find the differential of f(x,y)= sqrt(x^3 + y^2) at the point (1,2)

The line integral ∫C xey dx on the arc of the curve x = ey from (1, 0) to (e^9, 9) is (1/3)(e^27 - 1).

Hi! I'd be happy to help you evaluate the line integral on the given curve. To evaluate the line integral ∫C xey dx, where C is the arc of the curve x = ey from (1, 0) to (e^9, 9), follow these steps:

1. Parameterize the curve: Since x = ey, let y = t, so x = e^t. Thus, the parameterization of the curve is r(t) = (e^t, t), with t ranging from 0 to 9.

2. Compute the derivative of the parameterization: dr/dt = (de^t/dt, dt/dt) = (e^t, 1).

3. Substitute the parameterization into the integrand: xey = (e^t)(e^t) = e^(2t).

4. Compute the dot product of the integrand and dr/dt: (e^(2t)) * (e^t, 1) = e^(3t).

5. Integrate the dot product with respect to t from 0 to 9: ∫(e^(3t)) dt from t = 0 to t = 9.

6. Evaluate the integral: [1/3 * e^(3t)] from t = 0 to t = 9 = [1/3 * e^(27)] - [1/3 * e^0] = (1/3)(e^27 - 1).

So, the line integral ∫C xey dx on the arc of the curve x = ey from (1, 0) to (e^9, 9) is (1/3)(e^27 - 1).

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The derivative of the function y = (x^3 + 2)^2(x^4 + 4)^4 using logarithmic differentiation is: y' = 2(x^3 + 2)(x^4 + 4)^3[3x^2(x^4 + 4) + 8x(x^3 + 2)^2]

To use logarithmic differentiation, we take the natural logarithm of both sides of the equation and then differentiate with respect to x using the rules of logarithmic differentiation.

ln(y) = ln[(x^3 + 2)^2(x^4 + 4)^4]

Now, we use the product rule and chain rule to differentiate ln(y):

d/dx [ln(y)] = d/dx [2ln(x^3 + 2) + 4ln(x^4 + 4)]

Using the chain rule, we get:

d/dx [ln(y)] = 2(1/(x^3 + 2))(3x^2) + 4(1/(x^4 + 4))(4x^3)

Simplifying this expression, we get:

d/dx [ln(y)] = 6x^2/(x^3 + 2) + 16x^3/(x^4 + 4)

Finally, we use the fact that d/dx [ln(y)] = y'/y to solve for y':

y' = y(d/dx [ln(y)])

Substituting in the expression for d/dx [ln(y)], we get:

y' = (x^3 + 2)^2(x^4 + 4)^4 [6x^2/(x^3 + 2) + 16x^3/(x^4 + 4)]

Simplifying this expression, we get:

y' = 2(x^3 + 2)(x^4 + 4)^3[3x^2(x^4 + 4) + 8x(x^3 + 2)^2]

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Answer:

get a ruler, draw a line trough a to b and count the rise and run

Step-by-step explanation:

The angle of elevation is a function of the height of the space shuttle given by θ = arctan(h / 2).

(a) To express the height of the space shuttle as a function of the angle of elevation θ, we can use trigonometry. Let h be the height of the space shuttle above the launch pad. Then, we have:

tan(θ) = h / x

Solving for h, we get:

h = x * tan(θ)

Substituting x = 2 mi, we get:

h = 2 * tan(θ) mi

Therefore, the height of the space shuttle is a function of the angle of elevation θ given by h = 2 * tan(θ) mi.

(b) To express the angle of elevation as a function of the height h of the space shuttle, we rearrange the equation we found in part (a) as follows:

tan(θ) = h / x

tan(θ) = h / 2

Taking the inverse tangent of both sides, we get:

θ = arctan(h / 2)

Therefore, the angle of elevation is a function of the height of the space shuttle given by θ = arctan(h / 2).

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The parametrization of the tangent line to the function f(x) = ln(x) - 7/15x^3 at the point (1,-46/15) is r(t) = <1, -46/15> + t<1, -2/3>.

To find the tangent line at a point, we need the slope of the tangent line, which is the derivative of the function evaluated at that point. So, we first find the derivative of f(x):

f'(x) = 1/x - 7/5 x^2

Then, we evaluate f'(1) to find the slope at x = 1:

f'(1) = 1/1 - 7/5(1)^2 = -2/5

Thus, the slope of the tangent line is -2/5. We also know that the point of tangency is (1,-46/15), so we can use the point-slope form to find the equation of the tangent line:

y - (-46/15) = (-2/5)(x - 1)

Simplifying, we get:

y = (-2/5)x - 16/3

Now we can write the parametrization of the tangent line as r(t) = <1, -46/15> + t<1, -2/3>. This is because the direction vector of the tangent line is <1, -2/3>, which is the same as the slope of the line, and the point on the line is (1,-46/15).

So, to get the equation of the line in vector form, we start with the point <1, -46/15>, and add a scalar multiple of the direction vector <1, -2/3>. Thus, the parametrization of the tangent line is r(t) = <1, -46/15> + t<1, -2/3>.

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The parametrization of the tangent line to the function f(x) = ln(x) - 7/15x^3 at the point (1,-46/15) is r(t) = <1, -46/15> + t<1, -2/3>.

To find the tangent line at a point, we need the slope of the tangent line, which is the derivative of the function evaluated at that point. So, we first find the derivative of f(x):

f'(x) = 1/x - 7/5 x^2

Then, we evaluate f'(1) to find the slope at x = 1:

f'(1) = 1/1 - 7/5(1)^2 = -2/5

Thus, the slope of the tangent line is -2/5. We also know that the point of tangency is (1,-46/15), so we can use the point-slope form to find the equation of the tangent line:

y - (-46/15) = (-2/5)(x - 1)

Simplifying, we get:

y = (-2/5)x - 16/3

Now we can write the parametrization of the tangent line as r(t) = <1, -46/15> + t<1, -2/3>. This is because the direction vector of the tangent line is <1, -2/3>, which is the same as the slope of the line, and the point on the line is (1,-46/15).

So, to get the equation of the line in vector form, we start with the point <1, -46/15>, and add a scalar multiple of the direction vector <1, -2/3>. Thus, the parametrization of the tangent line is r(t) = <1, -46/15> + t<1, -2/3>.

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Answer:

To solve the equation 2/t + 1/(1+t) = -3/(2+t), we first need to find the least common multiple (LCM) of the denominators, which are t and 1+t, and then rewrite each fraction with the LCM as its denominator.

The LCM of t and 1+t is (t)(1+t) or t(t+1). To rewrite the fractions with this common denominator, we need to multiply the first fraction by (t+1)/(t+1) and the second fraction by t/t:

2/t * (t+1)/(t+1) + 1/(1+t) * t/t = -3/(2+t)

Simplifying each fraction, we get:

2(t+1)/(t(t+1)) + t/(t(t+1)) = -3/(2+t)

Combining the fractions on the left side, we get:

(2t+2+t)/(t(t+1)) = -3/(2+t)

Simplifying further:

(3t+2)/(t(t+1)) = -3/(2+t)

Now, we can cross-multiply and simplify:

(3t+2)(2+t) = -3t(t+1)

6t^2 + 11t + 4 = -3t^2 - 3t

9t^2 + 14t + 4 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 9, b = 14, and c = 4.

Plugging in these values, we get:

t = (-14 ± sqrt(14^2 - 4(9)(4))) / 2(9)

t = (-14 ± sqrt(136)) / 18

t = (-14 ± 2sqrt(34)) / 18

Simplifying the expression, we get:

t = (-7 ± sqrt(34)) / 9

These are the two possible solutions for t that satisfy the original equation.

Answer:

Step-by-step explanation:

The identity is (d) sin(x).

We can start by expanding the numerator:

(1 - (sin(x) - cos(x))^2) = 1 - (sin^2(x) - 2sin(x)cos(x) + cos^2(x))

= 1 - (1 - sin(2x))

= sin(2x)

Therefore, the expression simplifies to:

sin(2x)/(2cos(x))

Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x), we get:

2sin(x)cos(x)/(2cos(x)) = sin(x)

So the identity is (d) sin(x).

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The solution of the given problem of Permutation and Combination is .There are38,760 different groups of 10 can be formed from the 18 volunteers.

A permutation is a way of arranging a set of objects or events in a specific order. The number of possible permutations of a set of n objects taken r at a time is given by the formula nPr = n!/(n-r)!, where n! (n factorial) is the product of all positive integers up to n.

A combination, on the other hand, is a way of selecting a subset of objects or events from a larger set, where the order of the elements does not matter. The number of possible combinations of a set of n objects taken r at a time is given by the formula nCr = n!/r!(n-r)!.

The number of different groups of 10 that can be formed from 18 volunteers can be calculated using the formula for combinations:

C(18, 10) = 18! / (10! * 8!)

where "C(18, 10)" represents the number of ways to choose 10 volunteers out of 18.

Simplifying the expression:

C(18, 10) = (18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10!) / (10! * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)

The 10! in the numerator and denominator cancel out, leaving:

C(18, 10) = (18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)

Simplifying further, we get:

C(18, 10) = 38,760

Therefore, there are 38,760 different groups of 10 that can be formed from 18 volunteers.

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We need to roll the die at least 5 times to ensure that the probability is at least 2/3 that at least one 2 will appear.

To calculate the probability of rolling a 2 on a fair six-sided die, we first need to know the probability of rolling any number on a single roll, which is 1/6.

Since each roll of the die is independent of the previous roll, we can use the formula for the probability of independent events occurring together to find the probability of rolling a 2 at least once in a certain number of rolls.

Let's call the probability of rolling a 2 at least once in n rolls "P(n)". We can find P(n) using the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. So, the probability of not rolling a 2 in n rolls is (5/6)^n, since there are 5 possible outcomes (1, 3, 4, 5, or 6) on each roll that is not 2. Therefore, we can write:

P(n) = 1 - (5/6)^n

We want to find the minimum number of rolls needed to ensure that P(n) is at least 2/3, or 0.667. In other words, we want to find the smallest value of n that satisfies the inequality:

P(n) ≥ 2/3

Substituting the formula for P(n), we get:

1 - (5/6)^n ≥ 2/3

By multiplying both sides by -1 and rearranging, we get:

(5/6)^n ≤ 1/3

Taking the natural logarithm of both sides, we get:

n ln(5/6) ≤ ln(1/3)

Dividing both sides by ln(5/6), we get:

n ≥ ln(1/3) / ln(5/6)

Using a calculator, we find that:

n ≥ 4.81

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In a case whereby There are seven people fishing at Lake Connor three have fishing license and four do not an inspector chooses to do two of the people are random probability that the first person chosen does not have a license and the second one does is 2/7

Based on the given information, total number of the people = 7

those with fishing license =3

those without fishing license =4

chance of choosing someone without a license=4/7

chance of choosing someone with a license=3/6

Theerefore probability that the first person chosen does not have a license and the second one does= 4/7 * 3/6 =2/7

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(a) Yes, there is a smallest real number a for which x^26 is big-O of ax. To find this value, we can use the limit definition of big-O notation.

We want to find a value of a such that x^26 is less than or equal to ax multiplied by some constant C, for all x greater than some value N. Mathematically, we can write this as:

x^26 <= Cax, for all x >= N

Dividing both sides by x and taking the limit as x approaches infinity, we get:

lim x->inf (x^25 / a) <= C

This limit exists only if a is greater than zero, so let's assume that. Then we can simplify the left-hand side of the inequality as:

lim x->inf x^25 / a = inf

So for any value of C, we can always find a value of N such that x^26 is less than or equal to ax multiplied by C, for all x greater than or equal to N. Therefore, we can say that x^26 is big-O of ax, for any positive real number a, and there is no smallest such value of a.

(b) No, there is no smallest integer number a for which x^26 is big-O of ax. The proof is similar to part (a), but we need to show that for any positive integer a, there exists a constant C such that x^26 is not less than or equal to ax multiplied by C, for infinitely many values of x.

To do this, we can choose x to be a power of 2, say x = 2^k. Then we have:

x^26 = (2^k)^26 = 2^(26k)

ax = a * 2^k

So we want to find a value of a and a constant C such that:

2^(26k) > Ca * 2^k, for infinitely many values of k

Dividing both sides by 2^k, we get:

2^(25k) > Ca, for infinitely many values of k

But this is true for any value of a greater than 2^(25), since 2^(25k) grows faster than Ca for large enough values of k. Therefore, for any integer value of a greater than 2^(25), there exist infinitely many values of k for which x^26 is not less than or equal to ax multiplied by some constant C. Hence, x^26 is not big-O of ax for any integer value of a less than or equal to 2^(25), and there is no smallest such value of a.

(a) No, there isn't a smallest real number 'a' for which x^26x is big-O of ax. This is because x^26x has a higher growth rate than ax for any real number 'a'. As 'x' becomes larger, the term x^26x will always grow faster than ax, no matter the value of 'a'.

(b) Yes, there is a smallest integer number 'a' for which x^26x is big-O of ax. The smallest integer 'a' would be 1, because if we let 'a' be any integer smaller than 1, ax will have a lower growth rate than x^26x. When 'a' is equal to 1, we have x^26x = O(x), which means x^26x grows at most as fast as x, and there's no smaller integer 'a' for which this is true.

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Answer:

Step-by-step explanation:

1. B. lateral faces

2. D. surface area

3. B. lateral faces

4. A. lateral area

5. E. cylinder

Answer:

14 hamburgers

Step-by-step explanation:

The problem uses division, but we can create a proportion to see how the division works.

Since we know that Chelsea can make 1 hamburger with 0.16 pounds and allow x to represent the number of burgers Chelsea can make with 2.24 lbs of meat, we have:

[tex]\frac{2.24}{x}=\frac{0.16}{1}[/tex]

[tex]2.24=0.16x[/tex]

As the proportion shows, we can divide 2.24 by 0.16 to get x = 14.

Check: 0.16 lbs * 14 patties = 2.24 lbs

Answer:

If the new baseball cards represent 30% of David's special edition baseball card collection, then the original collection represents 70%. Let's represent the total number of cards in David's collection with the variable x. Then we can set up the following equation:

6 = 0.3x

To solve for x, we can divide both sides by 0.3:

x = 6 ÷ 0.3 = 20

Therefore, the total number of cards in David's baseball card collection is 20.

The angle 7π/12 is coterminal with 465.5 degrees.

To find the coterminal angle of 7π/12, we can add or subtract any multiple of 2π until we get an angle between 0 and 2π.

First, we can convert 7π/12 to degrees:

7π/12 = (7/12) * 180 ≈ 105.5 degrees

Next, we can add or subtract 360 degrees to get an angle between 0 and 360 degrees:

105.5 + 360 = 465.5

So the angle 7π/12 is coterminal with 465.5 degrees.

Therefore, the answer is d. 465 degrees.

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The correct answer is b. It would become wider with fewer values. This is because as the sample size decreases, the variability of the sample mean increases, leading to a wider confidence interval.

The distribution used for the confidence interval calculation (whether z or t) is not impacted by the sample size, only the size of the sample itself affects the confidence interval.

In a situation where the sample size was decreased from 39 to 29, the impact on the confidence interval would be (b) It would become wider with fewer values.

A smaller sample size generally leads to a wider confidence interval, as the decreased sample size provides less information about the overall distribution.

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True. This is a true statement known as the invertible matrix theorem. If a square matrix is invertible, then there exists a matrix b such that ab equals the identity matrix. However, not all square matrices are invertible.

True. If matrix A is a square matrix and has an inverse matrix B, then the product of A and B (AB) equals the identity matrix. In other words, if A is invertible, there exists a matrix B such that AB = BA = I, where I is the identity matrix. This is a true statement known as the invertible matrix theorem. If a square matrix is invertible, then there exists a matrix b such that ab equals the identity matrix. However, not all square matrices are invertible.
True. If matrix A is a square matrix and has an inverse matrix B, then the product of A and B (AB) equals the identity matrix. In other words, if A is invertible, there exists a matrix B such that AB = BA = I, where I is the identity matrix.

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Answer:

common ratio r = - 2

Step-by-step explanation:

the common ratio r is calculated as

r = [tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{-32}{16}[/tex] = - 2

Answer:

-2

Check:

16*-2 is -32

-32 * -2 is 64

The formula for p(n) is not valid for n = 1.

For each positive integer n, using the formula given, we can find p(1) by plugging in n = 1:

p(1) = 1(1-1)(2(1)-1)/6 = 0/6 = 0

So, according to the formula, p(1) is equal to 0.

However, we can see that this is not a true statement.

Because the product in the formula is defined as the product of the squares of the odd integers from 1 to n, and when n = 1, there is only one odd integer, which is 1.

Thus, p(1) should be equal to [tex]1^2 = 1.[/tex]

Therefore, the formula for p(n) is not valid for n = 1.

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Answer:

12.08 years

Step-by-step explanation:

to overcome this problem we will have to use the exponential growth formula A = P(1+r)^t

where a is the final amount

where p is the initial amount

where r is the rate per year

where t is the number of years

we can say

20= 2(1+0.21)^t

solve the equation for t

20/2 = 2(1.21)^t/2

10 = 1.21^t

take the log of both sides we get that

t = 12.08 years

1. The volume of the prism is 144 cubic units

2. The area of the shaded base is 16units²

A prism is a solid shape that is bound on all its sides by plane faces. A prism can have a rectangular base( rectangular prism) or a triangular base( triangular prism) or a circular base ( cylinder) e.t.c

Generally the volume of a prism is expressed as;

V = base area × height.

base area = l × w

therefore volume = l× w ×h

The base area = l× w

= 8× 2 = 16 square units

therefore the volume of the prism = 16 × 9

= 144 cubic units

Therefore the volume of the prism is 144 cubic units and the shaded base area is 16 units².

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Answer:

A. 2

Step-by-step explanation:

It would be a triangle. There's no other way unless you used a point in between a and b

The correct equation is D. To find the number of cupcakes baked by Mrs. Brown on the mth day, add 5 to the number of cupcakes baked on the (m-1)th day 20th of December. Am = A(m-1) + 5, where Ao = 30.

This is because Mrs. Brown increases the number of cupcakes by 5 over the previous day, so each day the number of cupcakes baked increases by 5. The initial value is 30, which is Ao. Therefore, to find the number of cupcakes baked on any given day, we add 5 to the number baked on the previous day.

Therefore, the correct answer is D.

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A. Two-way radio A covers 113 square miles.

B. Rounded to the nearest whole number, two-way radio B covers 523 square kilometers.

C. Comparing the areas, we can see that radio B covers the larger area with 523 square kilometers.

D. The coverage area of radio B is approximately 1.33 times larger than the coverage area of radio A.

Radius is a term used in geometry to describe the distance from the center of a circle or sphere to any point on its circumference or surface, respectively. It is usually denoted by the letter "r" and is measured in units of length, such as inches, centimeters, or meters. The radius of a circle is half of its diameter, while the radius of a sphere is one-half of its diameter.

Part A:

The area covered by two-way radio A can be calculated using the formula for the area of a circle:

Area = π x radius²

Radius of radio A = 6 miles

Area = 3.14 x 6²

Area = 113.04 square miles

Rounded to the nearest whole number, two-way radio A covers 113 square miles.

Part B:

The area covered by two-way radio B can also be calculated using the same formula:

Area = π x radius²

Radius of radio B = 12.88 kilometers

Area = 3.14 x (12.88)²

Area = 523.14 square kilometers

Rounded to the nearest whole number, two-way radio B covers 523 square kilometers.

Part C:

To compare the areas covered by the two-way radios, we need to convert the area covered by radio A from square miles to square kilometers, using the conversion factor given:

1 mile = 1.61 kilometers

Therefore, 1 square mile = (1.61)² square kilometers

Area covered by radio A = 113 square miles

Area covered by radio A in square kilometers = 113 x (1.61)²

Area covered by radio A in square kilometers = 290.22 square kilometers

Comparing the areas, we can see that radio B covers the larger area with 523 square kilometers.

Part D:

To determine the scale factor relationship between the radio coverages, we can divide the radius of radio B by the radius of radio A:

Scale factor = radius of radio B / radius of radio A

Scale factor = 12.88 kilometers / 6 miles

Scale factor = 12.88 kilometers / 9.66 kilometers (since 1 mile = 1.61 kilometers)

Scale factor = 1.33

This means that the coverage area of radio B is approximately 1.33 times larger than the coverage area of radio A.

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The given vectors are:
Vector A = (-11, b, 2)
Vector B = (b, b^2, b)

The values of b for which the given vectors are orthogonal are 0, -3, and 3.

To find the values of b for which the given vectors are orthogonal, we need to use the dot product of the vectors.

To determine if two vectors are orthogonal, their dot product should be equal to zero.

The dot product is calculated as follows:

Dot Product (A, B) = A1 * B1 + A2 * B2 + A3 * B3

Substituting the components of Vector A and Vector B:

(-11 * b) + (b * b^2) + (2 * b) = 0

Now, simplify the equation:

-11b + b^3 + 2b = 0
b^3 - 9b = 0

Factor the equation:

b(b^2 - 9) = 0

Now, we can find the values of b:

b = 0
b^2 - 9 = 0
b^2 = 9
b = ±3

So, the values of b for which the given vectors are orthogonal are 0, -3, and 3.

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The area of the region in the given quadrant i is 1/3 square units.

To find the area of the region in quadrant i bounded by y = 2x2 and y = 2x3, we need to first find the x-coordinates where these two curves intersect.

Setting 2x2 equal to 2x3, we get:
2x2 = 2x3

Dividing both sides by 2x2 (which is non-zero since we are only considering quadrant i), we get:
x3 = x2

So the curves intersect at the point (0,0) and (1,2).

To find the area of the region between these curves in quadrant i, we can integrate the difference between the two curves with respect to x, from x = 0 to x = 1:

∫[0,1] (2x3 - 2x2) dx
= [x4 - 2/3 x3] from 0 to 1
= (1 - 2/3) - (0 - 0)
= 1/3

Therefore, the area of the region in quadrant i bounded by y = 2x2 and y = 2x3 is 1/3 square units.

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Answer :-The interval of convergence for the given power series is (4, 14).

The power series in question is ∑n=1 to infinity [(x−9)^n]/[n(-5)^n].

To find the interval of convergence, we will use the Ratio Test:

1. Compute the absolute value of the ratio between the (n+1)th term and the nth term:

|(a_(n+1))/a_n| = |[((x-9)^(n+1))/((n+1)(-5)^(n+1))]/[((x-9)^n)/(n(-5)^n)]|

2. Simplify the ratio:

|(a_(n+1))/a_n| = |(x-9)/((-5)(n+1))|

3. Take the limit as n approaches infinity:

lim (n→∞) |(x-9)/((-5)(n+1))|

4. For the Ratio Test, if the limit is less than 1, then the series converges. In this case:

|(x-9)/(-5)| < 1

5. Solve the inequality to find the interval of convergence:

-1 < (x-9)/(-5) < 1

Multiply each side by -5 (and reverse the inequalities since we're multiplying by a negative number):

5 > x-9 > -5

Add 9 to each side:

14 > x > 4

So, the interval of convergence for the given power series is (4, 14).

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