Consider the function y = e⁻ˣ² and y = cos(x)
Write the Taylor expansions for the two functions about x = 0. What is similar about the two series? What is different? Looking at the series, which function do you predict will be greater over the interval (-1, 1)?
a. e⁻ˣ²
b. cos(x)
Answers
Looking at the two series, we can predict that the function e⁻ˣ² will be greater than cos(x) over the interval (-1, 1), since the terms in the series for e⁻ˣ² decay more rapidly than the terms in the series for cos(x).
The Taylor series expansion for y = e⁻ˣ² about x = 0 is:
e⁻ˣ² = 1 - x² + (x⁴ / 2!) - (x⁶ / 3!) + (x⁸ / 4!) - ...
The Taylor series expansion for y = cos(x) about x = 0 is:
cos(x) = 1 - (x² / 2!) + (x⁴ / 4!) - (x⁶ / 6!) + (x⁸ / 8!) - ...
Both series have even powers of x, indicating that the functions are even functions (i.e., symmetric about the y-axis).
However, the coefficients in the two series are different. Specifically, the coefficients in the series for e⁻ˣ² are given by the terms in the power series expansion of e⁻x², while the coefficients in the series for cos(x) are given by the terms in the power series expansion of cos(x).
Since the coefficients in the series for e⁻ˣ² are always positive, we can see that the function e⁻ˣ² is always greater than or equal to its Taylor series approximation. On the other hand, the coefficients in the series for cos(x) alternate in sign, so the function cos(x) oscillates around its Taylor series approximation.
Looking at the two series, we can predict that the function e⁻ˣ² will be greater than cos(x) over the interval (-1, 1), since the terms in the series for e⁻ˣ² decay more rapidly than the terms in the series for cos(x).
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Related Questions
Which value of x is the solution of the equation 2(x – 4) + 7 = 3? (a) 1 (b) 6 (c) 2 (d) 0
Answers
Answer:
(c) 2
Step-by-step explanation:
2(x - 4) + 7 = 3
2x + 2(-4) + 7 = 3
2x - 8 + 7 = 3
2x - 1 = 3
2x - 1 + 1 = 3 + 1
2x = 4
2x/2 = 4/2
x = 2
Hope this helps!!!
The correct value of x is 2 in the given equation.
What is an equation?An equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given is an equation, 2(x – 4) + 7 = 3
Solving for x,
2(x – 4) + 7 = 3
2x-8+7 = 3
2x-1 = 3
2x = 4
x = 2
Hence, the value of x is 2.
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knowing the difference between an experimental group and a control group is most relevant to understanding the nature of: group of answer choices random sampling. replication. hindsight bias. independent variables.
Answers
The experimental group is given the experimental treatment, and the comparison group is statistically significant.
Which three attitudes are the basis of the scientific viewpoint?Curiosity, scepticism, and humility are described by historians of science as the three attitudes that form the bedrock of the scientific viewpoint. Larger, more representative sample sizes aid in the production of accurate and valid data.When random sampling is used, every member of the population has an equal chance of being chosen, resulting in the collection of a representative sample. The odds of generalising the study's findings to the population increase with the use of a random sample. The study's margin of error is reduced with higher study sample sizes.) Researcher control over the possibility of reporting false-negative or false-positive findings is made possible by larger sample sizes. Results will be more precisely predicted with a larger sample size.To learn more about random sampling refer to :
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Will give brainliest
Answers
Answer:
[2(x+10)]°+(3x-30)°=180°[vertically opposite angle]
2x+20+3x-30=180°
5x-10=180°
5x=180-10
5x=170
x=170/5
x=34
solve for measure of arc dg and the angle DHG
Answers
what is the quotient of the expression
\( \frac{21a {}^{3} b - 14ab {}^{2} + 7ab}{7ab} \)
Answers
The scatterplot shows the monthly high temperatures for Austin, Texas, in
degrees Fahrenheit over a 12-month period.
Answers
The function most accurately represents the data from Month 1 through Month 9 is y = 3.5x + 85.
Given that,
The scatterplot displays the monthly high temperatures for Austin, Texas, over a 12-month period in Fahrenheit.
We have to find which function most accurately represents the data from Month 1 through Month 9.
We know that,
What is the Function of a Line of Best Fit on a Scatter Plot?Pick two locations on the scatter plot, determine the slope (m), determine the value of b (the y-intercept), and then enter the values of m and b into the equation y = mx + b to find the function's equation.
Using (5, 102) and (7, 109) as starting points:
Slope (m) is equal to (109 – 102)/(7 – 5) = 7/2 = 3.5.
Put the numbers m = 3.5 and (5, 102) into the equation y - b = m.
y - 102 = 3.5(x - 5)
Rewrite as y = mx + b
y - 102 = 3.5x - 17.5
y = 3.5x - 17.5 + 102
y = 3.5x + 84.5
y = 3.5x + 85
Therefore, The function most accurately represents the data from Month 1 through Month 9 is y = 3.5x + 85.
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What is the slope of the line that passes through the points (−3,−3) and (−9,5)?
Answers
A block i pulled 0. 90 meter to the right in 2. 5 econd. What i the block' average peed to the nearet hundredth of a meter/econd (m/)?
Answers
If a block is pulled 0.90 meter to the right in 2.5 second, then the average speed is 0.36 meter per second
The distance that he pulled the block = 0.90 meter
The time taken = 2.5 second
The average speed is defined as the total distance travelled by the total time taken
The average speed = Total distance / Time taken
Substitute the values in the equation
The average speed = 0.90 / 2.5
Divide the terms
= 0.90 / 2.5
= 0.36 meter per second
Hence, if a block is pulled 0.90 meter to the right in 2.5 second, then the average speed is 0.36 meter per second
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Soon-yi’s house has a hexagonal window made up of six identical equilateral triangles as pictured below. a triangular section has a height of 14 inches and a base of 16 inches. which expression is equivalent to the area of one triangular section? one-half times 7 times 8 one-half times 14 times 16 one-sixth times 14 times 16 one-half times 7 times 7
Answers
Answer:
16*14*1/2
Step-by-step explanation:
The area of a triangle can be found in 1/2bh.
Therefore, 16 is our base, and 14 is our height.
16*14*1/2 should be your answer, from the choices that I see.
Answer:
\(\frac{1}{2} *14*16\)
Correct Answer 100%. Pls. give me brainliest. Thank You.
bakit po ala kapong jowa
Answers
Answer:
tanong mopo sa pagong!
シ︎シ︎シ︎
Answer:
enewan nya na q :(((
Step-by-step explanation:
First step tingin salamin
sabay miss u balik kana
ahuhuhuhuh
Help please with this. Thanks <3
Answers
a flomming tile has the shape of a parallelogram whose base is 24cm and height is 10 cm how many tiles are required to cove the floor of area 1080m2
Answers
Answer:
4 1/2 tiles are needed to cover this area
Step-by-step explanation:
Mathematically, the area of a parallelogram is the product of the base and the height
We have this as;
24 * 10 = 240 cm^2
So the area of a tile is this
The number of tiles that will cover the given area will be;
1080/240 = 4.5 tiles
A cylinder has a base radius of 6ft and a height of 18ft. What is its volume in cubic ft, to the nearest tenths place?.
Answers
2034.72ft³ is the required cylinder volume.
What do we mean by the volume of the cylinder?The capacity of a cylinder determines how much material it can carry, or we may say that the volume of a cylinder is the total number of unit cubes that can fit inside of it.
The volume of a cylinder is r²h, and its surface area is 2πrh+2πr².
The volume of cylinder Formula: V = A × h
We will assume the cylinder is a right circular cylinder.
Then, we have the formula:
V = πr²h
We already know:
Radius = 6ft
Height = 18ft
Now, calculate the volume as follows:
(V) = π× 6² × 18
(V) = 2034.72ft³
Therefore, 2034.72ft³ is the required cylinder volume.
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pls help right now I need help
giving brainliest
this is reading btw
Answers
Answer:
The answer is A
Step-by-step explanation: I'm so smart
Answer the question in the picture below
Answers
Answer:
LN = 4
Step-by-step explanation:
given M lies on LN , then
LN = LM + MN ( substitute values )
4x = 3 + x ( subtract x from both sides )
3x = 3 ( divide both sides by 3 )
x = 1
Then
LN = 4x = 4 × 1 = 4
Which expression is equivalent to [CHECK IMAGE BELOW], if x > 0?
![Which expression is equivalent to [CHECK IMAGE BELOW], if x > 0?](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/LGYxxIdIfdPFKYPHOUw3cbDkjnzQdNdb.png)
Answers
Answer:
C. 4x²√(3x)
Step-by-step explanation:
You want to simplify √(48x⁵).
Simplifying RadicalsSquare root expressions can be simplified by removing perfect squares from under the radical:
\(\sqrt{48x^5}=\sqrt{16x^4\cdot3x}=\sqrt{(4x^2)^2}\cdot\sqrt{3x}=\boxed{4x^2\sqrt{3x}}\)
If the two figures are congruent, which statement is true?
A. BCDA ≅ FEHG
B. ABCD ≅ EFGH
C. BADC ≅ EFGH
D. ADCB ≅ HGFE
Answers
Answer:
A
Step-by-step explanation:
the order of letter should resemble the same shape
Type the correct answer in the box. Use numerals instead of wc
The length of each side of a square increases by 2.5 inches to
original square was
inches.
Question serial: FIB-3A42-E542-4CD6
P
Answers
Answer:
me dont know solve it your self HAHAHHH loser
Step-by-step explanation:
Given: (GH) is a midsegment of ∆DEF. Show all work.
a. Find the value of x and y.
b. What is the length of DF?
Answers
Applying the triangle midsegment theorem:
a. x = 17; y = 2
b. Length of DF = 20 units.
What is the Triangle Midsegment Theorem?The triangle midsegment theorem states that the midsegment of a triangle is always parallel to the third side, and has a length that is half the length of the third side of the triangle.
GH is said to be the midsegment of triangle DEF, therefore:
a. GH = 1/2(EF) [triangle midsegment theorem]
Substitute
x - 3 = 1/2(28)
2(x - 3) = 28
2x - 6 = 28
2x = 28 + 6
2x = 34
2x/2 = 34/2
x = 17
HF = x - 7
Plug in the value of x
HF = 17 - 7
HF = 10
HF = DH = 10
y + 8 = DH
y + 8 = 10
y = 10 - 8
y = 2
b. Length of DF = HF + DH
Length of DF = 10 + 10 = 20 units.
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Decide whether the following statement is compound if lana wins the election then mary will smile
Answers
The correct answer is OD. Although the word "then" appears in the statement, it is not used as a logical connective. So the statement is not compound.
The statement "If Laura sells her quota, then Marie will be happy" is a single declarative sentence. It consists of a conditional clause ("If Laura sells her quota") and a consequent clause ("then Marie will be happy"). However, these two clauses are not independent statements that can stand alone. Instead, they are connected in a cause-and-effect relationship. The word "then" in this context is not functioning as a logical connective, but rather as an indicator of the consequent clause.
A compound statement is formed by combining two or more independent statements using logical connectives such as "and," "or," or "if...then." In the given statement, there is no logical connective joining two independent statements.
Therefore, the statement is not compound.
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Who is correct, Casey or Jacinta ?
Answers
Answer: Jacinta
Step-by-step explanation: But the fraction still can be simplified to 1/7
Answer: j is correct
Step-by-step explanation:
An educational psychologist wishes to know the mean number of words a third grader can read per minute. 5 he wants to ensure that the estimate has an error of at most 0.2 words per minute. A previous study found that the mean was 24 words per minute. Assuming that the standard deviation is 5.4. What is the minimum number of third groders that must be included in a sample to construct the 98 कf confidence interval Round your answer up to the next integer.
Answers
The minimum number of third graders that must be included in a sample to construct a 98% confidence interval with an error of at most 0.2 words per minute can be determined using the formula:
n = (Z * σ / E)^2
Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (in this case, 98% confidence level)
σ = standard deviation of the population
E = maximum allowable error
In this case, the Z-score for a 98% confidence level can be found using a standard normal distribution table or a statistical calculator. The Z-score for a 98% confidence level is approximately 2.33.
Plugging in the given values:
σ = 5.4 (standard deviation)
E = 0.2 (maximum allowable error)
Z = 2.33 (Z-score for 98% confidence level)
n = (2.33 * 5.4 / 0.2)^2
n = 128.4881
Since the sample size must be a whole number, we round up to the next integer:
n = 129
Therefore, the minimum number of third graders that must be included in a sample to construct a 98% confidence interval with an error of at most 0.2 words per minute is 129.
To estimate the mean number of words a third grader can read per minute with an error of at most 0.2 words per minute and a 98% confidence level, a sample size of 129 third graders should be included in the study.
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K-7: Write a conversation with your friend in Telugu explaining the symmetry you observe in nature
Answers
A conversation with your friend explaining the symmetry you observe in nature is given below.
What is the conversation about?Me: Hey, have you ever noticed how symmetrical nature can be?
Friend: What do you mean?
Me: Well, think about things like snowflakes, leaves, and flowers. They all have a certain amount of symmetry to them.
Friend: Yeah, I guess I've noticed that before. But why is that?
Me: It's actually because of the way that nature grows and develops. The growth of these things is guided by physical laws and processes that tend to produce symmetrical shapes.
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which equation does the graph below represent?108642-5 -4 -3 2-1021 23 454-6-8-10Ly= 2x1уX212.y. 2x
Answers
The line passes through the point (0,0) and point (1,2).
Which expression is equivalent to 2w?
Answers
Answer:
w + w
Step-by-step explanation:
w + w * 1 is the coefficent of w *
1w + 1w
(1+1) w
2w
Answer:
w+w
Step-by-step explanation:
Will give brainliet answer my question if there are 25 frogs and 6 kitten how many animals I have in a fraction. 20 points
Answers
Answer:
2/3
Step-by-step explanation:
Because there are two types of animals, you have 2. And then you do 25+6, which is the sum of all animals.
Answer:
Frogs: 25/31 Cats: 6/31
Step-by-step explanation:
Pls, choose me as brainliest!
y=3.8x10 \(x^{5}\)
Answers
the length of a rectangle is 2cm longer than its width. If the perimeter of the rectangle is 60 cm, find its length and width
Answers
Answer:
Let the Width(W) = X cm
then the length(L)= (2+X)cm
Perimeter(P)= 60cm
P =L+W
L+W =P
(2+X)cm +X cm = 60 cm
2X+2=60
2X=60-2
2X=58
dividing both sides by 2
X=29cm
Width =X
Width =29cm
the width is 29cm
Length= (2+X)cm
= (2+29)cm
= 31cm
the length is 31cm
We will let w represent the width of the rectangle (in centimeters).
We know that the length is 2 cm longer than the width.
So the length is w + 2 .
We can label the rectangle as follows.
The perimeter is given as 60 cm.
So we solve for w as follows.
w + (w + 2) + w + (w + 2) = 60
w + w + 2 + w + w + 2 = 60
4w + 4 = 60
4w = 56
w = 14
Using this value of w, we can find w + 2.
w + 2 = 14 + 2 = 16
ANSWER
length: 16 cm
width: 14 cm
PLEASE HELP ME ANSWER ASAP
Answers
L = k/f, where k is the variational constant, is the formula for the inverse variation.
Inverse proportionsA mathematical relationship between two variables in which they vary in opposing directions is referred to as an inverse proportion, also known as an inverse relationship. When one variable increases while the other decreases, this is known as having inverse proportions.
Using the variables length of violin 'l' and frequency of vibration 'f'
If the length of violin 'l' is inversely proportional to the frequency of vibration 'f', this is expressed as:
l α 1/f
l = k/f
Hence the formula for the inverse variation is l = k/f where k is the constant of variation.
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What is the last step when finding the area of a compound shape?
Answers
Using the formula then calculator to get the ans and we must never forget to write the unit in squared form.