - 1)

As for the number, you already took 9 out because it's common for both. As for the m, m^4 is the same as m×m×m×m. So the common between both is m×m×m = m^3.

B) 25x^9y^10 - 35x^7y^5 umm are you sure it's well written? How do you have a power in a power?

C) (x-1)(x-1)-3(x-1) = (x²-1x-1x+1) - (3x-3)

= x² - 2x + 1 - 3x + 3

= x² - 5x + 4

Therefore, the probability that a randomly selected point within the square falls in the red-shaded area is 68%.

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. The probability of an event A is denoted as P(A). To calculate the probability of an event, we divide the number of favorable outcomes by the total number of possible outcomes.

Here,

The area of the red-shaded region is the area of the square minus the area of the white right-angled triangle. The area of the square is the length of one side squared, which is:

Area of square = 5 cm × 5 cm

= 25 cm²

The area of the right-angled triangle is one-half the base times the perpendicular height, which is:

Area of triangle = (1/2) × base × height

= (1/2) × 4 cm × 4 cm

= 8 cm²

Therefore, the area of the red-shaded region is:

Area of red-shaded region = Area of square - Area of triangle

= 25 cm² - 8 cm²

= 17 cm²

To find the probability that a randomly selected point within the square falls in the red-shaded area, we need to divide the area of the red-shaded region by the total area of the square, which is:

Probability = Area of red-shaded region / Area of square

Probability = 17 cm² / 25 cm²

= 0.68 or 68%

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The absolute value inequality in form |x-b| < c that has the solution set x<-5 or x>7 is |x-1| < 7

To write an absolute value inequality in the form |x-b| < c, we need to think about what this form means.

The expression |x-b| represents the distance between x and b on the number line. Therefore, |x-b| < c means that the distance between x and b is less than c.

Now, let's consider the given solution set: x < -5 or x > 7. We can see that the midpoint between -5 and 7 is 1, so we choose b = 1. Then, we need to determine c, which is the maximum distance between b and any of the solutions.

If we take x = -6 (which is less than -5), then |x-b| = |-6-1| = 7. Similarly, if we take x = 8 (which is greater than 7), then |x-b| = |8-1| = 7. Therefore, the maximum distance is 7.

Putting it all together, we get the absolute value inequality:

|x-1| < 7

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Answer:

Combining like terms,

(4x - 6) + (5x + 1) = 9x - 5

The ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is y=-4x+20.

Now put x=6 and y=4 then,

=> 4=-4(6)+20

=> 4 = -24+20

=> 4 ≠ -4.

Then the coordinate (6,4) dost not satisfy the function.

Put x=0 and y=20 then,

=> 20 = -4(0)+20

=> 20= 0+20

=> 20=20

Hence the coordinate (0,20) satisfy the function.

Now put x=-4 and y=20 then,

=> 20 = -4(-4)+20

=> 20 = 16+20

=> 20 ≠ 36

Hence the coordinate (-4,20) does not satisfy the function.

Now put x=10 and y=-20 then,

=> -20 = -4(10)+20

=> -20 = -40+20

=> -20=-20

Then the coordinate (10,-20) satisfy the function.

Hence the ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).

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The null hypothesis (H0) is that there is no significant difference in mean weight loss between men and women, or μm - μw = 0. The alternative hypothesis (Ha) is that there is a significant difference in mean weight loss between men and women, or μm - μw ≠ 0.

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The Gross Profit Margin Ratio for Frontier Art Gallery is 69.38%, calculated by dividing the Gross Profit by Net Sales and multiplying the result by 100 to get the percentage.

Gross Profit Margin Ratio is calculated by dividing the Gross Profit by Net Sales and multiplying the result by 100 to get the percentage

Gross Profit = Net Sales - Cost of Merchandise Sold

Gross Profit = $62,481.45 - $19,123.49

Gross Profit = $43,357.96

Gross Profit Margin Ratio = (Gross Profit / Net Sales) x 100

Gross Profit Margin Ratio = ($43,357.96 / $62,481.45) x 100

Gross Profit Margin Ratio = 69.38%

Therefore, the Gross Profit Margin Ratio for Frontier Art Gallery is 69.38%.

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Por lo tanto, el tercer carro que recibirá boletos de ambos colores ocupará la posición número 18 en la fila.

Hola, entiendo que quieres saber en qué posición de la fila se encontrará el tercer carro que recibirá boletos de ambos colores (verde y azul). Para esto, vamos a analizar la situación:

- La primera persona entrega un boleto verde cada 2 carros.
- La segunda persona entrega un boleto azul cada 3 carros.

Un carro que recibe boletos de ambos colores será aquel que ocupa una posición que es múltiplo común de 2 y 3. El mínimo común múltiplo (MCM) de 2 y 3 es 6. Por lo tanto, cada 6 carros, habrá uno que reciba boletos de ambos colores.

Para encontrar el tercer carro que recibirá boletos de ambos colores, simplemente multiplicamos el MCM (6) por la cantidad de carros que buscamos (3):

6 × 3 = 18

Por lo tanto, el tercer carro que recibirá boletos ición número 18 en la fila.

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The population in this study is the jars of marinara sauce produced at the Barilla plant in Ames, IA.

The population in this study refers to all jars of marinara sauce produced at the Barilla plant in Ames, IA in the last week. It represents the entire set of jars that the quality assurance engineer could potentially sample from.

The population mean is stated as 26.2 ounces, indicating the average amount of sauce per jar for the entire population. The population standard deviation is given as 0.04 ounces, representing the variability in the amount of sauce across all jars produced.

The quality assurance engineer randomly selected a sample of 34 jars from this population and found the mean amount of sauce per jar to be 26.197 ounces.

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This procedure is not appropriate because (A) the two sample proportions were not computed from independent samples.

Independent samples are those chosen at random such that their observations do not depend on the values of other observations. Many statistical analyses are predicated on the assumption of independent samples. Others are intended to evaluate non-independent samples.

Assume that quality inspectors want to compare two laboratories to see if their blood tests produce identical results. Both labs receive blood samples drawn from the same ten children for analysis.

The test results are not independent because both labs analyzed blood samples from the same ten youngsters. The inspectors would need to perform a paired t-test, which is based on the assumption that samples are dependent, to compare the average blood test results from the two labs.

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Correct question:

A polling organization asks a random sample of 1,000 registered voters which of two candidates they plan to vote for in an upcoming election. Candidate A is preferred by 400 respondents, Candidate B is preferred by 500 respondents, and 100 respondents are undecided. George uses a large sample confidence interval for two proportions to estimate the difference in the population proportions favoring the two candidates. This procedure is not appropriate because

(A) the two sample proportions were not computed from independent samples

(B) the sample size was too small

(C) the third category, undecided, makes the procedure invalid

(D) the sample proportions are different: therefore the variances are probably different as well

(E) George should have taken the difference interval for a single proportion instead 500-400 1,000 and then used a large sample confidence

The measurement of angle 2 is 55°, if the measurement of angle 1 and angle 2 equal 180° and measurement of angle 1 is 125° in the drawing of the bridge design.

The measurement of angle 1 is 125°, and the sum of angle 1 and angle 2 is 180°. The relationship between angle 1 and angle 2 is supplementary since their sum is equal to 180°. To find the measurement of angle 2,

Therefore, angle 2 has a measurement of 55°.

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