The radius of convergence is 5/7. The interval of convergence is (-5/7, 5/7) in interval notation.
To express the function f(x) = 7x² - 3x - 10 as the sum of a power series, we can start by factoring the quadratic term in the numerator:
f(x) = (7x² - 3x - 10)
The quadratic expression can be factored as follows:
f(x) = (7x + 5)(x - 2)
Now we can write the function f(x) as a sum of partial fractions:
f(x) = A/(7x + 5) + B/(x - 2)
To find the values of A and B, we can multiply both sides of the equation by the denominators and equate the coefficients of corresponding powers of x:
(7x + 5)(x - 2) = A(x - 2) + B(7x + 5)
Expanding both sides of the equation:
7x² - 14x + 5x - 10 = Ax - 2A + 7Bx + 5B
Grouping the terms with the same power of x:
(7x² + (5 - 14)x - 10) = (A + 7B)x + (-2A + 5B)
Equating the coefficients of corresponding powers of x:
7x² + (5 - 14)x - 10 = (A + 7B)x + (-2A + 5B)
Comparing the coefficients:
7 = A + 7B
5 - 14 = -2A + 5B
-10 = -2A
From the first equation, we can solve for A:
A = 7 - 7B
Substituting this value of A into the second equation:
-10 = -2(7 - 7B)
Simplifying:
-10 = -14 + 14B
14B = -10 + 14
B = 4/14
B = 2/7
Now we have the values of A and B:
A = 7 - 7B = 7 - 7(2/7) = 7 - 2 = 5
Therefore, the function f(x) can be expressed as:
f(x) = 5/(7x + 5) + 2/(x - 2)
Now, to find the interval of convergence for the power series representation of f(x), we need to determine the radius of convergence. The power series representation will converge within the interval (-r, r), where r is the radius of convergence.
In this case, since we have a rational function, the interval of convergence will be determined by the denominator with the smallest radius of convergence.
The denominators in the partial fractions are (7x + 5) and (x - 2). The radius of convergence for a power series centered at a point c is the distance from c to the nearest singularity.
For (7x + 5), the singularity occurs when 7x + 5 = 0, which gives x = -5/7.
For (x - 2), the singularity occurs when x - 2 = 0, which gives x = 2.
The distance from the center (c = 0) to the nearest singularity is the minimum of the absolute values of the two singularities: min(|-5/7|, |2|) = 5/7.
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HELP !!
In a sequence of numbers, a4=9, a5=15, a6=21, a7=27, and a8=33.
Which recursive rule can be used to find the nth term of the sequence, an?
a1=−9; an=an−1+6
a1=−9; an=6an−1
a1=9; an=an−1+6
a1=9; an=6an−1
Answer:
Given sequence:
a₄=9, a₅=15, a₆=21, a₇=27, and a₈=33Common difference is:
33-27 = 27 - 21 = 21 - 15 = 15 - 9 = 6Find the first term:
a₃ = 9 - 6 = 3a₂ = 3 - 6 = - 3a₁ = -3 - 6 = -9Recursive formula:
a₁ = -9 and aₙ = aₙ₋₁ + 6Correct choice is A
Find the total surface area.
Answer:
SA = 48 cm²
Step-by-step explanation:
SA = (3x12) + (4x3) = 48 cm²
An electrician bent a section of copper wire into a partial circle as shown. The dimensions are
given in feet (ft).
2.5 f
880
2.5
ft
What is the length of the section of wire to the nearest hundredth of a foot?
Answer:
Length = 3.84 feets (nearest hundredth)
Step-by-step explanation:
The length of section of wire can be obtyaiejd using the length of of a arc formular :
Length of arc = θ/360° * 2πr
Radius, r = 2.5 feets
Length of arc = (88/360) *2πr
Length of arc = (88/360) * 2π*2.5
Length of arc = 0.244444 * 15.707963
Length of arc = 3.8397
Length = 3.84 feets (nearest hundredth)
Answer:
Step-by-step explanation:
3.84
Today the high tide in Matheshan's Cove Lakeshore, is at midnight. The water level at high tide is 12.5 m. The depth, d metres, of the water in the cove at time t hours is modelled by the equation d(t)= 8+ 4.5sinl ő t).Kairvi is planning a day trip to the cove tomorrow, but the water needs to be at least 5 m deep for her to manoeuvre her sailboat safely. How can Kairvi determine the times when it will be safe for her to sail into Matheshan's Cove?
Solve the inequality 8 + 4.5sin(πt) ≥ 5 to find the times when it is safe for Kairvi to sail into Matheshan's Cove.
How can Kairvi determine the safe times to sail into Matheshan's Cove based on the water depth equation and the condition for safe sailing?To determine the times when it will be safe for Kairvi to sail into Matheshan's Cove, we need to solve the inequality 8 + 4.5sin(πt) ≥ 5, where t represents time in hours.
Here are the steps to solve the inequality:
Subtract 8 from both sides of the inequality:
4.5sin(πt) ≥ 5 - 8
4.5sin(πt) ≥ -3
Divide both sides of the inequality by 4.5 to isolate the sine function:
sin(πt) ≥ -3/4.5
sin(πt) ≥ -2/3
To find the values of t that satisfy this inequality, we need to consider the inverse sine (arcsine) function. Taking the inverse sine of both sides, we get:
πt ≥ arcsin(-2/3)
Solve for t by dividing both sides by π:
t ≥ (1/π)arcsin(-2/3)
The resulting expression (1/π)arcsin(-2/3) represents the minimum time at which the water depth will be at least 5 meters. To determine the specific times, you can use a calculator or reference table to evaluate the arcsin function and calculate the corresponding time values.
Note: Keep in mind that this solution assumes a 24-hour time format and that the given water depth equation and conditions are accurate.
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Which of the following are necessary conditions for the hypothesis test for the slope of the least squares regression line? (Select all that apply.) There is equal variance around the regression line for all x. The distribution of x is normal. The observations are independent. The sampling distribution of x is approximately normal. Data are from a random sample or experiment. The responses, y, for any value of x vary according to a normal distribution. The true relationship between the variables is linear. The parameter of interest is the true slope, ß
The conditions for hypothesis test for slope of least squares regression line are observations are independent, data is from random sample, the true relationship between variables is linear. So, correct options are c, e, f, g, h.
c) The observations are independent: This condition is necessary to ensure that the observations are not influenced by each other and that the regression estimates are not biased.
e) Data are from a random sample or experiment: Random sampling helps to ensure that the sample is representative of the population and allows for generalization of the results. In an experiment, random assignment helps establish causal relationships.
f) The responses, y, for any value of x vary according to a normal distribution: This assumption is needed to perform hypothesis tests and construct confidence intervals for the slope. It is usually assumed that the errors or residuals in the regression model are normally distributed.
g) The true relationship between the variables is linear: This assumption assumes that the relationship between the independent variable (x) and the dependent variable (y) can be adequately represented by a straight line.
h) The parameter of interest is the true slope, β: The hypothesis test focuses on testing whether the estimated slope coefficient significantly differs from zero, which represents the null hypothesis.
The remaining options (a, b, d) are not necessary conditions for the hypothesis test for the slope of the least squares regression line. They may be assumptions or conditions related to the regression model but are not directly tied to the hypothesis test for the slope.
So, correct options are c, e, f, g, h.
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At 2 pm, the temperature was -9 degrees. If the temperature drops 8 degrees, what is the new temperature?
Answer:
At 2 pm, the temperature was -9 degrees. If the temperature drops 8 degrees, what is the new temperature?
Step-by-step explanation:
sorry I can't delete it
Answer:
-15
Step-by-step explanation:
-7 Dropping 8 would be -15 because 8+7 is 15 but its negative so its -15 (im terrible at explaining)
Euler's formula, v − e + f = 2, relates the number of vertices (v), the number of edges (e), and the number of faces (f) of a polyhedron. Solve Euler's formula for v.
a) v = e + f + 2
b) v = e + f - 2
c) v = e - f - 2
d) v = e - f + 2
Euler's formula, v − e + f = 2, relates the number of vertices (v), the number of edges (e), and the number of faces (f) of a polyhedron.
Therefore, the correct option is (d) v = e - f + 2.
To solve Euler's formula for v, we'll have to isolate v on one side of the equation. The first step is to add e to both sides of the equation:
v − e + f + e = 2 + e
v + f = e + 2
Now subtract f from both sides of the equation:
v + f - f = e + 2 - f
v = e + 2 - f
Hence, the solution for Euler's formula for v is:
v = e + 2 - f
Therefore, the correct option is (d) v = e - f + 2.
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Simplify: 121/11 + 3(4)/2
Please and thank you. :)
Answer:
17
Step-by-step explanation:
I don't know the explanation, but hoped this helped.
SOMEONE HELPPP
What is the classification of the figure?
raindrops are falling at an average rate of 20 drops per square inch per minute. what would be a reasonable distribution to use for the number of raindrops hitting a particular region measuring 5 square inches in a minute? why? using your chosen distribution, compute the probability that the region has no rain drops in a given 6-second time interval.
The number of raindrops hitting a particular region can be modeled using the Poisson distribution. The Poisson distribution is commonly used to describe the number of events occurring in a fixed interval of time or space when these events happen randomly and independently with a known average rate. In this case, the average rate is given as 20 drops per square inch per minute. The probability that the region has no raindrops in a given 6-second time interval is approximately 0.7165
To compute the probability that the region has no raindrops in a given 6-second time interval, we need to convert the rate to match the time interval. Since the rate is given per minute, we can divide it by 60 to get the rate per second: λ = 20/60 = 1/3 drops per square inch per second.
Using the Poisson distribution, the probability of observing exactly k events in a given time interval is given by the formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where e is Euler's number (approximately 2.71828), λ is the average rate, and k is the number of events.
In our case, we want to find the probability that the region has no raindrops in a 6-second time interval, which corresponds to k = 0.
P(X = 0) = (e^(-1/3) * (1/3)^0) / 0! = e^(-1/3)
Using a calculator, we can evaluate e^(-1/3) ≈ 0.7165.
Therefore, the probability that the region has no raindrops in a given 6-second time interval is approximately 0.7165 or 71.65%.
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How many 5 bills seth has
Answer:
22= 6(1x+5x)
Step-by-step explanation:
He could have 4 $5 bills and 2 $1 bills.
Answer:
he has 4 $5 bills
Step-by-step explanation:
22 - 2 = 20
20 divided by 4 = 5
so 4 $5 bills and 2 $1 bills
An ocean liner leaves port at 9:00 a.m. traveling directly out to sea at 22 miles per hour. A ferry leaves port at 9:30
a.m. traveling the same direction at 28 miles per hour. Use this information to complete the sentences.
A pair of parametric equations representing the paths of the boats are
From the time the ocean liner leaves until the ferry overtakes it is minutes at a distance from the port of
approximately miles.
Answer:
x=22t and y=28(t-0.5)
140
51
Step-by-step explanation:
right on edg
Answer:
A pair of parametric equations representing the paths of the boats are
✔ x = 22t and y = 28(t – 0.5) (Negative 0.5).
From the time the ocean liner leaves until the ferry overtakes it is
✔ 140.
minutes at a distance from the port of approximately
✔ 51 miles.
Step-by-step explanation:
Edge 2022
Para resolver un sistema de ecuaciones lineales 2x2 por el método de sustitución se debe tener en cuenta:
A) consiste en despejar la misma incógnita en las dos ecuaciones y después igualar los resultados. En primer lugar, elegimos la incógnita que deseamos despejar. En este caso, empezaré por la «x» y despejo la misma en ambas ecuaciones.
B) debemos saber representar las gráficas de las rectas. Nosotros lo haremos uniendo puntos calculados previamente. Terminaremos con un sistema de dos inecuaciones (o desigualdades). En este caso, la solución del sistema es la intersección de dos regiones del plano.
C) es un método lineal ya que no se basa en despejes, se utilizan procesos algebraicos estructurados.
D) consiste en despejar o aislar una de las incógnitas (por ejemplo, x ) y sustituir su expresión en la otra ecuación. De este modo, obtendremos una ecuación de primer grado con la otra incógnita, y . Una vez resuelta, calculamos el valor de x sustituyendo el valor de y que ya conocemos.
AYUDA PLS
Answer:
El método de igualación consiste en despejar la misma incógnita en las dos ecuaciones y después igualar los resultados.
Los pasos a seguir son los siguientes:
sistema de ecuaciones
En primer lugar, elegimos la incógnita que deseamos despejar. En este caso, empezaré por la «x» y despejo la misma en ambas ecuaciones.
x+y=7; x=7-y
5x-2y=-7; 5x=2y-7
x=(2y-7)/5
Una vez hemos despejado, igualamos:
7-y = (2y-7)/5
5.( 7-y) = (2y -7)
35 -5y= +2y -7
42=7y
y=42/7=6
y=6
Por último, sustituimos el valor que hemos calculado despejando la otra incógnita en una de las ecuaciones iniciales.
Nick is curious about which cell phone provider is most used by his neighbors. He asks several neighbors about their provider and draws a conclusion based on the answers he received. What kind of statistical study did Nick conduct? A. survey B. experiment C. observational study D. theoretical study
Answer:
A. Survey
Step-by-step explanation:
Answer:
Option A. Survey
Step-by-step explanation:
What is survey?
Survey is defined as the act of examining a process or questioning a selected sample of individuals to obtain data about a service, product, or process.
Nick collected samples and then concluded his answer which is a survey he conducted.
Correct answer is Option A.
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multiply
12 times 1/3?
Answer:
4
Step-by-step explanation:
12(1/3)
so it is just 12/3 which is 4
Hope that helps :)
The length of a rectangular frame is 15 inches and the width of the frame is 8 inches. What
is the length of the diagonal of this frame in inches?
Record your answer. Be sure to use the correct place value.
How many residuals lie outside the 95% prediction bands? According to the SRM, how many of these should lie above and how many should lie below the estimated regression line?
The number of residuals outside the 95% prediction bands depends on the specific data and regression model. The explanation below provides general insights.
The number of residuals lying outside the 95% prediction bands can vary depending on the data and the estimated regression model. In a simple linear regression, the prediction bands represent the range within which future observations are expected to fall with 95% confidence.
Ideally, if the model assumptions are met and the regression is a good fit, we would expect only about 5% of the residuals to fall outside the prediction bands by chance. However, if the assumptions are violated or the model is not appropriate, more residuals may deviate beyond the bands.
The distribution of residuals above or below the estimated regression line depends on the symmetry of the errors. If the errors are normally distributed and the model is unbiased, roughly half of the residuals lying outside the prediction bands should be above the line, and the other half should be below the line.
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What is the value of 3^2? (3^2 means 3 raised to the second power.)
Answer:
9
Step-by-step explanation:
What is the perimeter of AOJL?
3
Ρ 2
K
M
9
Answer:
dunno.
Step-by-step explanation:
duno
what is the are of this?
Answer:
Ok first of all you start out splitting the shape into seperate rectangles, in this case 2. take those 2 rectangles and find the area of them. Say we start from the smaller rectangle. 2 x 4 = 8 so the area of the smaller rectangle is 8. Keep that in mind. Next we have to multiply 10 x 5 because that is the labeled number of meters and that equals 50. So add 50 + 8 = 58
Step-by-step explanation:
Your answer is 58 meters
Really hope i helped and have a wonderful day! :)
Answer:
58 m³
Step-by-step explanation:
To find the area of irregular shapes like this, it is easy to split the shape into 2 individual shapes, find the area of both, then add the areas to find the total.
Now, we'll split it into 2 rectangles with areas of:
2 x 4
10 x 5
To find area, you multiply the length of one side by the other. The first rectangle's area is (2 x 4) and the other rectangle's area is (10 x 5).
10 x 5 = 50
2 x 4 = 8
50 + 8 = 58
The area of the figure is 58 m³.
Create a real-world application and its complete solution that requires concepts of linear algebra.
Real-World Application: Image Compression: Image compression is a fundamental concept in the field of computer graphics and image processing.
Linear algebra plays a crucial role in various image compression techniques. Let's consider a complete solution for image compression using concepts of linear algebra.
Image Representation:Linear algebra concepts, such as matrix operations and transformations, are fundamental to every step of the image compression process outlined above. By applying these techniques, we can achieve efficient storage and transmission of images while balancing the trade-off between image quality and compression ratio.
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A box contains 3 red and 7 green marbles. If 9 marbles are drawn without replacement, what is the expected number of red marbles?
The expected number of red marbles is 197/240.
Given that a box contains 3 red and 7 green marbles. If 9 marbles are drawn without replacement, we need to determine the expected number of red marbles.
The total number of marbles in the box = 3 + 7 = 10The probability of selecting a red marble at the first draw = 3/10
The probability of selecting a red marble at the second draw = 2/9
The probability of selecting a red marble at the third draw = 1/8
∴ The expected number of red marbles = Probability of selecting a red marble at the first draw + Probability of selecting a red marble at the second draw + Probability of selecting a red marble at the third draw= 3/10 + 2/9 + 1/8= (216 + 240 + 135) / (10 * 9 * 8)= 591/720= 197/240
Hence, the expected number of red marbles is 197/240.
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The given information is: A box contains 3 red and 7 green marbles. If 9 marbles are drawn without replacement, what is the expected number of red marbles?The expected number of red marbles can be found as follows:
Let us assume that X is a random variable, that represents the number of red marbles drawn from the box.
If we want to calculate the expected value of X, then we can use the formula:E(X) = Σ(xP(x))Where Σ(xP(x)) is the sum of all possible outcomes, multiplied by their respective probabilities.
The probability of drawing a red marble on the first draw is 3/10.
The probability of drawing a red marble on the second draw is 2/9, since there are only 2 red marbles left out of 9 marbles total.
The probability of drawing a red marble on the third draw is 1/8, since there are only 1 red marble left out of 8 marbles total.
Since we are drawing 9 marbles without replacement, we need to multiply all these probabilities together to get the probability of drawing a certain sequence of red and green marbles.
P(X = k) represents the probability of drawing k red marbles out of 9 marbles.Using the formula: E(X) = Σ(xP(x))E(X) = Σ(kP(X = k))k ranges from 0 to 3.
So, the expected value of red marbles is:E(X) = 0P(X = 0) + 1P(X = 1) + 2P(X = 2) + 3P(X = 3)E(X) = 0 + 1(3/10)(6/9)(5/8) + 2(3/10)(2/9)(5/8) + 3(3/10)(2/9)(1/8)E(X) = 0 + 0.27917 + 0.0625 + 0.01406E(X) = 0.35573
Therefore, the expected number of red marbles is approximately 0.356.
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What is the probability of rolling a number less than 4 on a number cube labeled 1 trough 6?!
Answer:
3/6
Step-by-step explanation:
1 2 and 3 are less than 4 so only 3 numbers
Suppose we are given a series [=1(-1)+1 9n(x), where for each fixed x € R, we have 91() > 92(x) > 93(x) > ..0. Assume furthermore that 91(2) is bounded on R, and that the In(x) converge pointwise to 0. Prove that the series converges uniformly on R.
|In(x)| < ε/M for all x ∈ R, we can bound the above expression by:|91(n+1)(x) + 91(n+2)(x) + ...| ≤ 91(n+1)ε/M + 91(n+2)ε/M + ... = ε.This shows that the series converges uniformly on R.
We must demonstrate that for any given > 0, there exists a positive integer N such that the difference between the partial sum Sn(x) and the limit L(x) for all x R is less than to demonstrate that the series converges uniformly on R.
Since 91(2) is limited on R, let M be an upper headed for 91(2). Since the In(x) unite pointwise to 0, for any ε > 0, there exists a positive number N with the end goal that for all n > N, |In(x)| < ε/M for all x ∈ R.
Presently, for n > N and for all x ∈ R, we have:
|Sn(x) - L(x)| = |(∑ i=1 to n 91(i)(x)) - 0| = |91(n+1)(x) + 91(n+2)(x) + ...|
Since |In(x)| < ε/M for all x ∈ R, we can bound the above articulation by:
|91(n+1)(x) + 91(n+2)(x) + ...| ≤ 91(n+1)ε/M + 91(n+2)ε/M + ... = ε
This shows that the series combines consistently on R.
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Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. f(x) = 4x (3) = f(5) = f(-2) = f(0.4) =
Using the function f(x) = 4x, we find that f(3) = 12, f(5) = 20, f(-2) = -8, and f(0.4) = 1.6.
To evaluate the function f(x) = 4x at the indicated values, we can use a calculator to perform the calculations.
f(3):
Multiply 3 by 4: 3 * 4 = 12
The value of f(3) is 12.
f(5):
Multiply 5 by 4: 5 * 4 = 20
The value of f(5) is 20.
f(-2):
Multiply -2 by 4: -2 * 4 = -8
The value of f(-2) is -8.
f(0.4):
Multiply 0.4 by 4: 0.4 * 4 = 1.6
The value of f(0.4) is 1.6.
The function simply multiplies the input value by 4 to determine the output value.
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give a database of the results of an election, find the number of seats won by each party
To find the number of seats won by each party from a database of election results, we need the specific information about the parties, the candidates, and the corresponding vote counts or seat allocations.
With that information, we can perform calculations or queries to determine the number of seats won by each party.
Here's a general outline of the steps involved:
Obtain the election database or data that includes information on parties, candidates, and their respective vote counts or seat allocations.
Analyze the database structure to identify the relevant tables or fields
that store the necessary information.
Use database query or analysis tools to extract the relevant data. Write a query or use filtering mechanisms to retrieve the party names, candidate information, and corresponding vote counts or seat allocations.
Perform calculations or aggregations based on the data retrieved to determine the total number of seats won by each party. This can involve summing the vote counts or seat allocations for each party.
Present or display the results, showing the number of seats won by each party in the election.
Please provide the specific database structure or information about the parties, candidates, and vote counts if you have them, and I can assist you further in generating the desired results.
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What are the first, second, and third quartiles from the set of
data: (3,4,5,6,7,37,100)?
The first, second, and third quartiles from the set of data (3,4,5,6,7,37,100) include the following:
Q₁ = 4.
Q₂ = 6.
Q₃ = 37
How to determine the statistical measure of the data set?Based on the data set, the first quartile (Q₁) can be calculated as follows;
Q₁ = [(n + 1)/4]th term
Q₁ = (7 + 1)/4
Q₁ = 2nd term
Q₁ = 4.
For the second quartile (Q₂), median, or 50th percentile, we have the following:
second quartile (Q₂) = 4th term
second quartile (Q₂) = 6.
For the third quartile (Q₃), we have:
Q₃ = [3(n + 1)/4]th term
Q₃ = 3 × 2nd term
Q₃ = 6th term
Q₃ = 37
In conclusion, a box plot for the given data set is shown in the image attached below.
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i need help please and thanks
Answer:
[tex]48cm^{2}[/tex]
Step-by-step explanation:
Split the shape up into two rectangles
(1) rectangle would be 2 x 10 = 20
(2) rectangle would be 4 x 7 = 28
Add them up = [tex]48cm^{2}[/tex]
a sine function has an amplitude of 3, a period of π, and a phase shift of pi over 2 period what is the y-intercept of the function?
The y-intercept of the sine function with an amplitude of 3, a period of π, and a phase shift of π/2 is 0.
The general form of a sine function is y = A×sin(Bx - C) + D, where A represents the amplitude, B determines the period, C is the phase shift, and D is the vertical shift.
In this case, the given amplitude is 3, indicating that the maximum value of the function is 3 and the minimum value is -3.
The period is π, which means the function completes one full cycle in π units of x.
The phase shift is π/2 period, which shifts the graph to the right by π/2 units.
Since the y-intercept is the point where the graph intersects the y-axis (x = 0), and the sine function passes through the origin (0, 0), the y-intercept is 0.
Therefore, the y-intercept of the given sine function is 0.
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Use the four-step process to find f'(x) and then find f'(1), f'(2), and f'(3). f(x) = -x² + 4x-9 f'(x) =
The derivative of f(x) = -x² + 4x - 9 is f'(x) = -2x + 4. Evaluating f'(x) at x = 1, 2, and 3 gives f'(1) = 2, f'(2) = 0, and f'(3) = -2. To find the derivative of the function f(x) = -x² + 4x - 9, we will use the four-step process.
After applying the process, we obtain the derivative f'(x) = -2x + 4. Evaluating this derivative at x = 1, x = 2, and x = 3 gives us f'(1) = 2, f'(2) = 0, and f'(3) = -2.
The four-step process involves the following steps:
1. Begin by applying the power rule, which states that the derivative of [tex]x^n[/tex] is [tex]nx^{(n-1)[/tex], where n is a constant. In this case, we have -x², so the derivative becomes -2x.
2. Apply the power rule to the next term, which is 4x. The derivative of 4x is 4.
3. Since -9 is a constant term, its derivative is zero.
4. Combine the derivatives obtained in steps 1, 2, and 3 to find the overall derivative of the function f(x). In this case, f'(x) = -2x + 4.
To find the values of f'(1), f'(2), and f'(3), we substitute the corresponding values of x into the derivative function.
When x = 1, f'(1) = -2(1) + 4 = 2.
When x = 2, f'(2) = -2(2) + 4 = 0.
When x = 3, f'(3) = -2(3) + 4 = -2.
Therefore, the derivative of f(x) is f'(x) = -2x + 4, and the values of f'(1), f'(2), and f'(3) are 2, 0, and -2, respectively.
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