Use the definition of a Taylor series to find the first three non zero terms of the Taylor series for the given function centered at a=1. Write the Taylor series in summation notation
Answers
Answer:
\(e^{4x}=e^4+4e^4(x-1)+8e^4(x-1)^2+...\)
\(\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n\)
Step-by-step explanation:
Taylor series expansions of f(x) at the point x = a
\(\text{f}(x)=\text{f}(a)+\text{f}\:'(a)(x-a)+\dfrac{\text{f}\:''(a)}{2!}(x-a)^2+\dfrac{\text{f}\:'''(a)}{3!}(x-a)^3+...+\dfrac{\text{f}\:^{(r)}(a)}{r!}(x-a)^r+...\)
This expansion is valid only if \(\text{f}\:^{(n)}(a)\) exists and is finite for all \(n \in \mathbb{N}\), and for values of x for which the infinite series converges.
\(\textsf{Let }\text{f}(x)=e^{4x} \textsf{ and }a=1\)
\(\text{f}(x)=\text{f}(1)+\text{f}\:'(1)(x-1)+\dfrac{\text{f}\:''(1)}{2!}(x-1)^2+...\)
\(\boxed{\begin{minipage}{5.5 cm}\underline{Differentiating $e^{f(x)}$}\\\\If $y=e^{f(x)}$, then $\dfrac{\text{d}y}{\text{d}x}=f\:'(x)e^{f(x)}$\\\end{minipage}}\)
\(\text{f}(x)=e^{4x} \implies \text{f}(1)=e^4\)
\(\text{f}\:'(x)=4e^{4x} \implies \text{f}\:'(1)=4e^4\)
\(\text{f}\:''(x)=16e^{4x} \implies \text{f}\:''(1)=16e^4\)
Substituting the values in the series expansion gives:
\(e^{4x}=e^4+4e^4(x-1)+\dfrac{16e^4}{2}(x-1)^2+...\)
Factoring out e⁴:
\(e^{4x}=e^4\left[1+4(x-1)+8}(x-1)^2+...\right]\)
Taylor Series summation notation:
\(\displaystyle \text{f}(x)=\sum^{\infty}_{n=0} \dfrac{\text{f}\:^{(n)}(a)}{n!}(x-a)^n\)
Therefore:
\(\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n\)
Related Questions
What is the difference between repressed and recovered memories?
Answers
Recovered memory in CSA refers to the new development of abusive memories. Repressed memory is when the patient believes the CSA occurred, even though they have no specific memory of the event.
Recovered memory:
The term "recovered memory" implies that at some point the memory must become inaccessible to conscious awareness (as opposed to "continuous memory"). Although the term is not ideal, it is clear that people often fail to report important events, such as known hospitalizations (Loftus, 1993). In 1995, the debate over reclaiming memory was near its most violent climax. Hundreds of people have recovered memories of child sexual abuse (CSA), and sometimes in therapy it is thought that repressed or dissociative memories need to be recovered for a person to 'heal'.
Repressed Memory:
Repressed memory is a purported psychiatric phenomenon involving the inability to recall autobiographical information, usually traumatic or stressful. The concept has its origins in psychoanalytic theory, where repression is understood as a defense mechanism that keeps painful experiences and unacceptable impulses out of awareness. Repressed memory is a controversial concept, especially in a legal context, where it is used to unjustly and inaccurately accuse individuals, causing significant harm. Meanwhile, a working group from the American Psychological Association has stated that while "most people who were sexually abused in childhood remember some or all of what happened to them, it is possible to remember long-forgotten abuses.
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Consider the following vectors: v1 = 1 2 1 ; v2 = 1 3 2 ; v3 = 1 0 4 ; (a) Determine if these vectors are linearly independent or dependent. (b) Is is possible to express v = 1 2 −3 as a linear combination of v1, v2, and v3?
Answers
By solving the system of equations, we find that there is no solution. Therefore, it is not possible to express v = [1 2 -3] as a linear combination of v1, v2, and v3.
(a) To determine if the vectors v1, v2, and v3 are linearly independent or dependent, we can form a matrix A by placing the vectors as columns:
A = [v1 v2 v3]
| 1 1 1 |
| 2 3 0 |
| 1 2 4 |
Next, we can perform row operations to check if the matrix A is row equivalent to the identity matrix. If we can row reduce A to the identity matrix, then the vectors are linearly independent. Otherwise, they are linearly dependent.
Performing row operations on matrix A, we can obtain the following row-echelon form:
| 1 1 1 |
| 0 1 -2 |
| 0 0 0 |
Since there is a row of zeros in the row-echelon form, we can conclude that the vectors v1, v2, and v3 are linearly dependent.
(b) To determine if it is possible to express v = [1 2 -3] as a linear combination of v1, v2, and v3, we can set up the equation:
x1v1 + x2v2 + x3*v3 = v
This leads to the system of equations:
x1 + x2 + x3 = 1
2x1 + 3x2 + 2x3 = 2
x1 + 2x2 + 4x3 = -3
We can solve this system of equations using various methods such as Gaussian elimination or matrix inversion. After solving the system, if there exists a solution for x1, x2, and x3, then it is possible to express v as a linear combination of v1, v2, and v3. Otherwise, it is not possible.
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Jackiehas 7dresses, 4hats, and 8pairs of shoes. She will choose an outfit that has one dress, one hat, and one pair of shoes. How many different outfits are possible?
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formula of finding rate of discount when discount amt and marked price is given
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Can I have brain list pleaseeeee
this my last question for the day
Answers
Answer:
x=72
Step-by-step explanation:
2. A sport trainer wants to know whether the true average time of his athlete who do 100-meter sprint is 98 seconds. He recorded 18 trials of his team and found that the average time is 98. 2 seconds with a standard deviation of 0. 4 second. Is there sufficient evidence to reject the null hypothesis if μ = 98 seconds at the 0. 05 level of significance?
A. Test statistic t = 2. 1213 is a value greater than the critical value to = + 2. 110. Hence, the test value is clearly in the critical region. Thus, we decide to reject the null hypothesis.
B. Test statistic t = -2. 1213 is a value lesser than the critical value to = + 2. 110. Hence, the test value is clearly outside the critical region. Thus, we decide to reject the null hypothesis.
C. Test statistic t = 2. 1213 is a value greater than the critical value to = + 2. 110. Hence, the test value is clearly in the critical region. Thus, we decide to accept the null hypothesis.
D. Test statistic t = - 2. 1213 is a value lesser than the critical value to = +2. 110. Hence, the test value is clearly outside the critical region. Thus, we decide to accept the null hypothesis.
Solution:
Answers
The correct answer is A. Test statistic t = 2.1213 is a value greater than the critical value to = +2.110. Hence, the test value is clearly in the critical region. Thus, we decide to reject the null hypothesis.
To determine whether there is sufficient evidence to reject the null hypothesis that the true average time of the athlete's 100-meter sprint is 98 seconds, we need to perform a hypothesis test using the given data.
The test statistic for this scenario is calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is 98.2 seconds, the hypothesized mean is 98 seconds, the sample standard deviation is 0.4 seconds, and the sample size is 18.
Substituting the values into the formula, we get:
t = (98.2 - 98) / (0.4 / sqrt(18)) ≈ 2.1213
To make a decision, we compare the test statistic to the critical value at the given significance level of 0.05. The critical value for a two-tailed test with a significance level of 0.05 and 17 degrees of freedom is approximately ±2.110.
Since the test statistic (2.1213) is greater than the critical value (+2.110), it falls into the critical region. Therefore, we reject the null hypothesis.
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Please help me with this i really need help.
Answers
Answer:
Step-by-step explanation:
We are given an equation( a math expression with an equal sign)
5+x-12=x-7
First things first, we need to combine any like terms(add/subtract) on the same side
A term is a number, variable, that is separated by a plus symbol or minus symbol.
Here 5 and -12 are similar terms so let combine them
5-12=-7 So the equation become
x-7=x-7
Step 2: Notice how the left hand and right-hand side are the same thus we do not need to solve this equation. But let do the work instead
x-7=x-7
x=x
Thus, the solution is All Real Numbers
Part B:
Substitute -4 for x in the original equation.
5+(-4)-12=-4-7
1-12=-4-7
-11=-11
Thus, the equation is true here.
Substitute 0 for x in the original equation
5+0-12=0-7
-7=-7
Equation is True Here
Try 5 for x as well
5+5-12=5-7
10-12=5-7
-2=-2
So equation is true as well.
Let me know if you have any questions or concerns.
Venetta buys 2 pounds of pistachios and 3 pounds of almonds. The pistachios cost $4 more per pound than the almonds. She pays a total of $48. Which of the following are true? Select all that apply.
Answers
The true statements are:
Reducing the number of pounds of almonds by one results in a total cost of $40
The cost in dollars, of 1 pound of almonds is modeled by the equation 2(a−4)+3a=48
What are the true statements?The first step to take in order to determine the cost of one pound of pistachios and almonds is to form a system of linear equations:
2p + 3a = $48 equation 1
p - a = $4 equation 2
Where:
p = cost of one pound of pistachios
a = cost of one pound of almonds
The elimination method would be used:
Multiply equation 2 by 2
2p - 2a = $8 equation 3
Subtract equation 3 from equation 2 :
5a = $40
Divide both sides of the equation by 5
a = $40 / 5
a = $8
Substitute for a in equation 1:
2p + 3(8) = $48
2p + 24 = 48
2p = 48 - 24
2p = 24
p = 24 / 2
p = $12
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I’m looking for what kind of angles 2 and 6 are?
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Answer:
Angles 2 and 11 are alternate interior angles I believe
Hope this helped!
100 POINTS! HELP ON 9 AND 10 PLEASE
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Answer:
ok so each little piece of a cube would be .05 mile if each grid piece is .05 mile than mulitply the amount of grid pieces by the numner 0.05
Step-by-step explanation:
Hope this helps, i can give you this much, but the full answer you need to find and understand on your own!
Answer:
I would assume you have finsished this by now.
Step-by-step explanation:
Thank you for the free points!
Construct the graph of the direct proportion y=kx for each value of k k=3x
Answers
the answer is 3x²
Step-by-step explanation:
we know that k = 3x
when we look at the actual equasion we can se that y = kx, meaning we have to multiply x by 3x which would give us 3x²
Answer: the answer is 3x^2 three x to the power of 2
The distribution of intelligence test scores in the general population forms a bell-shaped pattern. This pattern is called a
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The distribution of intelligence test scores in the general population forms a bell-shaped pattern is known as a normal distribution.
Normal distribution, also known as Gaussian distribution, is a continuous probability distribution used in statistics. The normal distribution has a bell-shaped curve that is symmetrical about the mean (average).The normal distribution has some important properties, including that its mean, median, and mode are equal.
Furthermore, the total area under the curve is 1. A normally distributed population is one in which the majority of the data is clustered around the mean, with fewer data points further from the mean. It is a common distribution for a variety of natural phenomena, and many statistical analyses depend on it.
The normal distribution plays a critical role in statistical hypothesis testing and other statistical analyses because of its properties, and it has numerous applications in science, engineering, and finance.
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What is the probability that when you roll a standard 6-sided number cube, you roll a number more than 6?
A. Certain
B. Impossible
C. Likely
D. Unlikely
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A pyramid has a square base of 8cm height of the pyramid is half the side of pyramid find the slant height
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The slant height of the pyramid is approximately 5.657 cm.
Let's denote the side length of the square base of the pyramid as 's'.
According to the given information, the height of the pyramid is half the side length of the pyramid's base. This means the height (h) is equal to (1/2) * s.
To find the slant height (l) of the pyramid, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (in this case, the slant height) is equal to the sum of the squares of the other two sides.
In our case, the slant height (l) is the hypotenuse, and the height (h) and half the side length of the base (s/2) are the other two sides.
Using the Pythagorean theorem:
l^2 = h^2 + (s/2)^2
l^2 = [(1/2) * s]^2 + (s/2)^2
l^2 = (1/4) * s^2 + (1/4) * s^2
l^2 = (1/2) * s^2
Taking the square root of both sides:
l = √[(1/2) * s^2]
l = (1/√2) * s
Substituting the value of s = 8cm into the equation:
l = (1/√2) * 8
l ≈ 5.657 cm
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Benji is a good boyBenji is a good boyBenji is a good boy
Answers
Answer:
A
Step-by-step explanation:
A2 Shape C is similar to shape D
9 cm
C
x cm
9.6 cm
Work out the value of x.
D
6 cm
T
10
Answers
Answer:
If Shape C is similar to Shape D, then we know that the ratio of corresponding side lengths is the same. In this case, we can set up the following proportion:
9 / 6 = x / 10
To solve for x, we can cross-multiply and simplify:
9 * 10 = 6 * x
90 = 6x
x = 15
Therefore, the value of x is 15 cm.
Answer:
The answer for x is approximately 5.6
Step-by-step explanation:
9/9.6=x/6
cross multiply
9.6x=9×6
9.6x=54
divide both sides by 9.6
9.6x/9.6=54/9.6
x=5.625
x≈5.6
someone help me plz!!!
Answers
Answer:
The answer is b
Step-by-step explanation:
you look at the x factor first then the y factor
how do u get 598 x 9 when ur estimating numbers to find the product
Answers
The estimated product is 6000
What is estimation?
To make calculations simpler and more realistic, estimation of a number refers to a plausible assumption of the actual value.
Estimation is the process of approximating a quantity with the necessary accuracy.
The result is quickly and roughly determined by rounding off the numbers used in the calculation.
To estimate the product, we first round off the multiplier and the multiplicand to the nearest tens, hundreds, or thousands and then multiply the rounded numbers.
598 ≈ 600
9≈ 10
600*10
= 6000
The estimated product is 6000
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two banks in the area offer 20-year, $210,000 mortgages at 4.4 percent and charge a $3,100 loan application fee. however, the application fee charged by insecurity bank and trust is refundable if the loan application is denied, whereas that charged by i. m. greedy and sons mortgage bank is not. the current disclosure law requires that any fees that will be refunded if the applicant is rejected be included in calculating the apr, but this is not required with nonrefundable fees (presumably because refundable fees are part of the loan rather than a fee).
Answers
Borrowers can make more accurate comparisons between different loan options and understand the overall cost they will incur over the life of the loan.
In the given scenario, there are two banks offering 20-year, $210,000 mortgages at 4.4 percent interest. The banks charge a $3,100 loan application fee. However, there is a difference between the banks regarding the refundability of the application fee.
Insecurity Bank and Trust: This bank charges a $3,100 loan application fee, which is refundable if the loan application is denied. In this case, the fee is considered part of the loan rather than a separate fee.
I. M. Greedy and Sons Mortgage Bank: This bank also charges a $3,100 loan application fee, but it is nonrefundable, meaning it will not be returned if the loan application is rejected.
The disclosure law requires that any fees that will be refunded if the applicant is rejected must be included in calculating the Annual Percentage Rate (APR). This is because these fees are considered part of the loan amount and have an impact on the overall cost of borrowing.
On the other hand, nonrefundable fees like the one charged by I. M. Greedy and Sons Mortgage Bank are not required to be included in the APR calculation. This is because these fees are treated as separate costs that are not directly part of the loan.
When comparing mortgage offers and determining the true cost of borrowing, it is important to consider not only the interest rate but also any associated fees. By including refundable fees in the APR calculation, borrowers can make more accurate comparisons between different loan options and understand the overall cost they will incur over the life of the loan.
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There are two buildings that you want to have in the amusement park, but the size hasn’t been determined yet. Although you don’t know the specific dimensions, you do know the relationships between the sides.
The first is the rectangular gift shop. You know that the length will be 20x+24 feet and the width will be 36x-20 feet.
Write the expression that represents the area of the gift shop, in terms of x.
Write the expression that represents the perimeter of the gift shop, in terms of x.
If the perimeter is going to be 176 feet, what are the dimensions of the building?
Answers
An expression that represents the area of the gift shop, in terms of x is 720x² + 464x - 480.
An expression that represents the perimeter of the gift shop, in terms of x is 720x² + 464x - 480.
If the perimeter is going to be 176 feet, the dimensions of the building are 54 feet by 34 feet.
How to calculate the area of a rectangle?In Mathematics and Geometry, the area of a rectangle can be calculated by using the following mathematical equation:
A = LB
Where:
A represent the area of a rectangle.B represent the breadth of a rectangle.L represent the length of a rectangle.By substituting the given parameters into the formula for the area of a rectangle, we have the following;
Area of rectangular gift shop = (20x + 24) × (36x - 20)
Area of rectangular gift shop = 720x² - 400x + 864x - 480
Area of rectangular gift shop = 720x² + 464x - 480
Perimeter of rectangular gift shop = 2(20x + 24 + 36x - 20)
Perimeter of rectangular gift shop = 2(56x + 4)
Perimeter of rectangular gift shop = 112x + 8
176 = 112x + 8
112x = 168
x = 1.5
Length, L = 20(1.5) + 24 = 54 feet.
Width, W = 36(1.5) - 20 = 34 feet.
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Given that Justin is collecting data on reaction time, what type of data is he working with?
Select the correct answer below:
A. qualitative
B. discrete quantitative
C. continuous quantitative
D. none of the above
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Since reaction time is measured and not constrained to a specific range of numbers, it is continuous quantitative data.
'What is continuous quantitative?'
A continuous data set is a quantitative data set that represents a scale of measurement that includes fractions and decimals in addition to whole integers. Values like height, weight, length, temperature, and other similar metrics would be included in continuous data sets. They are items that can be quantified in decimals and fractions. A continuous data set typically requires the use of a tool, such as a ruler, measuring tape, scale, or thermometer, to create the numbers.
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2. 5 x 10^22. 5 × 10 2 and 3. 7 x 10^53. 7 × 10 5
Answers
The first number, 5 x \(10^22,\) can be written as 5 followed by 22 zeros:
5,000,000,000,000,000,000,000
To simplify means to make something easier to understand or do by reducing complexity, removing unnecessary details, or using simpler language or concepts.
The second number , 5 × 10², can be written as 5 followed by 2 zeros: 500
The third number, 3.7 x 10⁵³, can be written as 3.7 followed by 53 zeros:
370,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
or in scientific notation as 3.7 x 10⁵³
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Simplify
2. 5 x 10^22. 5 × 10 2 and 3. 7 x 10^53. 7 × 10 5
What is the length of the vinyl needed for the outer edges of the table top?
Answers
the perimeter of the triangle is 10.2ft (option A)
Explanation:Given:
A triangle with two angles and a side opposite one of the angles is given
To find:
the approximate length of the vinyl needed
The triangle is right-angled and isosceles as the base angles are equal. To get the approximate length of the vinyl needed, we will be assuming the whole triangle is the table top and vinyl will cover the whole of it
We will use Pythagoras theorem to get the longest side:
Hypotenuse² = opposite² + adjacent²
The two legs are equal since the base angles are equal
opposite = adjacent = 3 ft
Hypotenuse² = 3² + 3²
Hypotenuse² = 9 + 9 = 18
Hypotenuse = √18
Hypotenuse = 4.24
The perimeter of the triangle = Length of the outer edges of the tabletop
The perimeter of the triangle = 3ft + 3ft + 4.24 ft = 10.24 ft
From the options, the perimeter of the triangle is 10.2ft (option A)
for #13-14 determine if the lines are parallel, perpendicular or neither 10x +5y=-5 and y=-2x+6
Answers
Answer:
48
because. Of the time
Find the midpoint of the segment with the following endpoints. (
10
,
5
)
and
(
6
,
9
)
(10,5) and (6,9)
Answers
Step-by-step explanation:
like they don't know, and so we don't know and so what do we do when ask that question as overtime and somewhere else
Your friend took an introductory statistics class last year and learned all about density curves. she tells you that two requirements of a density curve are:______.
Answers
The required requirement of the density curve is should be equal to 1 and the ordinate should not be zero.
Given that,
Two requirements of a density curve are to be determined.
A density curve is defined as a curve on a graph that represents the distribution of values in a dataset.
Here,
The two requirements of a density curve should be,
a. The area under the density curve should be equal to 1.
b. The ordinate of the curve should not be zero i.e. they should always have the vertical height.
Thus, the required requirement of the density curve is mentioned above.
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abc is a right triangle with ab=ac. bisector of <a meets bc at d. prove that bc = 2ad.
Answers
Answer:
Let ac=ab=5
With this, bc= 5√2
Step-by-step explanation:
So to find ad, Let ad be x
5√2=(2)(x)
(5√2/2)= x
This proves that bc=2ad
Consider the following equation.
9x + 3y = 6
On a recent Algebra test, Brian was asked to rewrite the equation in slope-intercept form
The slope of the equation is
The y-intercept is
Answers
Answer: The slope is -3 and the y- intercept is (0,2)
Step-by-step explanation:
I need help with this practice problem solving Read the instructions, then answer by filling in the three boxes
Answers
Mr. Johnson's desk is 2 feet tall. How tall is the desk in meters? Hint: 1 m equals 3.3 feet
Answers
Answer:
0.61 meter
Step-by-step explanation:
Answer:
2*3.3=6.6
you bumbling idiot how can you make a post about this and not know how to use a calculator or even just do 2*3.3
Step-by-step explanation:Thats how