The solution would be like this for this specific problem:
f(x) = x^2 + sin(x)
f '(x) = 2x + cos(x)
The minimum value is at f '(x) = 0,
So, let g(x) = 2x + cos(x)
Thus, g '(x) = 2 - sin(x)
x(new) = x - g(x) / g '(x)
or
x(new) = x - [2x + cos(x)] / [2 - sin(x)]
Calculation
x1 = -0.5 - [2 * -0.5 + cos(-0.5)] / [2 - sin(-0.5)]
= -0.4506266931
x2 = -0.4501836476
x3 = -0.4501836113
x4 = -0.4501836113
This value for x, f(x) = -0.2324655752.
After converting to 6 decimal places: the minimum point is (-0.450184,
-0.232466).
Final answer:
To find the absolute minimum value of f(x)=x^2+sin(x), use Newton's method with Newton-Raphson iteration x_{n+1} = x_n - f(x_n)/f'(x_n). Start with an initial guess and iterate until the result converges to six decimal places, ensuring the second derivative at the critical point is positive.
Explanation:
To find the absolute minimum value of the function f(x)=x^2+sin(x) correct to six decimal places, we can use Newton's method. Newton's method helps to find successively better approximations to the roots (or zeroes) of a real-valued function. First, we need to find the derivative of the function, which gives us f'(x) = 2x + cos(x). We are looking for a critical point where the derivative is zero because this could indicate a potential minimum (or maximum).
Starting with an initial guess, we can apply the Newton iteration formula x_{n+1} = x_n - f(x_n)/f'(x_n) to find a better approximation. In this case, let's choose an initial guess close to the root of f'(x). Since we do not have the specific initial guess, we would theoretically pick a value near the expected minimum and iterate until the difference between consecutive approximations is less than the desired tolerance, which is the change in six decimal places.
The Newton-Raphson iteration would be applied repeatedly until convergence is seen at six decimal places. Note that in practice, one must also check the second derivative f''(x) at the found critical point to confirm it is a minimum (it should be positive).
Prove abcd is a parallelogram
Since both pairs of opposite sides of quadrilateral ABCD are parallel and equal, by definition, ABCD is a parallelogram.
Given that in quadrilateral ABCD, side AD is equal to side BC (AD = BC) and side AD is parallel to side BC (AD || BC), we can use these properties to prove that ABCD is a parallelogram.
Here is a step-by-step proof:
1. Definition of a parallelogram: A quadrilateral is a parallelogram if both pairs of opposite sides are parallel.
2. Given: AD = BC (opposite sides are equal) and AD || BC (opposite sides are parallel).
3. To Prove: AB || CD and AB = CD (the other pair of opposite sides are also parallel and equal).
4. Proof:
- Since AD = BC and AD || BC, we have one pair of opposite sides that are equal and parallel.
- In a quadrilateral, if one pair of opposite sides is equal and parallel, the quadrilateral is a parallelogram (This is a theorem in Euclidean geometry).
- Therefore, AB must be parallel to CD (AB || CD) because the opposite sides of a parallelogram are parallel by definition.
- Also, in a parallelogram, opposite sides are equal (another property of parallelograms), so we can conclude that AB = CD.
5. **Conclusion**: Since both pairs of opposite sides of quadrilateral ABCD are parallel and equal, by definition, ABCD is a parallelogram.
This logical sequence uses the properties of parallelograms and the given information to prove that ABCD must be a parallelogram.
Identify the reflection of the figure with vertices
True or False
The geometric mean of 8 and 2 is 4.
TRUE IVE GOT IT WRONG ON MY ASSIGNMENT.
Answer:
True
Step-by-step explanation:
To find the geometric men you have to cross multiply the two given numbers and then find x squared for those numbers. So to find x squared you have to find the square root of the product from the two multiplied numbers.
8/x = x/2
8*2 = 16 = x squared
x= square root of 16 which equals 4
You can solve equations by graphing. Explain how to find the solutions by a given graph.
geometry help! this is my 2nd to last question
For the graph y= 1 find the slope of a line that is perpendicular to it and the slope of a line parallel to it. Explain your answer with two or more sentences
Answer:
Parallel: [tex]m = 0[/tex] (Horizontal), Perpendicular: [tex]m = -\frac{1}{0}[/tex] (Vertical)
Step-by-step explanation:
The equation y = 1 is an horizontal straight line ([tex]y = 0\cdot x + 1[/tex]). The slope of a line parallel to it is [tex]m = 0[/tex]. The slope of line perpendicular to it is [tex]m = -\frac{1}{0}[/tex], since slope is represented by tangent function, which is undefined at [tex]\theta = \pm 0.5\pi[/tex].
To which graph does the point (2, 4) belong?
y ≥ x + 3
y ≥ −x + 8
y ≥ 4x − 5
y ≥ −2x + 9
Answer:
Option (3) is correct.
(2,4) belongs to y ≥ 4x - 5
Step-by-step explanation:
Given : the point (2,4)
We have to find the equation of graph to which the point (2,4) belongs.
We will substitute the point in the each given equation and for which the point satisfies will contain the point.
For 1) y ≥ x + 3
Put x = 2 and y = 4
⇒ 4 ≥ 2 + 3
⇒ 4 ≥ 5 (false)
For 2) y ≥ -x + 8
Put x = 2 and y = 4
⇒ 4 ≥ -2 + 8
⇒ 4 ≥ 6 (false)
For 3) y ≥ 4x - 5
Put x = 2 and y = 4
⇒ 4 ≥ 4(2) - 5 = 8 - 5
⇒ 4 ≥ 3 (true)
For 4) y ≥ -2x + 9
Put x = 2 and y = 4
⇒ 4 ≥ -4 + 9
⇒ 4 ≥ 5 (false)
Since, the point (2,4) satisfies only inequality y ≥ 4x - 5.
Thus, (2,4) belongs to y ≥ 4x - 5
one number is 10 times another number. their difference is 72. what are the two numbers?
The two numbers are 80 and 8.
What is linear equation in two variables?An equation is said to be linear equation in two variables if it is written in the form of ax + by + c = 0, where a, b & c are real numbers and the coefficients of x and y, i.e. a and b respectively, are not equal to zero.
What is substitution method ?The substitution method can be defined as a way to solve a linear system algebraically. This method works by substituting one y-value with the other.
Let the two numbers be x and y.
According to the question.
x = 10y
And,
x - 10y = 72
Now, We have a linear equation in two variables. So, for finding the values of x and y we solve the above two equations by substitution method.
Substitute the value of x = 10y in x - y = 72
[tex]\implies 10y -y = 72 \\\implies 9y = 72\\\implies y = \frac{72}{9} \\\implies y = 8[/tex]
Therefore,
x = 10 × 8 = 80
Hence, the two numbers are 80 and 8.
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A collection of quarters and nickels contains at least 42 coins and is worth at most $8.00. If the collection contains 25 quarters, how many nickels can be in the collection?
The total number of nickels that can be collected is 17 and this can be determined by forming the inequality.
Given :
A collection of quarters and nickels contains at least 42 coins and is worth at most $8.00.
The inequality can be formed in order to determine the total number of nickels that can be collected.
Let the total number of quarters be 'x' and the total number of nickels be 'y'. Then the inequality that represents the total number of quarters and nickels contained in the collection is at least 42 is given by:
x + y [tex]\geq[/tex] 42 --- (1)
The inequality that represents that the worth of the coins is at most $8 is given by:
0.25x + 0.5y [tex]\leq[/tex] 8 --- (2)
Now, according to the given data there are a total of 25 quarters in the collection, so, the number of nickels contained in the collection is:
x + y [tex]\geq[/tex] 42
25 + y [tex]\geq[/tex] 42
y [tex]\geq[/tex] 17
So, the total number of nickels that can be collected is 17.
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The collection can contain up to 35 nickels, given that it already includes 25 quarters and the total value does not exceed $8.00.
Explanation:To determine how many nickels can be in the collection, we must first calculate the total value of the 25 quarters. Since each quarter is worth 25 cents, we multiply 25 quarters by 25 cents to get 625 cents, which is $6.25. The maximum value of the entire collection is $8.00. Thus, we subtract $6.25 from $8.00 to find the remaining value that can be occupied by nickels, which is $1.75. Each nickel is worth 5 cents, so we divide $1.75 by 0.05 (5 cents) to find the maximum number of nickels. $1.75 divided by 0.05 equals 35. Therefore, the collection can contain up to 35 nickels.
The graph below shows the value of Edna's profits f(t), in dollars, after t months:
graph of quadratic function f of t having x intercepts at 6, 0 and 18, 0, vertex at 12, negative 36, and passes through point 21, 41.25
What is the closest approximate average rate of change for Edna's profits from the 18th month to the 21st month?
a. Three dollars per month
b. Nine dollars per month
c. 13.75 dollars per month
d. 41.25 dollars per month
Answer:
C) 13.75 dollars per month
Step-by-step explanation:
i took the test and got it right
F r(x) = 3x – 1 and s(x) = 2x + 1, which expression is equivalent to ?
what is the circumference of a circle with a radius of 39
Solve the linear equation:
3.4 + 2(9.7 – 4.8x) = 61.2
The mean is defined as the
10(2y+2)−y=2(8y−8)
y= ???
An extension ladder leans agintst a biulding, making a 75 degree angle of elevation with the ground. The base of the ladder is 8 ft from the base of the building.
To the nearest tenth of a foot, how long is the ladder?
The ladder is 29.9 feet long.
For solving this problem we have to use trigonometric ratio
We need to use the tangent function:
What is the ratio of tangent function?
[tex]tan (\theta) = opposite/adjacent[/tex]
[tex]tan 75 = opposite/8[/tex]
multiply both side by 8 so we will get,
Here, the opposite side is the length of the ladder and angle [tex]\theta =75^0[/tex]
Therefore by using the tan ratio we have,
[tex]8 * tan 75 = opposite[/tex]
[tex]29.8564065 = opposite[/tex]
Therefore, the ladder is 29.9 feet.
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32, 40, 34, 31, 20, 36, 40, 31. How does the outlier affect the mean? If necessary, round to the nearest tenth.
Please help solve -2|5y-1|=-10
Four times the difference of a number and seven is 12
4(x-7)=12=
4x-28=12
add 28 to both sides
4x=40
x =40/4 = 10
x = 10
In the triangle below, what ratio represents cot θ?
(6u3 + 7u2 + 2) + (3u3 – 8u + 4)
All of the following are equivalent except _____.
x - (-2)
-2 + x
x - 2
x + (-2)
A.
Its correct i got 100% on a test
Simplify this problem please
Which one of the numbers in the series is wrong, and should be replaced? 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46?
The number series 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46 has a pattern whereby each number doubles the previous number. However, 20 does not follow this pattern, and the correct number should be 32 (being double of 16). Therefore, 20 is the incorrect number in this series.
Explanation:This question is a query regarding a number series, specifically asking for us to identify any number that does not fit the pattern of the series. The series given is: 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46. By examining the series closely, we can observe a clear pattern: each number is doubling the previous number. So, after 2 we have 4 (2x2), then 8 (4x2), then 16 (8x2) and so forth.
However, at the fifth place, we have the number 20, which doesn't follow this pattern. If we were to follow the established pattern, the fifth number should be 32 (16x2), not 20. Therefore, the incorrect number in this series is 20, and the correct series should be 2 - 4 - 8 - 16 - 32 - 22 - 44 - 46.
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The incorrect number in the series is 20, and it should be replaced with 32. The accurate sequence is 2, 4, 8, 16, 32, 64, 128, 256.
Let's examine the sequence: 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46.
At a glance, it appears each number is double its preceding number, except for a couple of deviations.
If we look at the pattern,
2 x 2 = 44 x 2 = 88 x 2 = 1616 x 2 = 32Here, we expect 32 instead of 20.
Therefore, 20 is the incorrect number in the series. When we continue correctly:
The correct sequence would be,
2 - 4 - 8 - 16 - 32 - 64 - 128 - 256.
So, the number that should be replaced is 20, and it should be 32.
Which expression is equivalent to complex fraction 3/x-1-4/2-2/x-1
NEED HELP PLEASE!!!!
The diameter of a hydrogen atom is about 5×10^-15 meter. Suppose 8.4×10^8 hydrogen atoms were arranged side by side in a straight line. Multiply these numbers to find the length of this line of atoms. What is the length in scientific notation?
Select one:
a. 4.2×10^−2 meter
b. 0.042 meter
c. 42×10^−3 meter
d. 4.2×10^−3meter
Answer:
I'm a few years late, lol, but um the answer is 42 x [tex]10^{-3}[/tex] meters
For all the people who still need the answer.
Step-by-step explanation:
5 x [tex]10^{-11}[/tex] and 8.4 x [tex]10^{8}[/tex]
5 x 8.4 = 42 (multiply like terms)
[tex]10^{-11}[/tex] + [tex]10^{8}[/tex] = [tex]10^{-3}[/tex] (add the exponents)
42 x [tex]10^{-3}[/tex]
The length in scientific notation is [tex]42\times10^{-7[/tex] meter.
What is scientific notation?Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power such as [tex]1.56\times10^7[/tex].
Given that, the diameter of a hydrogen atom is about [tex]5\times10^{-15}[/tex] meter. Suppose 8.4×10⁸ hydrogen atoms were arranged side by side in a straight line.
Now, multiply the numbers
[tex]5\times10^{-15}[/tex]×8.4×10⁸
= [tex]42\times10^{-7[/tex] meter
Therefore, the length in scientific notation is [tex]42\times10^{-7[/tex] meter.
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Mx = Nx - Pt isolate x
The standard formula for the volume of a cylinder is V = πr2h. If the cylinder is scaled proportionally by a factor of k, its volume becomes V' = V × k3. Use your algebra skills to derive the steps that lead from V = πr2h to V' = V × k3 for a scaled cylinder. Show your work.
Answer:
The answer in the procedure
Step-by-step explanation:
we know that
If two figures are similar, then the ratio of its corresponding sides is equal to the scale factor
so
[tex]k=r'/r[/tex] and [tex]k=h'/h[/tex]
The volume of the original cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
If the cylinder is scaled proportionally by a factor of k
then
the new radius is ------> [tex]r'=kr[/tex]
the new height is ------> [tex]h'=kh[/tex]
The volume of the scaled cylinder is equal to
[tex]V'=\pi r'^{2}h'[/tex]
substitute the values
[tex]V'=\pi (kr)^{2}(kh)[/tex]
[tex]V'=(k^{3})\pi r^{2}h[/tex]
Remember that
[tex]V=\pi r^{2}h[/tex]
so
substitute
[tex]V'=V(k^{3})[/tex]
The volume of the scaled cylinder is equal to the scale factor elevated to the cube multiplied by the volume of the original cylinder
Find the area of a sector with a central angle of 170° and a diameter of 9.1 cm. Round to the nearest tenth.
A. 122.9 cm2
B. 30.7 cm2
C. 8.6 cm2
D. 3.4 cm2
To find the area of a sector given a central angle and diameter, use the formula A = ([tex]\frac{\theta}{360}[/tex]) x π x r². In this case, with a central angle of 170° and a diameter of 9.1 cm, the area of the sector is approximately 30.7 cm².
The area of a sector with a central angle and a diameter can be calculated using the formula for the area of a sector:
A = [tex]\frac{\theta}{360}[/tex] x π x r². First, convert the diameter to radius (r = d/2), then substitute the values into the formula to find the area. In this case:
Central angle (θ) = 170°
Diameter (d) = 9.1 cm
Radius (r) = 4.55 cm
Area (A) = ([tex]\frac{170}{360}[/tex]) x π x (4.55)² ≈ 30.7 cm²
Therefore, the area of the sector is approximately 30.7 cm².
What shape will stay the same no matter how many degrees it is rotated?