The Maclaurin polynomial of degree 4 for [tex]g(x) = x sin(x)[/tex] is given by [tex]P4(x) = x^2 - (1/6)x^4[/tex], and the Maclaurin polynomial of degree 2 for [tex]g(x) = (sin(x))/x[/tex] is given by [tex]P2(x) = 1 + 1/x[/tex].
How to find the Maclaurin polynomial?(a) To find the Maclaurin polynomials for f(x) = ex and g(x) = xex, we need to calculate the derivatives of these functions and evaluate them at x = 0.
For f(x) = ex:
f'(x) = ex, evaluated at x = 0, gives [tex]f'(0) = e^0 = 1[/tex].
f''(x) = ex, evaluated at x = 0, gives [tex]f''(0) = e^0 = 1[/tex].
So the Maclaurin polynomial of degree 2 for f(x) = ex is given by:
[tex]P2(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 = 1 + 1x + (1/2)x^2 = 1 + x + (1/2)x^2[/tex].
For g(x) = xex:
g'(x) = (1 + x)ex, evaluated at x = 0, gives [tex]g'(0) = (1 + 0)e^0 = 1[/tex].
g''(x) = (2 + x)ex, evaluated at x = 0, gives [tex]g''(0) = (2 + 0)e^0 = 2[/tex].
g'''(x) = (3 + x)ex, evaluated at x = 0, gives [tex]g'''(0) = (3 + 0)e^0 = 3[/tex].
So the Maclaurin polynomial of degree 3 for g(x) = xex is given by:
[tex]P3(x) = g(0) + g'(0)x + (g''(0)/2!)x^2 + (g'''(0)/3!)x^3 = 0 + 1x + (2/2!)x^2 + (3/3!)x^3 = x + x^2 + (1/2)x^3[/tex]
The relationship between the Maclaurin polynomials of degree 2 for f(x) = ex and degree 3 for g(x) = xex is that the polynomial for g(x) contains an extra term of degree 3 compared to the polynomial for f(x).
(b) We can use the result from part (a) and the Maclaurin polynomial of degree 3 for f(x) = sin(x) to find a Maclaurin polynomial of degree 4 for g(x) = x sin(x).
From part (a), we have the Maclaurin polynomial of degree 3 for f(x) = sin(x) given by:
[tex]P3(x) = x - (1/6)x^3[/tex].
To find the Maclaurin polynomial of degree 4 for g(x) = x sin(x), we can multiply P3(x) by x:
[tex]P4(x) = x * P3(x) = x * (x - (1/6)x^3) = x^2 - (1/6)x^4[/tex].
So the Maclaurin polynomial of degree 4 for g(x) = x sin(x) is given by:
[tex]P4(x) = x^2 - (1/6)x^4[/tex].
(c) Using the result from part (a) and the Maclaurin polynomial of degree 3 for f(x) = sin(x), we can find a Maclaurin polynomial of degree 2 for g(x) = (sin(x))/x.
From part (a), we have the Maclaurin polynomial of degree 2 for f(x) = sin(x) given by:
P2(x) = 1 + x.
To find the Maclaurin polynomial of degree 2 for g(x) = (sin(x))/x, we can divide P2(x) by x:
[tex]P2(x) / x = (1 + x) / x = 1 + 1/x[/tex].
So the Maclaurin polynomial of degree 2 for g(x) = (sin(x))/x is given by:
[tex]P2(x) = 1 + 1/x[/tex].
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Which of the following is the first step in the decision-making process?
a. Analyze the problem and its causes
b. Generate alternatives
c. Solicit and analyze feedback
d. Identify the problem
The correct answer is D to identify the problem.
What is the decision-making process?
The decision-making process is the method by which a judgment or a decision is reached. It involves identifying the problem or decision to be made, analyzing potential courses of action, evaluating alternatives, and selecting the best possible solution.
It's a structured process that assists in making effective decisions, and it can be useful in both personal and professional contexts. What is the first step in the decision-making process?
The first step in the decision-making process is to identify the problem. This entails defining the issue that requires a decision to be made. It's a crucial step because without accurately identifying the issue or problem, it's impossible to make the best decision.
Analyzing the problem and its causes (A), generating alternatives (B), and soliciting and analyzing feedback (C) are all critical components of the decision-making process, but they come after the problem has been identified.
As a result, option D, identifying the problem, is the first step in the decision-making process.
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A circuit containing an electromotive force (a battery), a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω). The voltage drop across the capacitor is Q/C, where Q is the charge (in coulombs), so in this case Kirchhoff's Law gives
RI+(Q/C)=E(t).
Since the current is I=dQ/dt, we have
R(dQ/dt)+(1/C)Q=E(t).
Suppose the resistance is 20Ω, the capacitance is 0.1F, a battery gives a constant voltage of E(t)=60V, and the initial charge is Q(0)=0C.
Find the charge and the current at time t.
Q(t)= ,
I(t)= .
The charge on the capacitor at time t is given by Q(t) = 6C - 6C * e^(-t/2)---- Eqn. (1) and the current at time t is given by I(t) = 3A * e^(-t/2) --- Eqn. (2).
How to determine the charge on the capacitorR(dQ/dt) + (1/C)Q = E(t)
The general solution of the above equation (when E(t) is a constant E) is:
Q(t) = CE + (Q(0) - CE)e^(-t/RC)
Plugging in the given values:
Q(t) = 0.1F * 60V + (0C - 0.1F * 60V)e^(-t/(20Ω * 0.1F))
Simplify this to get:
Q(t) = 6C - 6C * e^(-t/2) Eqn. (1)
The current is the derivative of the charge with respect to time:
I(t) = dQ(t)/dt = d/dt [6C - 6C * e^(-t/2)]
Taking the derivative and simplifying gives:
I(t) = 3A * e^(-t/2) Eqn. (2)
So the charge on the capacitor at time t is given by Eqn. (1) and the current at time t is given by Eqn. (2).
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Suppose a brewery has a filing machine that is 12 ounce bottles of beer, it is known that the amount of beer poured by this filing machine follows a normal dutiniowa mean of 12.10 and a standard deviation of .05 ounce. Find the probability that the bottle contains between 12.00 and 12.06 ounces
Answer:
Let X be the random variable representing the amount of beer poured by the filling machine. Since X follows a normal distribution with mean μ = 12.10 and standard deviation σ = 0.05, we can use the standard normal distribution to find the probability that a bottle contains between 12.00 and 12.06 ounces.
First, we need to standardize the values 12.00 and 12.06 by subtracting the mean and dividing by the standard deviation:
z1 = (12.00 - 12.10) / 0.05 = -2 z2 = (12.06 - 12.10) / 0.05 = -0.8
Now we can use a standard normal distribution table to find the probability that a standard normal random variable Z is between -2 and -0.8:
P(-2 < Z < -0.8) = P(Z < -0.8) - P(Z < -2) ≈ 0.2119 - 0.0228 ≈ 0.1891
So, the probability that a bottle contains between 12.00 and 12.06 ounces of beer is approximately 0.1891.
Step-by-step explanation:
Guess the rule and add the next number in the sequence.
1 6 16 31 51
The graph compares the heights and arm spans of players on a basketball team. The equation of the trend line that best fits the data is y = x + 2. Predict the arm span for a player who is 66 inches tall.
A. 69 inches
B. 67 inches
C. 64 inches
D. 68 inches
The correct answer is D. 68 inches. The trend line equation y = x + 2 indicates that there is a linear relationship between height and arm span. The coefficient of 1 on x suggests that, on average, for every increase of 1 inch in height, the arm span increases by 1 inch as well.
The intercept of 2 indicates that even at a height of 0 inches, there is a minimum arm span of 2 inches. By substituting the given height value into the equation, we can accurately predict the corresponding arm span
The equation of the trend line given is y = x + 2, where y represents the arm span and x represents the height of the players. We need to predict the arm span for a player who is 66 inches tall.
To make the prediction, we substitute x = 66 into the equation and solve for y:
y = 66 + 2
y = 68
Therefore, the predicted arm span for a player who is 66 inches tall is 68 inches.
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solve the given differential equation. y(ln(x) − ln(y)) dx = (x ln(x) − x ln(y) − y) dy
To solve the given differential equation:
y(ln(x) - ln(y)) dx = (x ln(x) - x ln(y) - y) dy
We can start by rearranging the terms:
y ln(x) dx - y ln(y) dx = x ln(x) dy - x ln(y) dy - y dy
Next, we can integrate both sides of the equation:
∫ y ln(x) dx - ∫ y ln(y) dx = ∫ x ln(x) dy - ∫ x ln(y) dy - ∫ y dy
To integrate the left-hand side, we can use integration by parts. Let's denote u = ln(x) and dv = y dx. Then, du = (1/x) dx and v = xy. Applying integration by parts, we have:
∫ y ln(x) dx = xy ln(x) - ∫ (1/x)(xy) dx
= xy ln(x) - ∫ y dx
= xy ln(x) - yx + C1
where C1 is the constant of integration.
Similarly, integrating the other terms:
∫ y ln(y) dx = xy ln(y) - yx + C2
∫ x ln(x) dy = (x^2 ln(x))/2 - ∫ (x^2)(1/x) dy
= (x^2 ln(x))/2 - ∫ x dy
= (x^2 ln(x))/2 - (x^2)/2 + C3
∫ x ln(y) dy = (x^2 ln(y))/2 - ∫ (x^2)(1/y) dy
= (x^2 ln(y))/2 - ∫ (x^2/y) dy
= (x^2 ln(y))/2 - x^2 ln(y) + ∫ x dy
= (x^2 ln(y))/2 - x^2 ln(y) + (x^2)/2 + C4
∫ y dy = (y^2)/2 + C5
Substituting these results back into the original equation:
xy ln(x) - yx + C1 - xy ln(y) + yx - C2 = (x^2 ln(x))/2 - (x^2)/2 + C3 - (x^2 ln(y))/2 + x^2 ln(y) - (x^2)/2 + C4 - (y^2)/2 - C5
Simplifying:
xy ln(x) - xy ln(y) = (x^2 ln(x))/2 - (x^2 ln(y))/2 - (y^2)/2 + C
where C = C1 - C2 + C3 + C4 - C5.
We can further simplify this equation:
xy (ln(x) - ln(y)) = (x^2 ln(x) - x^2 ln(y) - y^2)/2 + C
Finally, dividing both sides by (ln(x) - ln(y)), we get:
xy = (x^2 ln(x) - x^2 ln(y) - y^2)/(2(ln(x) - ln(y))) + C
This is the general solution to the given differential equation.
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Which one of these is in scientific notation?
(47 points)
Answer:
8.98 * 10^6 is the scientific notation
Step-by-step explanation:
the quotient of 17 and z
Answer:
17÷z = quotient
Step-by-step explanation:
quotient = ÷
Use the binomial series to find a Taylor polynomial of degree 3 for 1 91 +32 T3(0) X + c? + 23
The Taylor polynomial of degree 3 for the function 1/(1-2x) centered at x=0 is (1+2x+4x²+8x³).
Explanation: Given, 1/(1-2x) = ∑n=0 to infinity of 2^n * x^n The above series is the binomial series for (1+x)^n where n=-1Using the binomial series for n=-1, we have1/(1-2x) = ∑n=0 to infinity of 2^n * x^n= ∑n=1 to infinity of 2^(n-1) * x^(n-1)= 1 + ∑n=1 to infinity of 2^n * x^nTaking up to degree 3, we get1/(1-2x) = 1 + 2x + 4x² + 8x³ + ...Therefore, the Taylor polynomial of degree 3 for 1/(1-2x) is 1 + 2x + 4x² + 8x³.
An infinite sum of words that are expressed in terms of a function's derivatives at a single point is known as the Taylor series or Taylor expansion of a function in mathematics. Near this point, the function and the sum of its Taylor series are equivalent for the majority of common functions. for Brook Taylor, who introduced the Taylor series in 1715, they are named for him. In honour of Colin Maclaurin, who made great use of this unique example of Taylor series in the middle of the 18th century, a Taylor series is sometimes known as a Maclaurin series where 0 is the point at which the derivatives are taken into account.
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Write the equation of the line in slope- intercept form(y=mx+b)
Answer:
-1/3x + 5 = y
Step-by-step explanation:
The length of a rectangle is two more than triple the width. If the perimeter is 166 inches, what are the dimensions of the rectangle?
The dimensions of the rectangle are length = 62.75 inches and width = 20.25 inches.
The given problem states that the length of a rectangle is two more than triple the width.
If the perimeter is 166 inches, what are the dimensions of the rectangle? Let's solve the problem,
Step 1
Given, The length of the rectangle = l
Width of the rectangle = w
The perimeter of the rectangle = 166 inches
The formula for the perimeter of a rectangle is,
Perimeter = 2(l + w)
So, 166 = 2(l + w)166/2 = l + w83 = l + w ----(1)
Step 2
According to the given problem, The length of a rectangle is two more than triple the width
Therefore,
l = 2 + 3w
Substitute this value in equation (1)
83 = (2 + 3w) + w
83 = 2 + 4w
83 - 2 = 4w
81 = 4w
w = 81/4
w = 20.25 (approx)
Step 3
We have width w = 20.25 inches.
We can find the length l by substituting w in l = 2 + 3w
So,
l = 2 + 3(20.25)
= 2 + 60.75
= 62.75
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What is the area?
O 90 square kilometers
O 45 square kilometers
O 27 square kilometers
O 36 square kilometers
Find the eighth term of a geometric sequence for which a,3 = 35 and r= 7.
Answer:
588245.
Step-by-step explanation:
nth term = an = a1 r^(n-1) where a1 = the first term
a3 = 35 = a1 7^(3 - 1)
35 = a1* 49
a1 = 35/49 = 5/7
So the 8th term = (5/7)* (7)^7
= 588245
Renee is a sales associate at a store. She earns $80 a week plus a 15% commission on her sales. Last week, she sold $200 worth of items. What is the total amount Renee earned for the week? How much did she earn from commission?
Renee earned a total of $
for the week. The amount she earned from commission was $
.
Answer:
The amount earned for the week=$110
Amount earned from commission =$30
Step-by-step explanation:
commission earned on sales = $200×15%= $30
total amount for the week=$80 +$30= $110
find vertex of this function
Answer:
Step-by-step explanation:
(-1,-9)
Work out the circumference of this circle.
Take a to be 3.142 and write down all the digits given by your calculator.
16.8 cm
Answer: 52.7856cm
Step-by-step explanation:
Circumference of a circle = 2πr
Note that radius = Diameter / 2 = 16.8/2 = 8.4cm
Circumference = 2πr
= 2 × 3.142 × 8.4
= 52.7856cm
Therefore, the circumference of the circle is 52.7856cm
A 28 ft tall house casts a shadow that is 35 ft long. A umbrella outside the house casts a shadow that is 16 7/8 long. How tall is the umbrella?
Answer:
The unbrella is 13.5 feet tall.
Step-by-step explanation:
35 / 28 = 1.25
1.25x = 16 7/8
x = (16 7/8) / 1.25
x = 13.5 ft
Which graph shows a function where f(2)=4?
Answer:
A
Step-by-step explanation:
Answer:
the one that is a horizontal straight line on x axis that lies on 2
The table below gives the distribution of milk
chocolate M&M's
Color
NA
DO
Probability
0.13
0.13
0.14
0.16
0.20
0.24
If a candy is drawn at random, what is the probability
that it is not orange or red?
Answer: .67
Step-by-step explanation:
got it right on acellus
8. (08.02 lc)complete the square to transform the expression x2 6x 5 into the form a(x − h)2 k. (1 point)(x 6)2 4(x 6)2 − 4(x 3)2 − 4(x 3)2 4
The expression [tex]x^{2}[/tex] + 6x + 5 can be completed by transforming it into the form a(x - h)^2 + k.
To complete the square, we want to rewrite the quadratic expression x^2 + 6x + 5 in a perfect square trinomial form. We can achieve this by adding and subtracting a constant term inside the parentheses.
Starting with the given expression: x^2 + 6x + 5
To complete the square, we need to take half of the coefficient of x and square it. Half of 6 is 3, and squaring 3 gives us 9. So, we add and subtract 9 inside the parentheses:
x^2 + 6x + 5 = (x^2 + 6x + 9 - 9) + 5
Now, we can group the first three terms as a perfect square trinomial and simplify:
(x^2 + 6x + 9 - 9) + 5 = (x + 3)^2 - 9 + 5
Simplifying further, we have:
(x + 3)^2 - 4
Therefore, the expression x^2 + 6x + 5 can be written in the form a(x - h)^2 + k as (x + 3)^2 - 4.
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Please just give me the equation no one helped me last time
AHH HELP ME PLS
Choose the values of x that are solutions to the inequality 5 <,
Select all that apply
Ax=2
= 6
E *=-7
F1 = 10
Answer:
x = 5
x = 6
x = 10
Step-by-step explanation:
NOTE: IT HAS TO BE MORE THAN 5 OR EQUAL TO 5
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A part manufactured at a factory is known to be 12.05 cm long on average, with a standard deviation of 0.350. One day you suspect that that the part is coming out a little longer than usual, but with the same deviation. You sample 14 at random and find an average length of 12.20. What is the z-score which would be used to test the hypothesis that the part is coming out longer than usual?
The z-score to test the hypothesis that the part is coming out longer than usual is approximately 1.61.
Sample mean = x = 12.20 cm
Population mean = μ = 12.05 cm
Standard deviation = σ = 0.350 cm
Sample size = n = 14
A hypothesis is an informed prediction regarding the solution to a scientific topic that is supported by sound reasoning. there is the expected result of the experimentation even if there is not proved in an experiment.
Calculating the z-score -
[tex]z = (x - u) / (\alpha / \sqrt n)[/tex]
Substituting the values -
[tex]z = (12.20 - 12.05) / (0.350 / \sqrt{14)[/tex]
= z = 0.15 / (0.350 / √14)
= 0.093
Substituting the value again into the formula:
z = 0.15 / 0.093
= 1.61
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What’s the answer to this?!!!
Answer:
77 meters
Step-by-step explanation:
For a right triangle, the formula for side length is
a^2 + b^2 = c^2 where c is the hypotenuse (opposite the right angle)
36^2 + b^2 = 85^2
1296 + b^2 = 7225
b^2 = 5,929
find the square root of both sides:
b = 77
Please let me know if you have questions.
Find the solution of the initial-value problem y'" – 84" + 16Y' – 128y = sec 4t, y(0) = 2, y'(0) = 2, y"0) = 88. A fundamental set of solutions of the homogeneous equation is given by the functions: yı(t) = eat, where a = yz(t) = yz(t) = A particular solution is given by: Y(t) = ds-yi(t) to + ]) •yz(t) + • Y3(t) t) Therefore the solution of the initial-value problem is: y(t)=___ +Y(t).
The solution of the initial-value problem is:
y(t) = C1e^(-4t) + C2e^(4t) + Y(t)
where C1 and C2 are constants determined by the initial conditions, and Y(t) is the particular solution given by the formula provided.
To find the solution of the initial-value problem, we can use the given fundamental set of solutions of the homogeneous equation and the particular solution.
The fundamental set of solutions is y1(t) = e^at, where a = -4 and y2(t) = e^bt, where b = 4.
The particular solution is Y(t) = ds-y1(t) to + y2(t) • y3(t), where y3(t) is another function that satisfies the non-homogeneous equation.
Combining the solutions, the general solution of the non-homogeneous equation is y(t) = C1e^(-4t) + C2e^(4t) + Y(t), where C1 and C2 are constants
To determine the specific solution, we need to use the initial conditions. Given y(0) = 2, y'(0) = 2, and y''(0) = 88, we can substitute these values into the general solution and solve for the constants C1 and C2.
Finally, the solution of the initial-value problem is y(t) = C1e^(-4t) + C2e^(4t) + Y(t), where C1 and C2 are the constants determined from the initial conditions and Y(t) is the particular solution.
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Cheryl's making trail mix for a friend she always uses three cups of almonds for every four cups of cashews if Cheryl wants to make a larger batch of trail mix which of the following has the same ratio of almonds cashews
A. 9 cups of almonds for every 20 cups of cashews
B. 6 cups of almonds for every 12 cups of cashews
C. 9 cup of almonds for every 12 cups of cashews
D. 12 cups of almonds for every 20 cups of cashews
Answer:
c
Step-by-step explanation:
3x3=9
4x3=12
Answer:
C
Step-by-step explanation:
3 cups of almonds / 4 cups of cashews
So to get 9 cups of almonds, we need to multiply almonds and cashews by 3
3*3 = 9 cups of almonds
4*3 = 12 cups of cashews
So the correct answer is C
Find the value of x. I WILL MARK YOU BRAINLIEST!!!
Answer:
4x+3x+2x=180
9x=180
x=20
Suppose f(x,y,z) = In(x + 2y2 + 3z"). Find the following partial derivatives. a. fx b. fz c.d2f/dzdx.
The partial derivatives are as follows :
(a) fx = 1 / (x + 2y^2 + 3z^3)
(b) fz = 3z^2 / (x + 2y^2 + 3z^3)
(c) d^2f/dzdx = -3z^2 / (x + 2y^2 + 3z^3)^2
To find the partial derivatives of the function f(x, y, z) = ln(x + 2y^2 + 3z^3), we differentiate with respect to each variable while treating the other variables as constants.
(a) Partial derivative with respect to x (fx):
To find fx, we differentiate the function f(x, y, z) with respect to x while treating y and z as constants. The derivative of ln(u) with respect to u is 1/u, so we have:
fx = d/dx ln(x + 2y^2 + 3z^3) = 1 / (x + 2y^2 + 3z^3)
(b) Partial derivative with respect to z (fz):
To find fz, we differentiate the function f(x, y, z) with respect to z while treating x and y as constants. Again, applying the derivative of ln(u), we get:
fz = d/dz ln(x + 2y^2 + 3z^3) = 3z^2 / (x + 2y^2 + 3z^3)
(c) Second partial derivative with respect to z and x (d^2f/dzdx):
To find d^2f/dzdx, we differentiate fz with respect to x while treating y and z as constants. We differentiate fx with respect to z while treating x and y as constants, and then take the derivative of the result with respect to z. It can be written as:
d^2f/dzdx = d/dx (d/dz ln(x + 2y^2 + 3z^3)) = d/dx (3z^2 / (x + 2y^2 + 3z^3))
= -3z^2 / (x + 2y^2 + 3z^3)^2
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Proving a parallelogram side theorem.
Given ABCD is a parallelogram.
Prove: AB≈CD and BC ≈ DA
Answer:
just did it on edg, 2021
Hence, AB≈CD and BC ≈ DA
What is a parallelogram?
A four sided closed figure with all its sides parallel to its opposite side.
Consider two triangles ΔABD and ΔBCD
∠BCD=∠BCA( opposite angles are always equal)
∠ABD=∠BDC (AD||BC)
∠ADB=∠DBC(AD||BC)
ΔABD ≅ ΔBCD
AB≈CD by CPCTC
BC ≈ DA by CPCTC
Hence, proved
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Select all the correct answers. 6^3×2^6 Which expressions are equal to 2^3 ? 2^6×3^3; 6^3; 12^3; 2^3×3^3; 12^6
Answer:
12^3
Step-by-step explanation:
Answer:
12 ^3
Step-by-step explanation: