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Taylor series is [tex]f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2} }[/tex]

To find the Taylor series for f(x) = ln(x) centering at 9, we need to observe the pattern for the first four derivatives of f(x). From there, we can create a general equation for f(n). Starting with f(x), we have

f(x) = ln(x)

[tex]f^{1}(x)= \frac{1}{x} \\f^{2}(x)= -\frac{1}{x^{2} }\\f^{3}(x)= -\frac{2}{x^{3} }\\f^{4}(x)= \frac{-6}{x^{4} }[/tex]

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Since we need to have it centered at 9, we must take the value of f(9), and so on.

f(9) = ln(9)

[tex]f^{1}(9)= \frac{1}{9} \\f^{2}(9)= -\frac{1}{9^{2} }\\f^{3}(x)= -\frac{1(2)}{9^{3} }\\f^{4}(x)= \frac{-1(2)(3)}{9^{4} }[/tex]

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Following the pattern, we can see that for [tex]f^{n}(x)[/tex],

[tex]f^{n}(x)=(-1)^{n-1}\frac{1.2.3.4.5...........(n-1)}{9^{n} } \\f^{n}(x)=(-1)^{n-1}\frac{(n-1)!}{9^{n}}[/tex]

This applies for n ≥ 1, Expressing f(x) in summation, we have

[tex]\sum_{n=0}^{\infinite} \frac{f^{n}(9) }{n!} (x-9)^{2}[/tex]

Combining ln2 with the rest of series, we have

[tex]f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2} }[/tex]

Taylor series is [tex]f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2} }[/tex]

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Answer:

15740000 million dollars in profit in year 4

Step-by-step explanation:

Now since you didn't provide the table, im going to have to assume the years are going to be the X while the Y is going to be profit.

Using the equation you provided y=-0.34x^2+4.43x +3.46, we can simply put 4 as our x value (since x represents years) and we would get 15.74. Now you have to remember to convert it into millions to show the profit so 15.74 times 1,000,000 would be 15740000 million dollars

Answer:

Given is radius of circle that is 5.2, you just need to find the area of the circle.

To find the are of circle is A=πr2

π value is 3.14 and radius value is 5.2 you just need to square the 5.2, answer will come 27.04

now multiply the πr and 27.04.

the answer will come 84.9 and question says to round to the nearest tenth, that will be 85.

Rounding to the nearest tenth, the surface area of the circle with a 5.2 cm radius is approximately 84.8 cm².

The surface area of a circle can be calculated using the formula:

Surface Area = π * radius²

where π (pi) is a mathematical constant approximately equal to 3.14159, and the radius is the distance from the center of the circle to any point on its edge.

Given the radius is 5.2 cm, we can now calculate the surface area:

Surface Area = 3.14159 * (5.2 cm)²

Surface Area = 3.14159 * 27.04 cm²

Surface Area ≈ 84.823 cm²

The surface area represents the total area of the circle's two-dimensional space. In this case, it gives us the total area of the circular region with a 5.2 cm radius.

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Answer: B

Step-by-step explanation:

We know N cannot be 0 as the value would not be negative.
We know N cannot be negative as that would make the expression positive.
B is the only answer that works.

A function of random variables utilized to calculate a parameter of distribution exists as an unbiased estimator.

A function of random variables utilized to calculate a parameter of distribution exists as an unbiased estimator.

An unbiased estimator exists in which the difference between the estimator and the population parameter grows smaller as the sample size grows larger. This simply indicates that an unbiased estimator catches the true population value of the parameter on average, this exists because the mean of its sampling distribution exists the truth.

Also, we comprehend that the bias of an estimator (b) that estimates a parameter (p) exists given by; E(b) - p

Therefore, an unbiased estimator exists as an estimator that contains an expected value that exists equivalent to the parameter i.e the value of its bias exists equivalent to zero.

Generally, in statistical analysis, the sample mean exists as an unbiased estimator of the population mean while the sample variance exists as an unbiased estimator of the population variance.

Therefore, the correct answer is an unbiased estimator.

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Answer:

(-3,1)

Step-by-step explanation:

9x+8y=-19

7x+9y=-12

I'll assume the question is to find the solution to these equations.  The solution will be the point (x,y) where the two lines intersect.  The intersection is the one point that satisfies both equations (the smae value of (x,y) works in both.

We can either solve matematically of graph to find the intersection.  I'll do both, and hope the answers are identical.

Matematically

Rearrange either equation to isolate one of the variables (either x or y).  I'll take the second and isolate x:

7x+9y=-12

7x = -9y - 12

x = (-9y - 12)/7

Now use this definition of x in the other equation:

9x+8y=-19

9((-9y - 12)/7) + 8y = -19

(-81y - 108)/7 + 8y = -19

-81y - 108 + 56y = - 133

-25y = -25

y = 1

If y = 1, then:

9x+8y=-19

9x+8(1)=-19

9x = -27

x = -3

The solution is (-3,1)

Graphing

Graph both lines and look for the intersection.  The attached graph shows the lines cross at (-3,1).

The solution, bu both approachjes, is (-3,1)

Answer:

Wouldn't she have had 5 cats before? Adding 2 other cats would make it 7, which fits the "more than 5 cats" requirement.

The time it was when Frank left the playground is; 6:10 PM.

We are given;

Time at which Frank went to the playground; 5:20 p.m

Amount of time that frank played on the slide = 45 minutes

Amount of time that frank swings on the play ground = 5 minutes

Now, we are told that after he played on the slide and swung on the playground, that he went home.

Thus, he spent a total time of 45 + 5 = 50 minutes on the playground before going home.

Thus, time he went home = 5:20 p.m + 50 minutes = 6:10 p.m

Thus, we can conclude that the time it was when Frank left the playground is; 6:10 PM.

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Answer:

800÷100 = 8

65×8=520

[tex]\frac{520}{800}[/tex]

another way:

100%-65%=35%

35% = 0.35

800×0.35= 280

800-280=520

another way:

65% = 0.65

0.65×800=520

Hope it helped! :)

Answer:

Positive

Step-by-step explanation:

5a < 2a

Divide 2 on both sides

5/2a < a

f(x) = 3^x

f(4) = 3^4

f(4) = 3 * 3 * 3 * 3

f(4) = 9 * 9

f(4) = 81

Hope this helps!

The identity Sin(α)/Tan(α) = Cos(α) is valid

Trigonometry is study of triangles. All trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. Three major of them are as follows :-

Sine Function:

sin(θ) = Opposite / Hypotenuse

Cosine Function:

cos(θ) = Adjacent / Hypotenuse

Tangent Function:

tan(θ) = Opposite / Adjacent

Lets prove this identity by proceeding with the LHS

= Sin(α)/Tan(α)

= Sin(α)/ (Sin(α)/Cos(α)) (Tan(α) = Sin(α)/Cos(α))

= Sin(α)xCos(α) / Sin(α)

= Cos(α)

Hence verified

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Answer:

8/15

Step-by-step explanation:

possibilities/ sample size=16 girls/30 "learners"=8/15

From the given options we can say that the only one that represents the area of the sector is; A = n/360 * πr²

In circles, a sector is said to be a part of a circle made of the arc of the circle together with its two radii. This means that it is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc.

The formula for Area of a sector is given as;

θ/360 * πr²

where;

θ is the central angle of the sector

r is radius

Now, looking at the given options we can say that the only one that represents the area of the sector is;

A = n/360 * πr²

where n is the central angle of the sector

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Answer:

A!

Step-by-step explanation:

Answer:

0.5%

Step-by-step explanation:

The length of one edge of the cube exists 6 centimeters.

The lateral surface area of a cube exists

[tex]$A=4 a^{5}$[/tex]

where a stands the length of one edge of a cube.

It exists given that the lateral surface area of the square stands at 144 square centimeters.

Substitute A = 144 in the above formula.

[tex]$144=4 a^{2}[/tex]

Divide both sides by  4.

[tex]$36=a^{2}[/tex]

Taking square root on both sides.

[tex]$\sqrt{36}=\sqrt{a^{2}}[/tex]

6 = a

The length of one edge of the cube exists 6 centimeters.

Therefore, the correct answer is option c. 6 cm.

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If a plot of land in the shape of a horizontal ellipse has a pole at each focus. the foci are 16 feet from the center. if the plot of land is 40 feet across one axis, how long is it across the other axis is: c. 24 feet.

Using Pythagoreans theorem formula

a²=b²+c²

Where:

c=16 feet

a=40/2=20 feet

Hence:

20²=b²+16²

b²=20²-16²

b²=400-256

b²=√144

b=12

2b=12×2

2b=24 feet

Therefore if the plot of land is 40 feet across one axis, how long is it across the other axis is: c. 24 feet.

The complete question is are:

A plot of land in the shape of a horizontal ellipse has a pole at each focus. The foci are 16 feet from the center. If the plot of land is 40 feet across one axis, how long is it across the other axis?

a. 34 feet

b. 46 feet

c. 24 feet

d. 30 feet

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The angles and side lengths for both triangles are;

3) A = 36°; B = 54°; C = 90°; a = 7; b = 9.63; c = 4.11

4) A = 64°; B = 26°; C = 90°; a = 1.798; b = 0.88; c = 2

3) From the diagram, we are already given;

B = 54°

C = 90°

a = 7

We know that sum of angles in a triangle is 180° and so;

A = 180 - (90 + 54)

A = 36°

By trigonometric ratios;

b/7 = tan 54

b = 7 * tan 54

b = 7 * 1.376

b = 9.63

7/c = cos 54

c = 7 * cos 54

c = 4.11

4) From the diagram, we are already given;

B = 26°

C = 90°

c = 2

We know that sum of angles in a triangle is 180° and so;

A = 180 - (90 + 26)

A = 64°

By trigonometric ratios;

b/2 = sin 26

b = 2 * sin 26

b = 2 * 0.4384

b = 0.88

a/2 = cos 26

a = 2 * cos 26

a = 1.798

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Answer:

14 children

Step-by-step explanation:

Half the children like pets and a quarter of the children do not, so another quarter does not have preference. Let x = total number of children, and we would have this equation.

0.25x = 7

x = 28

Since we are asked to give the number of children who like pets, 28/2 = 14 is the answer.

Answer:

0

Step-by-step explanation:

54m² - 54m² = 0

Water is 0

Answer:

5050

Step-by-step explanation:

we know that 100/2 is 50 and 50 x 100 is 5000

so now another 50 is remaining but we can't multiply but add so

1+2+3+4+5...100 = in simple form= 50x100+50=5050//

Answer:

336 ways

Step-by-step explanation:

Use nPk which is [tex]\frac{n!}{(n-k)!}[/tex]. This is [tex]\frac{8!}{5!}[/tex]. This becomes 8×7×6 as the 8! and 5! cancel out. 8×7×6 is 336.

Total number of possible 3-topping pizzas are 336 ways.

In more general terms, if we have n items total and want to pick k in a certain order, we get: n! / (n – k)! And this is the permutation formula: The number of ways k items can be ordered from n items: P(n,k) = n (n – k)!

Given that,

Total number of items n = 8

number of picking item k = 3

Now,

p(n,k) = [tex]\frac{n!}{(n -k)!}[/tex]

p(8,5) =  [tex]\frac{8!}{(8 -3)!}[/tex]

         = [tex]\frac{8!}{5!}[/tex]

         = [tex]\frac{8.7.6.5! }{5! }[/tex]

         = 8 × 7 ×6

p(8,5) = 336 ways

Hence, Total number of possible 3-topping pizzas are 336 ways.

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Answer: No solution

Distribute the parenthesis

3x-15=2x-10+x

Combine like terms

3x-15=3x-10

This answer has no solution for x.

Considering the equation of the parabola, the coefficients are given as follows: p = 5, q = -20, r = 19.

The graph is the one given at the side of the question.

The equation of a quadratic function, of vertex (h,k), is given by:

y = a(x - h)² + k

In which a is the leading coefficient.

In this problem, the turning point is (2,-1), hence h = 2, k = -1, and the equation has the following format:

y = a(x - 2)² - 1

The curve passes through (3,4), that is, when x = 3, y = 4, and we use it to find a.

y = a(x - 2)² - 1

4 = a(3 - 2)² - 1

a = 5.

Hence the equation is:

y = 5(x - 2)² - 1

Placing it into standard form, we have that:

y = 5(x² - 4x + 4) - 1

y = 5x² - 20x + 19

Hence the coefficients are:

p = 5, q = -20, r = 19.

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Answer:

$25.60

Step-by-step explanation:

1 kg = 1000 grams

peanuts = 6.40 per kg = 0.064 per grams

0.064*400 = 25.6

The difference between the 6th term and the 9th term of the sequence is 135

Given that the nth term is;

3n² + 11

For the 6th term, the value of n is 6

Let's solve for the 6th term

= 3( 6)^2 + 11

= 3 × 36 + 11

= 108 + 11

= 119

For the 9th term, n = 9

= 3 (9)^2 + 11

= 3( 81) + 11

= 243 + 11

= 254

The difference between the 6th and 9th term

= 254 - 119

= 135

Thus, the difference between the 6th term and the 9th term of the sequence is 135

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The system of equations is -x + y = 4 and -2x + y = 0

To do this, we make use of the following ordered pairs

(4, 8)

The above point represents April 2008.

Let the system of equations be

mx + ny = 4

2mx + ny = 0

Substitute (4, 8) for x and y in the above equations.

4m + 8n = 4

8m + 8n = 0

Subtract both equations

4m - 8m = 4

This gives

-4m = 4

Divide by 4

m = -1

Substitute m = -1 in 8m + 8n = 0

-8 + 8n = 0

This gives

8n = 8

Divide by 8

n = 1

So, the system of equations is -x + y = 4 and -2x + y = 0

The system of equations is -x + y = 4 and -2x + y = 0

See attachment for the graph of the system of equation

The above is represented in the (a) part of this solution

Verify that the solution is (4, 8)

We have:

-x + y = 4 and -2x + y = 0

Substitute (4, 8) for x and y in the above equations.

-4 + 8 = 4 --- true

-2*4 + 8 = 0 --- true

Both equations are true

Hence, the system of equations have been verified

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