The terms of this series may be arranged without changing the value of the series. The sum of the reciprocals of the squares of the odd positive integers is [tex]\pi ^{2} /8[/tex].
In mathematics, a sequence is the cumulative sum of a given collection of terms. Usually, those phrases are actual or complicated numbers, but plenty of extra generalities are feasible.
A series is described as an arrangement of numbers in a specific order. then again, a chain is described as the sum of the factors of a sequence.
In mathematics, a series is, more or less speaking, a description of the operation of including infinitely many quantities, one after the alternative, to a given beginning quantity. The look at of series is a primary part of calculus and its generalization, mathematical analysis.
k=1
1/(1)2+1/(2)2+1/(3)2+1/(4)2+1/(5)2+1/(6)2+1/(7)2+.
up to ∞ terms = 2/6
[1/(1)2+1/(3)2+1/(5)2+1/(7)2+]+[1/(2)²+1/(4)²+1/(6)²+
..∞0] = T²/6
→ [1/(1)² + 1/(3)² + 1/(5)2+1/(7)2+......00] + [1/4 (1)² + 1/4(2)²+
1/4(3)²+....0] =²/6
[1/(1)²+1/(3)²+1/(5)2+1/(7)2+.......)] + 1/4[1/(1)² + 1/(2)²+
1/(3)²+....x] = 2/6
⇒ [1/(1)² + 1/(3)² + 1/(5)²+1/(7)²+..] + 1/4 [π²/6] = 2/6
⇒ [1/(1)² + 1/(3)² + 1/(5)²+1/(7)²+] = (1-1/4)/6
⇒ [1/(1)²+1/(3)2+1/(5)2+1/(7)2+..∞ = 3/4 x π²/6
=
↑
[1/(1)2+1/(3)2+1/(5)2+1/(7)²+] = 2/8
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Rashon walked 10 miles in 4 hours. if he walked the same distance every hour, how far did he walk in one hour? write your answer in feet if 1 mile equals 5280 feet.
Answer:
Step-by-step explanation:
Miles per hour: [tex]\frac{10miles}{4hrs} = 2.5\frac{miles}{hour}[/tex]
Feet per hour:
[tex]2.5\frac{miles}{hours} (\frac{5280ft}{1mile} )[/tex]
[tex]=13,200\frac{ft\\}{hr}[/tex]
The law of Cosines Is......
Answer: [tex]\Large\boxed{C.~21}[/tex]
Step-by-step explanation:
The goal of the question
Find the value of 2abcosC
Given information
a = 4
b = 3
c = 2
Given formula (Law of Cosine)
[tex]c^2=a^2+b^2-2abcosC[/tex]
Substitute values into the formula (Leave 2abcosC as it is)
[tex](2)^2=(4)^2+(3)^2-2abcosC[/tex]
Simplify the exponents
[tex]4=16+9-2abcosC[/tex]
Simplify by addition
[tex]4=25-2abcosC[/tex]
Add 2abcosC on both sides
[tex]4+2abcosC=25-2abcosC+2abcosC[/tex]
[tex]4+2abcosC=25[/tex]
Subtract 4 on both sides
[tex]4 + 2abcosC-4=25-4[/tex]
[tex]\Large\boxed{2abcosC=21}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
C
Step-by-step explanation:
a² + b² - 2abcosC = c² ( subtract a² + b² from both sides )
- 2abcosC = c² - a² - b² ( multiply through by - 1 )
2abcosC = a² + b² - c²
a, b, c are the sides opposite vertices A, B, C
here a = 4 , b = 3 , c = 2 , then
a² + b² - c²
= 4² + 3² - 2²
= 16 + 9 - 4
= 25 - 4
= 21
Which of the following is a correct interpretation of the expression -4 + 13−4+13minus, 4, plus, 13?
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
The number that is 444 to the left of -13−13minus, 13 on the number line
(Choice B)
B
The number that is 444 to the right of -13−13minus, 13 on the number line
(Choice C)
C
The number that is 131313 to the left of -4−4minus, 4 on the number line
(Choice D)
D
the sum between negative 4 and 13, this can be tought as:
Stepping on -4 and moving 13 units to the right.Stepping on 13 and moving 4 units to the left.Which is the interpretation of the given expression?
The expression is just the sum:
-4 + 13
So we have the sum between negative 4 and 13, this can be tought as:
Stepping on -4 and moving 13 units to the right.Stepping on 13 and moving 4 units to the left.Now, none of the written options (a, b, and c) depict any of this, so I assume that the correct option is the one that you did not write, which is option D.
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It took Sandra 5 hours to drive to her father's house and back. She lives 120 miles from the city where her father lives. What was her per hour driving rate for the trip?
Which linear inequality is represented by the graph
Answer:[tex]\Large\boxed{First~choice.~y\leq \frac{1}{2}x+2 }[/tex]
Step-by-step explanation:
Determine the equation form of the inequalityGiven function form
Point-Slope form : y = mx + b
m = slopeb = y-interceptGiven the information from the graph
Since the graph crosses (0, 2), b = 2
Find the slope
Choose any two points from the graph
(x₁, y₁) = (0, 2)
(x₂, y₂) = (2, 3)
Slope = (y₂ - y₁) / (x₂ - x₁)
Slope = (3 - 2) / (2 - 0)
Slope = 1 / 2
m = 1 / 2
Therefore, the equation form of the inequality is y = (1/2)x + 2
Determine whether it is ≤ or ≥Since it is a solid line, the values on the line are also included. Thus, it must be either greater than or equal to, or less than or equal to.
The shaded region is below the solid line, therefore, it is ≤
Therefore, the inequality is [tex]\Large\boxed{y\leq \frac{1}{2}x+2 }[/tex]
To double-check the sign, we can substitute (0, 0) into the inequality to see whether it is true.
y ≤ (1/2) x + 2
0 ≤ (1/2) 0 + 2
0 ≤ 2 TRUE
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Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
[tex]m = m \\ \frac{97 - 68}{2 - 1} = \frac{134 - 97}{3 - 2} \\ 29 = 37 \\ this \: function \: is \: non \: linear[/tex]
Since the only two other options are quadratic and given that it must satisfy one of them i will assume the following general form of the function.[tex]f(1) = 68 \: \: \: f(2) = 97 \: \: \: \: f(3) = 134 \\ f(x) = ax {}^{2} + bx + c[/tex]
Substitute in the first function any point.[tex]f(1) = 68 \\ 7.8(1) {}^{2} - 2.9(1) + 67.1 = 68 \\ 7.8 - 2.9 + 67.1 = 68 \\ 72 = 68 \\ this \: is \: false[/tex]
I'm pretty sure something is wrong with the givenIn an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for Independent Practice For See Exercises Example 8-9102113124 Extra Practice Skills Practice p. S5 Application Practice p. S28 the translation that maps the rocket from position I to position 2.
The rule of the translation that maps the rocket from position I to position 2 is 4 units right and 4 units up
How to determine the rule for the translation?The translation is added as an attachment
From the attached figure, we have the following corresponding coordinates:
Figure 1 = (0, 0)
Figure 2 = (4, 4)
The rule of translation is calculated as:
(x, y) = T<Figure 2 - Figure 1>
This gives
(x, y) = T<4 - 0, 4 - 0>
Evaluate
(x, y) = T<4, 4>
Hence, the rule of the translation that maps the rocket from position I to position 2 is 4 units right and 4 units up
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Complete question
In an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for the translation that maps the rocket from position I to position 2.
Find the domain of the graphed function.
A. x is all real numbers.
B. 2≤x≤5
C. 1≤x≤5
D. x≥2
Answer:
Step-by-step explanation:
The domain is an x thing. We are being asked to determine what x values to function takes on. We see that the blue dot on the left is at the x-value of 2. It doesn't go any farther to the left of 2. Since the dot is closed, the inequality symbol will have the "equal to" line underneath.
The function then continues along the x axis and stops at 5, closed hole. This means that the function does not go past the x value of 5 but it is included in our domain.
2 ≤ x ≤ 5. That is choice B.
Out of the total respondents, the percentage of respondents from the 46–55 age group who rated the film excellent is %. write your answer up to two decimal places.
Missing data:
Viewer's Age Group Excellent Good Average Poor Marginal Total
16–25 52 42 12 7 113
26–35 33 50 5 9 97
36–45 58 12 28 34 132
46–55 25 17 22 12 76
56 + 12 5 3 8 28
Marginal Total 180 126 70 70 446
A rating of good or excellent indicates the audience liked the movie, while a rating of poor indicates the audience disliked the movie.
How to determine the rating of the film from the 46–55 age group?A movie producer accomplished a survey behind a preview screening of her latest movie to estimate how the film would be accepted by viewers from various age groups. The table displays the numbers of viewers in various age groups who ranked the film excellent, good, average, and poor.
25/446 = 0.05605
0.05605 [tex]*[/tex] 100% = 5.605%
Out of the entire respondents, the percentage of respondents from the 46–55 age group who ranked the film excellent exists at 5.605%.
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1,615×10 to the 2 power
simplifying 1,615×10 to the 2 power would give 161500
Simplifying the index form
Index notation is known as a way of representing numbers (constants) and variables (e.g. x and y) that have been multiplied by themselves a number of times.
Index notations, or indices are use to simplify expressions or solve equations involving powers.
For instance;
8 × 8 × 8 × 8
8 is multiplied by itself 4 times
In index form , it is written as 8 ^4, that is, 8 to the 4 power
From the information given, we have to simply 1,615×10 to the 2 power
It can be written as;
= 1, 615 × 10 × 10
= 1, 615 × 100
= 161500
Thus, simplifying 1,615×10 to the 2 power would give 161500
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The external and the internal radii of a hollow cylindrical metallic vessel 56 cm long are 10.5 cm and 10.1 cm respectively. Find the cost of the metal contained by the vessel at Rs 2 per cubic cm. Also, find the cost of polishing it's outer surface 20 paisa per square cm.
The total cost of polishing it's outer surface 20 paisa per square cm exists Rs. 738.58.
How to estimate the cost of polishing metallic vessel outer surface 20 paisa per square cm.?External radius of the hollow cylinder = 10.5 cm
Internal radius of the hollow cylinder = 10.1 cm
Height of the hollow cylinder = 56 cm
Cost to Polish Outer Surface: 20 paisa/sq. cm.
Surface Area of Hollow Cylinder = Circumference [tex]*[/tex] height
Surface area of hollow cylinder = 2 π r h
= 2 π (10.5 cm) 56 cm
= π (21)(56)
= 3692.64 [tex]cm^2[/tex] (Using π ≈ 3.14 rounding to 2 digits)
Total Cost of metallic cylinder [tex]$= 3692.64 cm^2 * 20 paisa/cm^2[/tex]
= 73,852.8 paisa
= Rs. 738.58
Therefore, the total cost of polishing it's outer surface 20 paisa per square cm exists Rs. 738.58.
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The figure shown is made up of a cone and a cylinder. The height of the cone is 5 ft and its diameter is 12 ft. The height of the cylinder is 20 ft.
Diagram shows a composite shape formed by placing a cone next to a cylinder that share a common circular base. The diameter of the cone is labeled 12 feet, the height of the cone is labeled 5 feet, and the height of the cylinder is labeled 20 feet.
Find the lateral surface area of the cone and the surface area of the sides and bottom of the cylinder.
The lateral surface area of the cone is about Blank ft2. The surface area of the sides and the bottom of the cylinder is about blank ft2. The total surface area of the figure is about blank ft2.
Options for blank 1
A.260
B.188
C. 147
Options for blank 2
A. 980
B. 867
C. 1,508
Options for blank 3
A. 1,168
B. 1,768
C.1,127
D. 1,014
The lateral area of a cone is 147ft².
The total surface area of the cylinder is 867ft².
The total surface area is 1014ft²
What is the lateral surface area of a cone?The lateral area of a cone is defined as the area covered by the curved surface of the cone.
Lateral area of a cone = πr x √(r² + h²)
Where:
r = radius = 12/2 = 6ft h = height = 5ftπ = 3.143.14 x 6 x √(36 + 25) = 147ft²
What is the total surface area of the figure ?
Total surface area of the cylinder : πr(r + 2h),
(3.14 x 6) x (6 + 40) = 867ft²
Total surface area = 867 + 147= 1014ft²
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I reallly badly need help
Answer:
You can find either x or y by applying the Theorem which states that sum of opposite angles of a cyclic quadrilateral is 180°
Step-by-step explanation:
The above is a cyclic quadrilateral.
What is a cyclic quadrilateral?A Cyclic quadrilateral is having all the vertices of a quadrilateral ( four sided figure) inside a circle, that is a quadrilateral inscribed in a circle.
To find either x or y, we should note:
The sum of opposite angles of a cyclic quadrilateral is 180° or Angles in opposite segment are supplementary.
Therefore,
2x + 2x + 4= 180
4x = 180 - 4
4x = 176
x = 44°
To find y
3y + 8 + y = 180°
4y = 180 - 8
4y = 172
y = 43°
Therefore, you can find either x or y by applying the Theorem which states that sum of opposite angles of a cyclic quadrilateral is 180°
A data set contains three points, and two of the residuals are 7 and -3. What 7
is the third residual?
The data set in discuss which contains three points and two of the residuals are 7 and -3 would have its third residual as; -4.
What is the third residual if the data set characterized as having 2 of its three residuals to be; 7 and -3?According.tinthe task content, the data set in discuss has three points and consequently should have three residuals.
Additionally, two of the three residuals have been given and on this note, it follows that the third residual must have a value such that the algebraic sum of all residuals is 0.
Hence, we have; -3 + 7 +x = 0;
x = 3-7 = 4.
Ultimately, the third residual is; -4.
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Talia buys a 6-ounce container of macadamia nuts for $5.94 what is the unit price per ounce?
Answer:
$0.99 or 99 cents
Step-by-step explanation:
Since 6 ounces is worth $5.94, divide 5.94 by 6 to get the unit price. This is $0.99 or 99 cents.
A quantity h varies directly with w and inversely with p. when w = 4 and p = 6, h = 2. what is the constant of variation?
The value of the constant of variation is 3.
The correct option is C.
What is variation?Direct variations occur when variables in a variation vary proportionately, that is, when they both increase or decrease at the same time. As a result, you can write X Y symbolically if X and Y are in direct variation. The variables alter disproportionally in inverse or indirect variations.
According to the given information:
Let the constant of variation be k
A quantity h varies directly with w and inversely with p
That is,
h∝ w/p
Introducing the constant of variation, k, we get
h = k(w/p)
Also, from the give information,
w = 4
p = 6
h = 2
Putting the parameters into the relation, we get
2 = K x 4/6
6 x 2= 4K
K = (6 x 2)/4
K = 3
Hence, the value of the constant of variation is 3.
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I understand that the question you are looking for is:
A quantity h varies directly with w and inversely with p. If h = 2, w = 4, and p = 6. What is the constant of variation?
A. 1/2
B. 4/3
C. 3
D. 12
Answer:
C
Step-by-step explanation:
The verified user said so
A man bought a car for 8500 and sold it for 9500. find his percentage gain
Answer:
11.7647%
Hope this helps!
A 5-pound box of raspberries costs $27.20. What is the price per ounce?
$
Answer:
$0.34 per ounce
Step-by-step explanation:
A 5 pound box can be changed to ounces. There are 16 ounces in a pound. So we have 5 groups of 16.
5 × 16 is 80 ounces.
There are 80 ounces in 5lbs.
To get a price per ounces, divide.
dollars/ounces
= 27.20/80
= 0.34 dollars/oz.
This is 34cents per ounce.
The sum of two numbers is 84. The square of the first number is 6 more than the second number.
Write a system of equations to find the value of x, the first number, and y, the second number.
y = -x +
y = x² +
The system of equations to find the value of x, the first number, and y, the second number are;
y = -x + 84
y = x² + 6
System of equationThe system of equation are equations that contains more than two equation and unknown.
Let the two unknown numbers x and y
If the sum of two numbers is 84, hence;
x +y = 84
y = -x + 84
If the square of the first number is 6 more than the second number, hence;
x² = y + 6
Substitute equation 2 into 2 to have:
x² = y + 6
x² = (-x+84) + 6
x² = -x+84 + 6
x² +x+84 + 6 = 0
x² +x + 90 = 0
Hence the system of equations to find the value of x, the first number, and y, the second number are;
y = -x + 84
y = x² + 6
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In which order do these stages occur? 4, 3, 2, 1, 3 3, 2, 1, 3, 4 4, 3, 2, 3, 1 4, 2, 3, 1, 3
The order of the stages of recession will be 4, 2, 3, 1, 3.
What happens in a recession?
This is the first stage, and it's characterized by a decrease in activity throughout the economy. This can manifest itself in different ways, like lower production levels, fewer jobs, and less spending by consumers and businesses.Who benefits in a recession?
Rental agents, landlords, and property management companies can thrive during a recession when renting is likely to become a more appealing option, if not the only one available.
So, these stages are occur -
4. peak production
2. falling production
3. falling demand (occurs twice)
1. job loss
3. falling demand (occurs twice)
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The complete question is-
There are five stages in a recession.
1. job loss
2. falling production
3. falling demand (occurs twice)
4. peak production
In which order do these stages occur?
4, 3, 2, 1, 3
3, 2, 1, 3, 4
4, 3, 2, 3, 1
4, 2, 3, 1, 3
4, 2, 3, 1, 3.Answer:4, 2, 3, 1, 3.
4, 2, 3, 1, 3.Step-by-step explanation:4, 2, 3, 1, 3.
In 1993, the moose population in a park was measured to be 4760. By 1999, the population was measured again to be 5960. If the population continues to change linearly:
The formula for moose population is P = 4760 + (number of years x 200).
The moose population in 2003 would be 6760.
What would be the moose population in 2003?When a population increases linearly, it means that it increases by the same amount each year.
Rate of linear increase: (population in 1999 - population in 1993) / difference in years
(5960 - 4760) / (1999 - 1993)
1200 / 6 = 200
Linear function : initial population + (rate of increase x number of years)
Population in 2003: 4760 + (200 x 10)
4760 + 2000 = 6760
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Help pls!!
Use 3.14 for pi.
To find the area, we take into account that your data is:
r = 6 cmπ = 3.14To calculate the area of the circle, we use the following formula:
Area: π * r²
We substitute the data in the formula, and we begin to solve.
A = 3.14 * (6 cm)²taking the square root
A = 3.14 * 36 cm²Multiply the two numbers
A = 113.04 cm²Therefore, the area of the circle is 113.04 cm². The correct alternative is option "A".
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In the question, it is given that a circle has a radius of 6 cm , we have to find the Area.
To Find the area of the circle , we must know this formula:
[tex] \quad\quad\quad \longrightarrow\underline{\boxed{\color{violet}{A=\pi*r^2}}}[/tex]
Now, we will substitute the values in the formula:
[tex]\rightarrow\sf{A=\pi*r^2} \: \: \: \: \: \: \: ( \red{\pi = 3.14})[/tex]
[tex]\rightarrow\sf{A = 3.14 * 6^2}[/tex]
[tex]\rightarrow\sf{A = 3.14 * 36}[/tex]
[tex]\rightarrow\sf{A = 113.0}[/tex]
The area of circle is 113.0 approx .
[tex] \color{blueviolet}{ \mathfrak{llNotYourll}}[/tex]
Find the ratio of the areas of to squares whose sides are 3cm and 5cm, respectively
Answer: 9 cm² : 25 cm²
Step-by-step explanation:
The area of a square can be found with x² where x is the length of one side.
x² : x²
3² : 5²
9 : 25
9 cm² : 25 cm²
A circle is described by the equation x2 y2 14x 2y 14 = 0. what are the coordinates for the center of the circle and the length of the radius?
Answer:
length is (-7,-1) and radius is 6
Step-by-step explanation:
We are given the expression of the equation of a circle that is
x2 + y2 + 14x + 2y + 14 = 0.
Using completing the squares:
x2 + y2 + 14x + 2y + 14 = 0(x+7)^2 + (y+1) ^2 = -14 + 49 + 1(x+7)^2 + (y+1) ^2 = 36 center thus is at (-7,-1) and the radius is equal to square root of 36 equal to 6.
What is the solution to this system of equations?
4x + 5y = 7
3x – 2y = –12
The solution is
The solution to the system of equations is (-2, 3)
Powers of 10 and Exponents (number 4 is a 10 incase u can’t see of the ripped page.)
Answer:
below
Step-by-step explanation:
2) 10², the second power of ten
3) 10⁴, the fourth power of ten
4) 1000
5) 400
6) 90,000
7) 10
8) 100,000
9) 50
10) 7000
11) 8
12) 24 x 10⁴
13) 93,000,000
Answer + Step-by-step explanation:
Question #1) x4 = 256 solve for x
In the given equation the value of x is 4.
According to the statement
we have given that the a equation and we have to solve that equation for x and find the value of the x.
So, For the value of x in given statement:
The given equation is
x⁴ = 256
we know that the 2 changes into the square root after rearrangement in the equation so, the equation become
(x²)² = 256
Now,
(x²) = √256
Then equation becomes
(x²) = 16
This is because the square root of the 256 is 16.
Now another square changes into square root then
x = √16
And
x = 4
So, In the given equation the value of x is 4.
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Sea World's dining card offers a choice of a three main entrees, a choice from five sides and a choice from four desserts. How many different meals can be selected?
Find the matrix for the linear transformation which rotates every vector in r2 through an angle of −π/3.
The matrix for the linear transformation for which rotates every vector in r2 is [tex]\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right][/tex].
In this question,
The angle is -π/3 = -60°
Clockwise rotations are denoted by negative numbers.
Matrix for counter-clockwise direction for an angle α is
[tex]\left[\begin{array}{cc}cos\alpha &-sin\alpha \\-sin\alpha &cos\alpha \end{array}\right][/tex]
Then, matrix for clockwise direction for an angle α
Put α = -α in the above matrix,
⇒ [tex]\left[\begin{array}{cc}cos(-\alpha) &-sin(-\alpha) \\-sin(-\alpha) &cos(-\alpha) \end{array}\right][/tex]
⇒ [tex]\left[\begin{array}{cc}cos\alpha &sin\alpha \\sin\alpha &cos\alpha \end{array}\right][/tex]
Now substitute α = 60°
Then matrix becomes,
⇒ [tex]\left[\begin{array}{cc}cos60 &sin60 \\sin60 &cos60 \end{array}\right][/tex]
⇒ [tex]\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right][/tex]
Hence we can conclude that the matrix for the linear transformation for which rotates every vector in r2 is [tex]\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right][/tex].
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Find the third order maclaurin polynomial. Use it to estimate the value of sqrt1.3
You can use the well-known binomial series,
[tex]\displaystyle (1+x)^\alpha = \sum_{k=0}^\infty \binom \alpha k x^k[/tex]
where
[tex]\dbinom \alpha k = \dfrac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-(k-1))}{k!} \text{ and } \dbinom \alpha0 = 1[/tex]
Let [tex]\alpha=\frac12[/tex] and replace [tex]x[/tex] with [tex]3x[/tex]; then the series expansion is
[tex]\displaystyle (1+3x)^{1/2} = \sum_{k=0}^\infty \binom{\frac12}k (3x)^k[/tex]
and the first 4 terms in the expansion are
[tex]\sqrt{1+3x} \approx 1 + \dfrac{\frac12}{1!}(3x) + \dfrac{\frac12\cdot\left(-\frac12\right)}{2!}(3x)^2 + \dfrac{\frac12\cdot\left(-\frac12\right)\cdot\left(-\frac32\right)}{3!}(3x)^3[/tex]
which simplify to
[tex]\sqrt{1+3x} \approx \boxed{1 + \dfrac32 x - \dfrac98 x^2 + \dfrac{27}{16} x^3}[/tex]
You can also use the standard Maclaurin coefficient derivation by differentiating [tex]f[/tex] a few times.
[tex]f(x) = (1+3x)^{1/2} \implies f(0) = 1[/tex]
[tex]f'(x) = \dfrac32 (1+3x)^{-1/2} \implies f'(0) = \dfrac32[/tex]
[tex]f''(x) = -\dfrac94 (1+3x)^{-3/2} \implies f''(0) = -\dfrac94[/tex]
[tex]f'''(x) = \dfrac{81}8 (1+3x)^{-5/2} \implies f'''(0) = \dfrac{81}8[/tex]
Then the 3rd order Maclaurin polynomial is the same as before,
[tex]\sqrt{1+3x} \approx f(0) + \dfrac{f'(0)}{1!} x + \dfrac{f''(0)}{2!} x^2 + \dfrac{f'''(0)}{3!} x^3 = 1 + \dfrac32 x - \dfrac98 x^2 + \dfrac{27}{16} x^3[/tex]
Now,
[tex]\sqrt{1.3} = \sqrt{1+3x} \bigg|_{x=\frac1{10}} \\\\ ~~~~~~~~ \approx 1 + \dfrac32 \left(\dfrac1{10}\right) - \dfrac98 \left(\dfrac1{10}\right)^2 + \dfrac{27}{16} \left(\dfrac1{10}\right)^3 \\\\ ~~~~~~~~ = \dfrac{18,247}{16,000} \approx \boxed{1.14044}[/tex]
Compare to the actual value which is closer to 1.14018.
[tex]\sqrt{1+3x}=1+\frac{3}{2} x-\frac{9}{8} x^{2} + \frac{81}{8}x^{3}[/tex] is the maclaurin polynomial and estimate value of [tex]\sqrt{1.3}[/tex] is 1.14. This can be obtained by using the formula to find the maclaurin polynomial.
Find the third order maclaurin polynomial:Given the polynomial,
[tex]f(x)=\sqrt{1+3x}=(1+3x)^{\frac{1}{2} }[/tex]
The formula to find the maclaurin polynomial,
[tex]f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^{2} + \frac{f'''(0)}{3!}x^{3}[/tex]
Next we have to find f'(x), f''(x) and f'''(x),
[tex]f'(x) = \frac{3}{2}(1+3x)^{-\frac{1}{2} }[/tex] [tex]f''(x) =-\frac{9}{4}(1+3x)^{-\frac{3}{2} }[/tex][tex]f'''(x) = \frac{81}{8}(1+3x)^{-\frac{5}{2} }[/tex]By putting x = 0 , we get,
[tex]f(0)=(1+3(0))^{\frac{1}{2} }=1[/tex] [tex]f'(0) = \frac{3}{2}(1+3(0))^{-\frac{1}{2} }=\frac{3}{2}[/tex][tex]f''(0) =-\frac{9}{4}(1+3(0))^{-\frac{3}{2} }=-\frac{9}{4}[/tex][tex]f'''(0) = \frac{81}{8}(1+3(0))^{-\frac{5}{2} }=\frac{81}{8}[/tex]Therefore the maclaurin polynomial by using the formula will be,
[tex]\sqrt{1+3x}=f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^{2} + \frac{f'''(0)}{3!}x^{3}[/tex]
[tex]\sqrt{1+3x}=1+\frac{3}{2} x-\frac{9}{8} x^{2} + \frac{81}{8}x^{3}[/tex]
To find the value of [tex]\sqrt{1.3}[/tex] we can use the maclaurin polynomial,
[tex]\sqrt{1.3}[/tex] is [tex]\sqrt{1+3x}[/tex] with x = 1/10,
[tex]\sqrt{1+3(1/10)}=1+\frac{3}{2} (1/10)-\frac{9}{8} (1/10)^{2} + \frac{81}{8}(1/10)^{3}[/tex]
[tex]\sqrt{1+3(1/10)}=\frac{18247}{16000} = 1.14[/tex]
Hence [tex]\sqrt{1+3x}=1+\frac{3}{2} x-\frac{9}{8} x^{2} + \frac{81}{8}x^{3}[/tex] is the maclaurin polynomial and estimate value of [tex]\sqrt{1.3}[/tex] is 1.14.
Learn more about maclaurin polynomial here:
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