Memoization is a technique used in programming to store the results of expensive function calls and return the cached result when the same inputs occur again. This approach can significantly improve the performance of recursive functions, such as the Legendre polynomials.
To define a recursive function L using memoization, we can create a dictionary or map that stores the previously calculated values of L(n, x). We can then check if the requested value of L(n, x) is already in the dictionary and return it if it is. If not, we can calculate the value using the recursive definition and store it in the dictionary for future use.
Here is an implementation of the memoized L function in Python:
```
memo = {}
def L(n, x):
if n == 0:
return 1
elif n == 1:
return x
elif (n, x) in memo:
return memo[(n, x)]
else:
result = ((2*n - 1)*L(n-1, x) - (n - 1)*L(n-2, x))/n
memo[(n, x)] = result
return result
```
In this implementation, we first check if n is 0 or 1 and return the appropriate base case values. If the requested value is already in the memoization cache, we return it directly. Otherwise, we calculate the value recursively using the provided formula and store it in the cache before returning it.
Using memoization, we can avoid recalculating the same values of L(n, x) multiple times and greatly speed up our computation.
To learn more about recursive function : brainly.com/question/30027987
#SPJ11
la suma de tres números consecutivos es 54.
Answer:
16, 18, 20
Step-by-step explanation:
16, 18, 20 es igual a 54, así que esa será tu respuesta.
¡espero que esto haya ayudado!
if the question is right you will be marked as branliest (guarantied)
Answer:
(a) 48 inches
(b) 3rd quarter
(c) 2nd quarter
Step-by-step explanation:
You want various descriptors of the data represented by the given box plot.
(a) MostThe greatest amount of snowfall is represented by the extreme value at the right end of the right whisker. The value there is 48 inches.
The greatest amount of snowfall is 48 inches.
(b) Most concentratedThe data is most concentrated where the limits of a quartile are closest together. Those limits are ...
Q1 — 5 to 13 (8 inch range)
Q2 — 13 to 35 (22 inch range)
Q3 — 35 to 40 (5 inch range)
Q4 — 40 to 48 (8 inch range)
The quartile with the least range is Q3. The data is most concentrated in the 3rd quarter.
(c) Most spread outAs we saw in the previous section, the second quartile has the greatest range.
The data is the most spread out in the 2nd quarter.
A publishing company printed 920,500 copies of a book. The company sold 843, 255 copies. How many books did not sell?
There are 77,245 books did not sell.
We have to given that;
A publishing company printed 920,500 copies of a book.
And, The company sold 843, 255 copies.
Hence, Number of books did not sell are,
⇒ 920,500 - 843,255
⇒ 77,245
Thus, Number of books did not sell are,
⇒ 77,245
Learn more about the subtraction visit:
https://brainly.com/question/17301989
#SPJ4
The equation of line EF is y = 1 over 2x + 6. Write an equation of a line parallel to line EF in slope-intercept form that contains point (0, −2).
Answer:
y = 1/2x -2
Step-by-step explanation:
You want the equation of a line parallel to y = 1/2x +6 that contains the point (0, -2), written in slope-intercept form.
SlopeThe slope of the line you want will be the same as the slope of the line you have. That is because parallel lines have the same slope.
The equation you are given is written in slope-intercept form:
y = mx + b . . . . . . where m is the slope and b is the y-intercept
Comparing this form to the given equation, you see that ...
m = 1/2 . . . . . the slope of the line you want
InterceptThe "intercept" in the "slope-intercept" form is the value of y when x=0. The point you are given, (0, -2), tells you that y = -2 when x = 0. So, the "intercept" in your slope-intercept equation is -2.
EquationThe equation you want is the equation of a line with slope 1/2 and a y-intercept of -2.
y = 1/2x -2
The equation of the line that is parallel to line EF and passes through point (0, -2) is y = (1/2)x - 2.To find an equation of a line that is parallel to line EF and passes through point (0, -2), we need to use the fact that parallel lines have the same slope. The slope of line EF is 1/2, so the slope of the parallel line will also be 1/2.
We can start by using the point-slope form of the equation of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Plugging in m = 1/2 and (x1, y1) = (0, -2), we get:
y - (-2) = (1/2)(x - 0)
Simplifying this equation gives:
y + 2 = (1/2)x
To get this equation in slope-intercept form (y = mx + b), we can isolate y:
y = (1/2)x - 2
So the equation of the line that is parallel to line EF and passes through point (0, -2) is y = (1/2)x - 2. Note that this line intersects the y-axis at y = -2, which is the y-intercept.
learn more about slope here: brainly.com/question/16949303
#SPJ11
9) Find a11 in an arithmetic sequence where a1 = −15 and a8 = 34
10) Find the arithmetic mean of the sequence -10,___,___,___,18
Karrie has 20 bags of prizes each bag has four prizes. She also has a red bag with 13. Prizes and a blue bag with the rest of the prizes. She has a total of 100 prizes how many prizes are in the blue bag write an equation use a variable for the unknown number
91 is 130% of what number?
Answer: 70
Step-by-step explanation: 70 X 1.30 =91
Convert the given rectangular
coordinates into polar
coordinates.
(-7, 4) = ([ ? ], [_])
Round your answer to the nearest hundredth.
Thx
Conversion of the given rectangular coordinates into polar coordinates gives: (8.06, -0.52)
How to convert rectangular coordinates into polar coordinates?To convert rectangular coordinates to polar coordinates, we will use the following formulas:
r = √(x² + y²)
θ = tan⁻¹(y/x)
where (r, θ) is the polar coordinates and (x, y) is the rectangular coordinates.
Using these formulas, we can convert (-7, 4) as follows:
r = √((-7)² + 4²)
r = √(49 + 16)
r = √65
r = 8.06
θ = tan⁻¹ (4/(-7)) = -0.52 radians
Therefore, the polar coordinates are (8.06, -0.52)
Learn more about polar coordinates on:
https://brainly.com/question/14965899
#SPJ1
find 1 3/4 × 3 3/7 record you answer in simplest form
Answer:
1 3/4 × 3 3/7 = 6 in simplest form.
Step-by-step explanation:
To multiply 1 3/4 by 3 3/7, we first convert the mixed numbers to improper fractions.
1 3/4 = (4 x 1 + 3)/4 = 7/4
3 3/7 = (7 x 3 + 3)/7 = 24/7
Now, we can multiply the two fractions:
(7/4) x (24/7) = (7 x 24)/(4 x 7) = 168/28
To simplify this fraction, we can divide both the numerator and denominator by their greatest common factor, which is 4.
168/28 = (4 x 42)/(4 x 7) = 42/7
Finally, we can simplify the improper fraction to the mixed number:
42/7 = 6
Therefore, 1 3/4 × 3 3/7 = 6 in simplest form.
Math 7 Test Prep Questions
1. An energy company graphs the average number of rotations that a windmill makes each minute for 4
minutes.
Number of Rotations
484
42+
36+
30+
24+
18+
12+
6+
Windmill Rotations
5
Time (minutes)
Which statement describes what the point (1, 12) means in terms of rotations and minutes?
a. 1 rotation occurs every 12 minutes.
b.
of a rotation occurs every minute.
c. The windmill rotates 12 times in 1 minute.
d. The windmill rotates 12 times in 12 minutes.
The point (1, 12) on the graph represents that the windmill rotates 12 times in 1 minute.
c. The windmill rotates 12 times in 1 minute.
The given graph represents the average number of rotations made by a windmill every minute for 4 minutes.
The horizontal axis represents time in minutes, while the vertical axis represents the number of rotations made by the windmill.
The point (1,12) represents the number of rotations made by the windmill in the first minute.
Option (a) suggests that one rotation occurs every 12 minutes, which is not true as the windmill makes 12 rotations in the first minute itself.
Option (b) suggests that a fraction of a rotation occurs every minute, which is not true as the number of rotations is always a whole number.
Option (c) states that the windmill rotates 12 times in 1 minute, which is the correct interpretation of the point (1,12).
Finally, option (d) suggests that the windmill rotates 12 times in 12 minutes, which is not true as the windmill makes 12 rotations in just 1 minute.
Thus, option (c) correctly describes what the point (1,12) means in terms of rotations and minutes.
It indicates that the windmill rotates 12 times in 1 minute.
The graph also shows that the windmill made 24 rotations in the first 2 minutes, 30 rotations in the first 3 minutes, and 36 rotations in the first 4 minutes.
The windmill's rotation speed appears to be decreasing with time.
In conclusion, interpreting graphs is an important skill in mathematics, and it helps us understand and analyze data.
In this case, the graph provides information about the number of rotations made by a windmill over a period of time, and we can use this information to make predictions or draw conclusions about the windmill's performance.
c. The windmill rotates 12 times in 1 minute.
For similar question on windmill rotates.
https://brainly.com/question/15321264
#SPJ11
The mean number of words per minute (WPM) typed by a speed typist is 7373 with a variance of 144144. What is the probability that the sample mean would differ from the population mean by greater than 0.60.6 WPM if 7777 speed typists are randomly selected
The probability that the sample mean of 77 speed typists would differ from the population mean by greater than 0.6 WPM is approximately 0.33 or 33%.
- Population mean (μ) = 73 WPM
- Population variance[tex](\sigma ^2)[/tex] = 144[tex]WPM^2[/tex]
- Sample size (n) = 77 typists
- Difference from the population mean (d) = 0.6 WPM
First, let's find the standard deviation of the population (σ) by taking the square root of the variance:
σ = √144 = 12 WPM
Next, we need to find the standard error (SE) of the sample mean:
SE = σ / √n = 12 / √77 ≈ 1.37 WPM.
Now, we'll convert the difference (d) to a z-score:
z = (d - 0) / SE = (0.6 - 0) / 1.37 ≈ 0.44
Finally, we'll find the probability that the sample mean differs from the population mean by greater than 0.6 WPM.
To do this, we'll use the z-score we calculated and consult a z-table to find the area under the curve corresponding to the z-score.
For a z-score of 0.44, the area is approximately 0.670.
Since we're looking for the probability that the sample mean differs by greater than 0.6 WPM, we need to find the area in both tails of the distribution.
The area in one tail is given by 0.5 - 0.670/2 = 0.165.
To find the total area in both tails, simply multiply by 2:
Probability = 2 * 0.165 = 0.33.
For similar question on probability.
https://brainly.com/question/28185528
#SPJ11
FILL IN THE BLANK. "Test yourself
The quotient-remainder theorem says that for all integers n and d with d ≤ 0, there exist ___________ q and r such that __________ and ______________."
The quotient-remainder theorem says that for all integers n and d with d > 0, there exist unique integers q and r such that n = qd + r and 0 ≤ r < d.
In other words, any integer n can be divided by a positive integer d to yield a quotient q and a remainder r, where the remainder is always strictly less than the divisor.
The quotient q represents the number of times that the divisor d "fits" into the dividend n, while the remainder r represents the amount that is left over after we have divided n by d as many times as possible. This theorem is also known as the division algorithm or Euclidean division, and is a fundamental result in elementary number theory.
The theorem can be used to solve many problems involving integers, such as determining whether a given integer is divisible by another integer, finding the greatest common divisor of two integers, and solving linear Diophantine equations.
To know more about integers refer to-
https://brainly.com/question/15276410
#SPJ11
Sample size in factor analysis/ PCA
Factor analysis and Principal Component Analysis (PCA) are statistical techniques used to reduce the dimensionality of data sets, identify underlying patterns, and simplify data interpretation. In both methods, sample size plays a critical role in determining the accuracy and reliability of the results.
An appropriate sample size is essential for several reasons:
1. Adequate Representation: A larger sample size helps ensure that the data is representative of the population, capturing a wide range of variability and improving the generalizability of the results.
2. Statistical Power: Larger sample sizes increase the statistical power of the analysis, enabling the detection of smaller effects and providing more reliable estimates of the underlying factors or components.
3. Stability: In factor analysis and PCA, larger sample sizes lead to more stable factor structures or component loadings, increasing the likelihood that the results are replicable in different samples or across time.
4. Error Minimization: A sufficient sample size helps minimize errors and distortions in the analysis, reducing the chance of overfitting the data or extracting spurious factors or components.
While there is no one-size-fits-all rule for determining an optimal sample size, some guidelines can help. A widely cited rule of thumb is a minimum of 10 to 20 observations per variable, with more complex models or data structures requiring larger sample sizes. Another approach is to perform a power analysis or Monte Carlo simulation, considering the desired level of statistical power, effect size, and specific model parameters.
In summary, selecting an appropriate sample size for factor analysis or PCA is crucial to ensure accurate, reliable, and generalizable results. Following guidelines and considering the context of the data and research question can help make informed decisions about sample size.
To learn more about factor analysis : brainly.com/question/30974361
#SPJ11
i need to find the area of this cylinder can you help me
The calculated value of the surface area of the cylinder is 652.94 square cm
From the question, we have the following parameters that can be used in our computation:
Height, h = 12 cm
Radius, r = 6 cm
Using the above as a guide, we have the following:
SA = 2πr(r + h) * 3/4 + 2 * area of rectanglt
Substitute the known values in the above equation, so, we have the following representation
SA = 2π * 6(6 + 12) * 3/4 + 2 * 6 * 12
Evaluate
SA = 652.94
Hence, the surface area of the cylinder is 652.94 square cm
Read more about surface area at
https://brainly.com/question/16519513
#SPJ1
T/F: To construct a binomial distribution, it is necessary to know the total number of trials and the probability of success on each trial.
The given statement is "To construct a binomial distribution, it is necessary to know the total number of trials and the probability of success on each trial." true because to construct a binomial distribution, it is necessary to know the total number of trials and the probability of success on each trial.
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success.
The parameters of a binomial distribution are n, the number of trials, and p, the probability of success on each trial. The distribution is then constructed by computing the probability of getting k successes in n trials, where k can be any integer from 0 to n.
The binomial distribution is widely used in many areas of statistics, including quality control, reliability engineering, and hypothesis testing. It is a fundamental concept in probability theory and serves as the basis for many other probability distributions.
To know more about probability refer to-
https://brainly.com/question/30034780
#SPJ11
A food truck sells hamburgers and hotdogs in the ratio of 6 to 8, meaning that for every 6 hamburgers sold, 8 hotdogs are sold. on a given day, there were 14 more hot dogs sold than hamburgers. given this information, determine how many of each were sold.
The food truck sold 42 hamburgers and 56 hotdogs.
Let's use algebra to solve this problem. Let x be the number of hamburgers sold and y be the number of hotdogs sold. We can set up a system of equations based on the information given:
The ratio of hamburgers to hotdogs sold is 6:8, or simplified, 3:4:
x/y = 3/4
There were 14 more hotdogs sold than hamburgers:
y = x + 14
Now we can substitute the second equation into the first equation and solve for x:
x/(x+14) = 3/4
4x = 3(x+14)
4x = 3x + 42
x = 42
So 42 hamburgers were sold. We can use the second equation to find the number of hotdogs sold:
y = x + 14
y = 42 + 14
y = 56
So 56 hotdogs were sold. Therefore, the food truck sold 42 hamburgers and 56 hotdogs.
Learn more about truck here
https://brainly.com/question/29149735
#SPJ11
whats the meaning of Angle Bisector Equidistant Theorem CONVERSE (ABET CONVERSE)
The Angle Bisector Equidistant Theorem Converse (ABET Converse) states that if a point is equidistant from the sides of an angle, then it lies on the angle bisector.
To understand the ABET Converse, we first need to understand the Angle Bisector Equidistant Theorem (ABET). The ABET states that if a point lies on the angle bisector of an angle, then it is equidistant from the sides of the angle. In other words, if we draw an angle and then bisect it (i.e., draw a line that splits the angle into two equal parts), any point on that line will be equidistant from the sides of the angle. The ABET Converse simply flips this idea around. It states that if a point is equidistant from the sides of an angle, then it must lie on the angle bisector. So, if we have a point that is the same distance from both sides of an angle, then that point must lie on the line that bisects the angle.
To learn more about angle bisector theorem : brainly.com/question/2478436
#SPJ11
the number 1 through 10 out in one each bag. The number 5 through 14 are put in another bag. When you pick one number from each bag, what is the probability you get the same number
Answer: 9/25
Step-by-step explanation:
What numbers do each bags have in common
5, 6, 7. 8, 9, 10 Tht's 6 numbers
the probability for each bag is the possibilities/total outcome
P(same from first bag)=6/10 = 3/5
P(same from other bag=6/10 = 3/5
Because you are looking for an "and" probability, you mulitiply
P(number from first and number from second) = 3/5 * 3/5 = 9/25
Which measure of central tendency is appropriate for nominal, ordinal, and ratio-interval level variables
The mean is the most appropriate measure of central tendency as it takes into account all values in the dataset and provides a useful representation of the average value. However, the median and mode can also be useful to consider, particularly if the dataset has extreme values or is not normally distributed.
The appropriate measure of central tendency depends on the level of measurement of the variable.
For nominal level variables, the mode is the most appropriate measure of central tendency. The mode represents the most frequently occurring value in the data and is a useful summary statistic for categorical data.
For ordinal level variables, the median is the most appropriate measure of central tendency. The median is the value that separates the data into two equal parts, with half of the observations above the median and half below. The median is appropriate for ordinal data because it does not assume that the intervals between values are equal.
For ratio and interval level variables, the mean is the most appropriate measure of central tendency. The mean is calculated by adding up all the values and dividing by the number of observations. It is appropriate for ratio and interval data because these data types have equal intervals between values, and the mean takes into account the magnitude of the values.
for such more question on central tendency
https://brainly.com/question/17631693
#SPJ11
The __________ (one word) Of two events A and B contains only those outcomes that are in both A and B
The intersection of two events A and B contains only those outcomes that are in both A and B.
The intersection of two events A and B, denoted by A ∩ B, is the set of all outcomes that are common to both A and B. In other words, it contains only those outcomes that are in both A and B.
For example, if event A is "rolling an even number on a fair six-sided die" and event B is "rolling a number greater than 3 on a fair six-sided die", then the intersection of A and B is the event "rolling an even number greater than 3 on a fair six-sided die", which contains the outcome 4 and has probability [tex]\frac{1}{6}[/tex].
To know more about "Intersection" refer here:
https://brainly.com/question/30748800#
#SPJ11
Why do we want homogeneity of variance?
a. to satisfy an assumption
b. it's considered politically correct.
c. we can't get a significant effect unless we do.
d. statement is false > we do not want homogeneity of variance.
In a random sample, 20 out of 50 students said they prefer chocolate ice cream. How many servings should the school cafeteria prepare for 1,000 students
Can someone please help me ASAP? It’s due today!! I will give brainliest if it’s all correct.
Please do part a, b, and c
Answer:
Step-by-step explanation:
Systematic sampling for part a
15%, 20%, 10%, 15%, 10% for red
30%, 20%, 20%, 25%, 25% for black
The mean percentage of black cars is greater than the mean percentage of red cars.
The range of both data is 10%.
Sorry if it's not in detail but I'm in a rush myself. If you need I could always answer in depth some other time.
Use your knowledge of the customary system to circle each equation that is true. A. 8ft=3yd B. 2mi=10,460ft C. 9ft=98in D. 5mi=8,800yd
Use your knowledge of the customary system to circle each equation that is true
B. 2mi = 10,460ft
D. 5mi = 8,800yd
Let's evaluate each equation:
A. 8ft = 3yd
To convert feet to yards, we know that 1 yard is equal to 3 feet. So, if we divide 8ft by 3, we get 2.67 yards, which is not equal to 3 yards. Therefore, equation A is not true.
B. 2mi = 10,460ft
To convert miles to feet, we know that 1 mile is equal to 5,280 feet. Multiplying 2 miles by 5,280 gives us 10,560 feet, which is equal to 10,460 feet. Therefore, equation B is true.
C. 9ft = 98in
To convert feet to inches, we know that 1 foot is equal to 12 inches. Multiplying 9 feet by 12 gives us 108 inches, which is not equal to 98 inches. Therefore, equation C is not true.
D. 5mi = 8,800yd
To convert miles to yards, we know that 1 mile is equal to 1,760 yards. Multiplying 5 miles by 1,760 gives us 8,800 yards, which is equal to 8,800 yards. Therefore, equation D is true.
Know more about equation here:
https://brainly.com/question/29657983
#SPJ11
what’s 9 plus 10????
Answer:
21
Step-by-step explanation:
the answer would be 21111. its19.
Answer: 19
Step-by-step explanation:
10
9
-----
19
Question number graph this piece wise function
The graph of the piece wise function is attached accordingly.
What is a piece wise function?A piecewise-defined function is one that is defined by numerous sub-functions, each of which applies to a distinct interval in the domain. Piecewise definition is a method of defining the function, not a feature of the function itself.
Piecewise specified functions can take many different shapes. Their "pieces" could be all linear or a mix of functional forms (constant, linear, quadratic, cubic, square root, cube root, exponential, and so on). Because of this variety, there is no such thing as a "parent function" for piecewise specified functions.
Learn more about piecewise functions;
https://brainly.com/question/31806828
#SPJ1
Which two angles are supplementary?
Angle D and Angle E
Angle C and Angle E
Angle D and Angle F
Angle A and Angle C
Angle ∠C and angle ∠E are supplementary angles. Then the correct option is B.
The sum of the supplementary angle is 180 degrees.
The sum of two acute angles will be less than 180 degrees.
Similarly, the sum of two obtuse angles is more than 180 degrees.
The addition of angle ∠C and angle ∠E is calculated as,
∠C + ∠E = 61° + 119°
∠C + ∠E = 180°
The addition of angle ∠C and angle ∠E is 180°. Thus, the correct option is B.
More about the angled link is given below.
https://brainly.com/question/15767203
#SPJ1
Given the equation 5x2 − 20x + 15 = 0, what are the values of h and k when the equation is written in vertex form a(x − h)2 + k = 0?
To write the equation 5x^2 - 20x + 15 = 0 in vertex form, we need to complete the square. First, we can factor out the common factor of 5:
5(x^2 - 4x + 3) = 0
Next, we can add and subtract the square of half the coefficient of x:
5(x^2 - 4x + 4 - 4 + 3) = 0
Simplifying this expression gives:
5((x - 2)^2 - 1) = 0
Finally, we can divide both sides by 5 to get the equation in vertex form:
(x - 2)^2 - 1 = 0
Comparing this equation to the vertex form a(x - h)^2 + k = 0, we can see that h = 2 and k = -1. Therefore, the values of h and k are 2 and -1, respectively.
Answer:
h = 2, k = −5
Step-by-step explanation:
Took the quiz and it's correct
Simplify (23)–2.
2
64
Playing tennis and playing golf are ____________ (choose between dependent and independent)
Playing tennis and playing golf are independent activities.
What is the relationship between playing tennis and playing golf?Playing tennis and playing golf are two distinct activities that do not rely on each other for completion.
Although they share some similarities such as requiring physical stamina and hand-eye coordination, they have different sets of rules, equipment, and playing surfaces.
While playing tennis involves hitting a ball over a net to the opponent's side of the court, golf requires hitting a ball into a series of holes with as few strokes as possible.
Therefore, one can engage in either activity without having to participate in the other.
Learn more about Independence
brainly.com/question/15375461
#SPJ11
