The area of R can be calculated using the definite integral of the function y = e^(2x-x^2) between x = 0 and x = 1.
The integral expression is ∫e^(2x-x^2)dx = ∫e^u du = e^u + C = e^(2x-x^2) + C. Evaluating the integral between x = 0 and x = 1 gives the area of R as e^2 - 1.
(b) The area of S can be calculated using the definite integral of the function y = e^(2x-x^2) between x = 0 and x = 1, minus the integral of the function y = 1 between x = 0 and x = 1. The integral expression for S is ∫e^(2x-x^2)dx - ∫1dx = ∫e^u du - x = e^u + C - x = e^(2x-x^2) + C - x. Evaluating the integral between x = 0 and x = 1 gives the area of S as e^2 - 1 - 1.
(c) The volume of the solid generated when R is rotated about the horizontal line y = 1 can be found using the integral expression ∫2πx(e^(2x-x^2) - 1)dx. This expression represents the area of the circular cross-sections of the solid multiplied by 2πx, which is the circumference of the circle. When evaluated between x = 0 and x = 1, this integral gives the volume of the solid generated when R is rotated about the horizontal line y = 1.
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write the equation of an ellipse with center at (2, 1), one vertex at (2, -4), and one focus at (2, -2)
An ellipse is defined as the set of all points such that the sum of the distances between the point and the two foci is constant. The equation of an ellipse can be represented in the standard form as:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) is the center of the ellipse, a is the distance from the center to a vertex, and b is the distance from the center to the focus.
Given that the center of the ellipse is at (2, 1), one vertex is at (2, -4), and one focus is at (2, -2), we can use this information to find the values of a and b, and represent the equation of the ellipse:
a = distance from center to vertex = |1 - (-4)|/2 = 2.5
b = distance from center to focus = |1 - (-2)|/2 = 1.5
The equation of the ellipse is:
(x-2)^2/2.5^2 + (y-1)^2/1.5^2 = 1
which can also be written as:
(x-2)^2/6.25 + (y-1)^2/2.25 = 1
This equation represents the ellipse with center at (2, 1), one vertex at (2, -4), and one focus at (2, -2)
(c) use the answers to parts (a) and (b) to estimate by how much the average monthly cell phone bill changed between 1990 and 2008. (round your answer to two decimal places.)
The area above x axis is 12.6 sq units while the area under the y-axis is 42.443 sq units. The change in the average phone bill is 29.843 units.
a) Here we see that the graph is in the shape of a right-angled triangle above the x-axis
The area of a triangle = 1/2 X Base X Height
Here the base will be the measure of the distance of co-ordinates on the x-axis
The height is the distance between the coordinates on the axis for that triangle
= 1/2 X (16 - 8) X (3.15 - 0)
= 1/2 X 8 X 3.15
= 12.60 sq units
b)
Below the x-axis, the graph is a rectangle and a triangle
The area of the rectangle = length X width
Here length is the distance of the co-ordinates of the y-axis and width is the distance between the coordinates for the x-axis
Hence we get
(0 - (-5.14))(8 - 0)
= 5.14 X 8
= 41.12 sq units for the rectangle.
The area of the triangle will be
1/2 X (18 - 15.48) X (0 - (-1.05))
1/2 X 2.52 X 1.05
= 1.323 sq unit
Hence, the total area is
41.12 + 1.323
= 42.443 sq unit
c)
Since the area under the graph represents change,
Here we see that the change in the average cell phone bill will be
42.443 + 12.6
= 55.043
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Complete Question
(Image Attached)
Work out 194.55 divided by 1.5
Answer:129.7
Step-by-step explanation:
Literally just used a calculator but if you need explanation lmk
Pls help me with questions, I will give brainliest
Cotangent is the inverse of tangent:
cotΘ = 1/tanΘ
cotΘ = [tex]1/\frac{\sqrt{6} }{2}[/tex]
=[tex]\frac{2}{\sqrt{6}}[/tex]
=[tex]\frac{2}{\sqrt{6}}*\frac{\sqrt{6}}{\sqrt{6}}=\frac{2\sqrt{6}}{6}[/tex] Rationalize the denominator
cotΘ = [tex]\frac{\sqrt{6}}{3}[/tex]
Plug and solve.
[tex]\frac{\frac{\sqrt{6}}{2}+\frac{\sqrt{6}}{3} }{\frac{\sqrt{6}}{2}-\frac{\sqrt{6}}{3}}[/tex]
We'll break the problem into 2, simplify the denominator, then simplify the numerator.
Simplify the denominator with the common denominator 6, of 2 and 3:
[tex]{\frac{\sqrt{6}}{2}-\frac{\sqrt{6}}{3} = {\frac{3\sqrt{6}}{6}-\frac{2\sqrt{6}}{6} = {\frac{3\sqrt{6}-2\sqrt{6}}{6} = {\frac{\sqrt{6}}{6}[/tex]
Simplify the Numerator with the common denominator 6, of 2 and 3:
[tex]{\frac{\sqrt{6}}{2}+\frac{\sqrt{6}}{3} = {\frac{3\sqrt{6}}{6}+\frac{2\sqrt{6}}{6} = {\frac{3\sqrt{6}+2\sqrt{6}}{6} = {\frac{5\sqrt{6}}{6}[/tex]
Back in original equation with simplified numerator and denominator:
[tex]\frac{\frac{5\sqrt{6}}{6}}{\frac{\sqrt{6}}{6}}[/tex]
Multiply by reciprocal and simplify:
[tex]\frac{5\sqrt{6}}{6}*\frac{6}{\sqrt{6}} = \frac{5\sqrt{6} * 6}{6\sqrt{6}} = 5*\frac{6\sqrt{6}}{6\sqrt{6}} = 5*1 = 5[/tex]
A graph the shows y= -3/2x - 2
The graph of the given equation is given at end.
A graph is a graphic depiction or diagram that displays data in an organized manner. The points on a graph are typically used to depict the relationships between two or more things.
Determine the slope and y-intercept first, then put the equation in y-intercept (y=mx+b) form to get the equation of a graphed line. The y-to-x change ratio is known as the slope. The two locations you locate on the line should be connected by a sloping triangle.
Common graph types include line graphs, bar graphs, pie charts, scatter plots, and histograms. Graphs are an excellent tool for visualizing data and presenting statistics. An example of displaying numerical data is with a bar graph or chart.
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A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side A C is six units. Side B C is three units.
The picture explains it all. We use tangent rule to find out the measure of angle A.
The measure of angle unknown angle is 30°.
Use the concept of the triangle defined as:
A triangle is a three-sided polygon, which has three vertices and three angles which has a sum of 180 degrees.
Given that,
Side AC = 6 units
Side BC = 3 units
Let ∠BAC = θ
Since we know that,
Sin θ = opposite/Hypotenuse
Sin θ = BC/AC
Put the values,
Sin θ = 3/6
Sin θ = 1/2
Since we know that,
sin(π/6) = 1/2
Therefore,
Sin θ = sin(π/6)
Take the inverse of sin on both sides we get,
θ = π/6 or 30°
Hence,
The measure of angle unknown angle is 30°.
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The picture shows two ladders propped against
facing walls. The end of the first ladder is 5m
above the base of the second wall. The second
ladder is 5m above the base of the first wall.
5m
Can you work out at what height the
ladders cross?
Are you surprised that we don't need to know
the distance between the two walls?
5m
5m
10m
By using Triangles ABC and ADC given in image, Ladders will cross at 10/3 m above the ground.
What are triangles?
Triangles are polygons in geometry that have three sides and three vertices. This figure is two-dimensional and has three straight sides. A triangle is a three-sided polygon. A triangle's three angles added together equals 180°. A single plane contains the triangle. The triangle is classified into six forms based on its sides and angles.
A triangle is categorized into three categories based on its sides, namely:
Scalene Triangle - Each side has a distinct length.
Isosceles Triangle - A triangle with two sides of equal length and one side of a different length.
Equilateral Triangle - A triangle has three sides that are of the same length.
A triangle is categorized into three categories based on its angles, namely:
Acute Angle Triangle - A triangle with all of its angles smaller than 90°.
Obtuse Angle Triangle - A triangle with one of its angles larger than 90°.
Triangle with a Right Angle - A triangle with one of its angles equal to 90°.
Now,
As shown in image that the angles ∠BAD and ∠ADC are alternate angles, and therefore equal. Likewise, angles ∠ABC and BCD are also alternate and therefore equal.
∠BEA and ∠CED are opposite, which means that they are equal. You can see these equal angles in the image.
This means that the triangles ABE and DCE have the same angles as each other, and are therefore similar.
This means:
AE/ED=AB/DC=5m/10m=1/2
Therefore AE=1/3AD
This means E is 1/3 of the way up from the floor to D, so is at a height of 10/3m=3 1/3m
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if trace of a matrix is postive and determinant is positive does it mean eigen values are all positive?
To have the determinant negative, at least one Eigenvalue has to be negative but the reverse of this statement may or may not be true.
If we add all the diagonal elements of the square matrix, we get the trace of the given matrix.
Eigenvalues is a scalar that associates with a set linear equation described as -
(A - λ I) X = 0
If we take the product of all the eigenvalues of a matrix, we'll get the determinant of the matrix. If we consider a case in which the trace of the matrix is +ve, and its determinant is -ve, then at least one or an odd no. of Eigenvalues must be negative, since their product has to be positive when we solve for the determinant of the matrix
Therefore, to have the determinant negative, at least one Eigenvalue has to be negative but reverse may or may not be true.
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Gabriel is deciding between two landscaping companies for his place of business. Company A charges $45 per hour and a $250 equipment fee. Company B charges $55 per hour and a $200 equipment fee. Let AA represent the amount Company A would charge for tt hours of landscaping, and let BB represent the amount Company B would charge for tt hours of landscaping. Write an equation for each situation, in terms of t,t, and determine the interval of hours, t,t, for which Company A is cheaper than Company B.
The linear functions for each situation are given as follows:
Company A: 45x + 250.Company B: 55x + 200.The interval of hours for which Company A is cheaper than Company B is given as follows:
x > 5.
How to model the linear functions?The linear functions are modeled in slope-intercept format, as follows:
y = mx + b.
In which:
The slope m represents the cost per hour.The intercept b represents the equipment fee.Hence the functions are defined as follows:
Company A: 45x + 250.Company B: 55x + 200.Company A will be cheaper when:
45x + 250 < 55x + 200
10x > 50
x > 5.
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A car dealership pays a wholesale price of $14,000 to purchase a vehicle. They sell the car for $16,100. If the dealership decided to use a 20% markup instead, how much MORE would they make?
If the dealership decided to use a 20% markup instead, then they earn $700 more.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given that A car dealership pays a wholesale price of $14,000 to purchase a vehicle.
They sell the car for $16,100
If the dealership decided to use a 20% markup instead, we have to find the hu=ow much more would they make.
Let us find 20% of 14000
20/100×14000
0.2×14000
2800
So with 20% the price will become $16800.
As they sell the car for $16100.
The difference between $16800-$16100 is $700
Hence, If the dealership decided to use a 20% markup instead, then they earn $700 more.
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A number is between 26 and 31. It has prime factors of 2 and 7. What is the number?
Answer:28
Step-by-step explanation:
Trick: just check if it's divisible by both numbers
PLEASE HELP! Find the slope of the secant line between x= 3 and x= 8 for the function f(x) = [tex]\sqrt{16x-47}[/tex]. (Enter your answer in either exact notation of round to four decimal places.
The slope of the secant line between x = 3 and x = 8 for the function [tex]f(x) = \sqrt{16x - 47}[/tex] is given as follows:
8/5.
How to obtain the slope of the secant line?The secant line intersects the function f(x) at the coordinates x = 3 and x = 8.
The numeric value for the function at each of these points is given as follows:
x = 3: [tex]f(x) = \sqrt{48 - 47} = 1[/tex].x = 8: [tex]f(x) = \sqrt{128 - 47} = \sqrt{81} = 9[/tex]Given two points, the slope of a line is calculated as the change in y divided by the change in x, hence:
m = (9 - 1)/(8 - 3)
m = 8/5.
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which of the following sampling strategies makes it difficult to discern what sub-population the sample represents?
Convenience sampling represents the sub-population of the sample strategies.
Convenience sampling is defined as a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand and is a type of sampling that is most useful for pilot testing.
Convenience sampling can be used by almost anyone and has been around for generations. One of the reasons that it is most often used is due to the numerous advantages it provides
Expedited data collection, Ease of research, Ready availability, Cost-effectiveness.
Bias, Power is the disadvantages of convenience sampling.
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Find the volume of the solid of revolution generated when the region in the first quadrant bounded by the graphs of y = x2 +1, y = 5, and the y-axis is revolved about the line x = -3. a. 8πb.12πc. 20π d. 40π
When the area in the first quadrant bordered by x=0, y=1, x=y4, and y=10 is rotated about the line y=10, the solid's volume is 11π/3
What is meant by volume?Every three-dimensional item takes up space in some way. The volume of this area is what is used to describe it. The area occupied within an object's three-dimensional bounds is referred to as its volume. The object's capacity is another name for it. Volume is a mathematical term that describes how much space in three dimensions is occupied by an object or a closed surface. The measurement of volume is done in cubic units, like m3, cm3, in3, etc.Volume is a measurement of how much room a thing occupies. To put it another way, just as height and width are terms used to describe size, volume is a measurement of the size of an object.We using,
V = π ∫|(f(x) - d) 2 - (g (x) - d) 2|dx Integrating from a to b
For this case, a = 0, b = 1, g(x) = 1, f(x) = x1/4
Simplifying the equation,
V = π∫|(x1/4 - 10) 2 - (1 - 10) 2dx = π∫|x1/2 - 20x1/4 + 100 - 81|dx
= π[(2/3) x 3/2 - 20(4/5) x 5/4 + 19x] evaluated from at 1 and 0
= π|(2/3 -16 +19) = 11π/3
The complete question is?
find the volume of the solid obtained by rotating the region in the first quadrant bounded by x=0, y=1, x=y^4 about the line y=10
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What is the degree of the polynomial
Answer:
A polynomial’s degree is the highest or the greatest power of a variable in a polynomial equation. The degree indicates the highest exponential power in the polynomial (ignoring the coefficients).
For example: 6x4 + 2x3+ 3 is a polynomial. Here 6x4, 2x3, 3 are the terms where 6x4 is a leading term and 3 is a constant term. The coefficients of the polynomial are 6 and 2.
The degree of the polynomial 6x4 + 2x3+ 3 is 4.
how many minutes are in 3 years
Answer:
1,576,800 minutes
Here are the weights (in pounds) of a sample of 13 male eleventh graders.
148, 140, 159, 141, 151, 156, 173, 167, 160, 159, 146, 174, 170
Send data to calculator
Find the median weight of these students.?
pounds
S
Weight of the 13 male eleventh graders is 157.23 pounds. Median is the number in the middle, ranked from least to greatest.
Median weight = 148+140+159+141+151+156+173+167+160+159+146+174+170
= 2044/13
= 157.23
Median :
The value that separates the upper half of a population, a data sample, or a probability distribution from the lower half is known as the median. It could be referred to as "the middle" value for a data set. The value in the middle of a data set is called the median. This means that 50% of the data points have values that are smaller than or equal to the median, and 50% of the data points have values that are higher than or equal to the median. First, you count the number of data points (n) and arrange them in ascending order for a small data set.
How is the Median Calculated?The middle value in a set of data is called the median. First, arrange the data in ascending order from smallest to largest. Divide the total number of observations by two to get the midpoint value. Round that number up if there are an odd number of observations, and the value in that position is the median. Take the average of the values found above and below that position if the number of observations is even.
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An EMT sees himself as a bit of a rebel, but is highly sociable and is able to keep calm in an emergency. This person would likely score in the lower range of which of the following traits? (A) Psychoticism (B) Neuroticism (C) Extraversion (D) Conscientiousness
The correct answer is option D, extraversion, neuroticism, and psychoticism
Eysenck's theory of personality is based on three logical attributes namely:
a) introversion vs. extroversion – Extroversion leads to sociable life , while introversion causes need of being alone and limited interactions
b) neuroticism vs. stability – Neuroticism leads to anxiousness and overactive sympathetic nervous system while stability leads to emotional stability
c) psychoticism vs. socialization. - psychoticism leads to independent thinking, and hostility. While socialization leads to co-operative and conventional behaviour.
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at a local veterinary office, 48% of dogs get their teeth cleaned, while 35% of cats get their teeth cleaned. let p hat subscript upper d and p hat subscript upper c be the sample proportions of dogs and cats at this veterinary office, respectively, who get their teeth cleaned. suppose 25 dogs and 32 cats from this veterinary office are selected at random to collect data on their teeth-cleaning history. which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of p hat subscript upper d baseline minus p hat subscript c ?
The difference (Dogs – Cats) in the sample proportions of those that get their teeth cleaned typically varies about 0.054 from the true difference in proportions.
The third option is correct.
What is the standard deviation?
Standard deviation is a statistical measure of the spread of a dataset, defined as the square root of its variance. It is a way to quantify the amount of variation or dispersion of a set of data values.
Percentage of Dogs getting their teeth cleaned = 48% = pD
Percentage of Cats getting their teeth cleaned = 35% = pC
Number of Dogs selected = nD = 25
Number of Cats selected = nc = 32
As the sample sizes are large enough, we can apply Central Limit Theorem
According to Central Limit Theorem for proportions –
pD-hat ~ Normal(pD, {pD*(1-pD)}/nD)
Thus, pD-hat ~ Normal(0.48, 0.009984)
Similarly,
pC-hat ~ Normal(pC, {pC*(1-pC)}/nC)
Thus, pC-hat ~ Normal(0.35, 0.0071094)
Now if X ~ N(E(X), Var(X)) and Y ~ N(E(Y), Var(Y))
then X -Y ~ N(E(X)-E(Y), Var(X)-Var(Y)) (If X and Y are Independent)
Thus,
pD-hat - pC-hat ~ Normal(0.48-0.35, 0.009984 – 0.0071094)
pD-hat - pC-hat ~ Normal(0.13, 0.0028746)
Thus, Standard Deviation of pD-hat - pC-hat = (0.0028746^0.5) = 0.0536 that is 0.054 approximately
Hence, the difference (Dogs – Cats) in the sample proportions of those that get their teeth cleaned typically varies about 0.054 from the true difference in proportions.
Third Option is correct.
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The scale factor is _______________ the original figure
Answer: A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller). Therefore, The scale factor would be 1/2
A client who weighs 70 kg is receiving a dopamine solution of 800 mg/500 ml normal saline at 5 ml/hour. How many mcg/kg/minute is the client receiving? (Enter the numeric value only. If rounding is required, round to the nearest tenth.)
To calculate the mcg/kg/minute, first divide 800 mg by 500 ml to get 1.6 mg/ml. Then, multiply 1.6 mg/ml by 5 ml/hour to get 8 mg/hour. Finally, divide 8 mg/hour by 70 kg to get 0.114 mcg/kg/minute.
To calculate the mcg/kg/minute, first divide 800 mg by 500 ml to get 1.6 mg/ml, which is the concentration of dopamine in the solution. Next, multiply 1.6 mg/ml by 5 ml/hour to get 8 mg/hour, which is the rate of infusion. Finally, divide 8 mg/hour by 70 kg to get 0.114 mcg/kg/minute, which is the mcg/kg/minute the client is receiving. This calculation can also be done by dividing 800 mg by 500 ml to get 1.6 mg/ml, then multiplying 1.6 mg/ml by 5 ml/hour to get 8 mg/hour, and finally dividing 8 mg/hour by 70 kg to get 0.114 mcg/kg/minute. By doing this calculation, the client is receiving 0.114 mcg/kg/minute of dopamine solution.
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how many different numbers can be represented in a 4 byte integer. (note that this is not the same as the above question.)
The range, based on the solution, is between -2,147,483,648 and 2,147,483,647.
Describe range:When the sample maximum as well as minimum are subtracted, the range of a collection of data seems to be the difference between the greatest and lowest values. It uses the same units as the data to express itself.
Why is math range?The distance between the greatest and smallest data sets in some kind of a set of numerical data is known as the range. The average of the greatest and smallest data points is the middle.
According to the given information:1 byte has 8 bits,
so 4 bytes have 32 bits.
Each bit can have 2 distinct states, so we have 2³² distinct combinations of bits (in 4 bytes) which correspond to 2³² distinct whole numbers.
Normally,
the range is 0 to 4,294,967,295 but if you want to allow negatives,
the range is -2,147,483,648 to 2,147,483,647
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prove mr. greedy that it is enough to work on 2 edges to find your circumcenter by providing a 2 columns proof of the circumcenter theorem
The proof of the circumcenter theorem states that given three points A, B and C, drawing lines AB and AC or AB and BC and finding the perpendicular bisector of the corresponding segment will intersect at the circumcenter of triangle ABC.
Column 1:
1. Given three points A, B and C, draw lines AB and AC.
2. A bisector of angle BAC is the perpendicular bisector of segment AC.
3. The perpendicular bisector of segment AC intersects line AB at the circumcenter of triangle ABC.
Column 2:
1. Given three points A, B and C, draw lines AB and BC.
2. A bisector of angle ABC is the perpendicular bisector of segment BC.
3. The perpendicular bisector of segment BC intersects line AB at the circumcenter of triangle ABC.
The proof of the circumcenter theorem states that given three points A, B and C, drawing lines AB and AC or AB and BC and finding the perpendicular bisector of the corresponding segment will intersect at the circumcenter of triangle ABC. Therefore, it is enough to work on two edges to find the circumcenter.
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C
Salida 7-8: Comprobacion rapida
What is the missing number in the following equation
33 +52 = 9/1/20
10
A 3
B.5
С. 6
D 8
The numbers that have been omitted from the given series of a number with comparable discrepancies between them are known as missing numbers.
What is the missing number?Finding similarities between those numbers and filling in the missing phrases in the designated series and locations is the strategy indicated for writing the missing numbers.We can solve one-step equations in four different ways: by adding, subtracting, multiplying, and dividing. Both sides of an equation will stay equal if we add the same amount to each side.The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal. Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the symbol "equal."Two expressions joined by an equal sign form a mathematical statement known as an equation.To learn more about missing number refer to:
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[tex]-\frac{11}{48} +\frac{1}{72} -\frac{3}{16}[/tex]
Answer:
[tex]\frac{-29}{72}[/tex]
Step-by-step explanation:
use a calculator
FILL IN THE BLANK When multiplying matrices, multiply the elements in each ____ of the first matrix time the corresponding elements in each column of the second matrix.
The answer to the problem is: row
Which sign makes the following statement true?
1/4 _____ 5/16
Answer:
2. <
Step-by-step explanation:
To compare these two fractions we can first find them a common denominator. A common denominator between 4 and 16 is 16.
To get from 16 from 4 we multiply by 4 so we do the same to the numerator, changing 1/4 to 4/16.
5/16 already has a denominator of 16 so we can leave that one as is.
4/16___5/16
In this form it is obvious that 5/16 is the larger fraction because 5 is larger than 4, meaning that 5/16 is larger than 1/4.
g write a method odd sum that accepts an array of numbers. the method should return the total sum of all odd numbers of the array. odd sum([5, 4, 6, 13, 1])
The odd_sum method accepts an array of numbers and returns the total sum of all odd numbers found in the array.
def odd_sum(numbers)
sum = 0
numbers.each do |num|
if num.odd?
sum += num
end
end
sum
end
1. We define a method called odd_sum that accepts an array of numbers as an argument.
2. We set a variable called sum and set it to 0.
3. We iterate through the array using the each method.
4. We check if the number is odd using the odd? method.
5. If the number is odd we add it to the sum variable.
6. We return the sum variable at the end of the method.
The odd_sum method accepts an array of numbers and returns the total sum of all odd numbers found in the array
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.
Use the result of Example 6 to compute arctan 0.2 correct to five decimal places.
The value of [tex]tan^{-1}(0.2)[/tex] = -0.19739 + ... i.e. [tex]tan^{-1} (0.2)= (0.2) - \frac{(0.2)^{3} }{3} +\frac{(0.2)^{5} }{5} -\frac{(0.2)^{7} }{7}+...[/tex]
The power series of [tex]tan^{-1}(x)[/tex] obtained in example 6 is called the Gregory's series. The Gregory series is an infinite series of fractions or rational numbers
After the Scottish mathematician James Gregory (1638-1675) who had anticipated some of Newton's discoveries we have shown that the Gregory's series is valid when -1 < x < 1.
But it turns out (although it isn't easy to prove) that it is also valid when:
x = 1 or ≠ 1.
According to Example 6, The radius of convergence of the series for
[tex]1/(1 +x^{2} )[/tex] is 1, the radius of convergence of this series for [tex]tan^{-1}(x)[/tex] is also 1.
[tex]tan^{-1} x = x - \frac{x^{3} }{3} +\frac{x^{5} }{y} -\frac{x^{7} }{y} +....[/tex]
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the polynomial 1-x x^2-x^3 \ x^{16}-x^{17} may be written in the form a 0 a 1y a 2y^2 \a {16}y^{16} a {17}y^{17}, where y
The polynomial[tex]1-x x^2-x^3 \ x^{16}-x^{17}[/tex] can be written in the form [tex]a0 + a1(x+1) + a2(x+1)^2 + ... + a16(x+1)^{16} + a17(x+1)^{17}[/tex] where y = x + 1 and a1's are constants.
a polynomial can be written in the form
[tex]a0 + a1x + a2x^2 + a3x^3 + ... + anx^n[/tex]
where a0, a1, a2, ..., an are constants (also known as coefficients) and x is the variable.
In this case, we are given a polynomial in the form [tex]1-x x^2-x^3 \ x^{16}-x^{17}[/tex] and we want to express it in the form [tex]a0 + a1y + a2y^2 + ... + a17y^{17}[/tex], where y = x + 1 and a1's are constants.
By substituting y = x + 1 into the polynomial, we get:
[tex]1 - (x + 1) + (x + 1)^2 - (x + 1)^3 + ... + (x + 1)^{16} - (x + 1)^{17}[/tex]
Each term in this polynomial is a function of y, which is in turn a function of x.
We can then match the coefficients a0, a1, a2, ..., a17 with the corresponding powers of y in the polynomial, i.e, a0 = 1, a1 = -1, a2 = -1, a3 = 0, ..., a16 = -1, a17 = -1, and we get the polynomial
It's important to note that the value of a1's are constants, meaning that they are not dependent on the variable x.
Did you mean: The polynomial [tex]1-x x^2-x^3 \ x^{16}-x^{17}[/tex] may be written in the form [tex]a 0 a 1y a 2y^2 \a {16}y^{16} a {17}y^{17}[/tex], where y = x + 1 and a1's are constants?
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