Answer:
[tex]\displaystyle \frac{x}{\sin x} = \sum^{\infty}_{n = 0} \frac{(2n + 1)!}{(-1)^nx^{2n}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationSequences
Series
Taylor Polynomials
MacLaurin Polynomial: [tex]\displaystyle P_n(x) = \frac{f(0)}{0!} + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots + \frac{f^n(0)}{n!}x^n + \cdots[/tex]Power Series
Power Series of Elementary FunctionsMacLaurin SeriesTaylor SeriesExplanation:
*Note:
You could derive the Taylor Series for sin(x) using Taylor polynomials by differentiation by usually you have it memorized.
We want to obtain an infinite amount of derivatives of the given:
[tex]\displaystyle \frac{x}{\sin x}[/tex]
We know that a power series is defined as an infinite derivatives of a function. We know that the power series for sin(x) is:
[tex]\displaystyle \sin x = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!}[/tex]
Divide x by the power series to obtain our derivatives:
[tex]\displaystyle \frac{x}{\sin x} = \sum^{\infty}_{n = 0} \frac{x}{\frac{(-1)^nx^{2n + 1}}{(2n + 1)!}}[/tex]
Simplify this down:
[tex]\displaystyle \frac{x}{\sin x} = \sum^{\infty}_{n = 0} \frac{(2n + 1)!}{(-1)^nx^{2n}}[/tex]
And we have our final answer.
Topic: AP Calculus BC (Calculus I + II)
Unit: Power Series
In his experiments on "cathode rays" during which he discovered the electron, J. J. Thomson showed that the same beam deflections resulted with tubes having cathodes made of different materials and containing various gases before evacuation.(d) Could Thomson observe any deflection of the beam due to gravitation? Do a calculation to argue for your answer. Note: To obtain a visibly glowing patch on the fluorescent screen, the potential difference between the slits and the cathode must be 100V or more.
Thomson could not observe any deflection of the beam due to gravitation.
What is a cathode ray?Cathode ray refers to the beam of electrons that moves from the negatively charged cathode to the positive anode at the other end of the pipe.
J. J. Thomson demonstrated that the identical beam deflections produced tubes with cathodes composed of varied materials and containing a range of gases prior to evacuation.
The gravitational force is approximately 10⁴⁰.
The movement of the light beam shouldn't be impacted because it is 10²⁸ times weaker than the electric force.
Thomson was unable to detect any gravitational beam deflection.
Thus, Thomson could not observe any deflection of the beam due to gravitation.
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How can you drop two eggs the fewest amount of times, without them breaking?
Answer:
Both eggs are identical. The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that's it for that egg.
Which astronomer measured the motion of stars in the sky?
A. William Herschel
B.Harlow Shapley
C.Vesto Slipher
D. Edwin Hubble
Edwin Hubble was the astronomer which measured the motion of stars in the sky and is therefore denoted as option D.
What is Astronomy?This refers to an aspect of a science called natural science and is the study of celestial bodies such as the moon, stars etc and the different phenomena which occurs between them.
An astronomer called Edwin Hubble was responsible for the measurement of the motion of stars through when he was in the process of the formulation of his law which involves the determination of distances to galaxies and quasars which is therefore the reason why option D was chosen as the correct choice.
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Consider (a) an electron (b) a photon, and (c) a proton, all moving in vacuum. Choose all correct answers for each question. (iii) Which carry energy?
Electron , photon and proton carry energy which is option (a) , (b) ,(c)
An electron is a solid negatively charged factor of an atom. Electrons exist outdoor of and surrounding the atom nucleus. Each electron includes one unit of bad charge (1.602 x 10-19 coulomb)A proton is a subatomic particle discovered withinside the nucleus of each atom. The particle has a advantageous electric charge, identical and contrary to that of the electron Photon is a subatomic particle, having strength and momentum however no mass or electric powered charge, this is the quantum unit of electromagnetic radiation, together with light.Photons are massless, but they have energy [tex]E = hv= \frac{hc}{\lambda}[/tex]
Here 'h' is called Planck's constant .
For electron and proton energy is given by
[tex]E=\frac{1}{2} \times mv^2[/tex]
Where m is the mass of the particle and v is the speed of particle .
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Under which condition is the resultanat vector equal to zero?
A linearly polarized microwave of wavelength 1.50cm is directed along the positive x axis. The electric field vector has a maximum value of 175V/m and vibrates in the x y plane. Assuming the magnetic field component of the wave can be written in the form B=Bmax sin (k x-Ω t) give values for (f) If this wave were directed at normal incidence onto a perfectly reflecting sheet, what radiation pressure would it exert?
The radiation pressure directed at normal incidence onto a perfectly reflecting sheet is 2.71 × 10⁻⁷ Pa
What radiation pressure would it exert?Given that a linearly polarized microwave of wavelength 1.50cm is directed along the positive x axis and the electric field vector has a maximum value of 175V/m and vibrates in the x y plane.
Also, assuming the magnetic field component of the wave can be written in the form B=Bmax sin (k x-Ω t) give values for (f) If this wave were directed at normal incidence onto a perfectly reflecting sheet, we desire to find the radiation pressure.
The radiation pressure for total reflection is given by P = 2I/c where
I = intensity = E²/2μ₀c where E = maximum electric field = 175 V/m, μ₀ = permeability of free space = 4π × 10⁻⁷ T/m and c = speed of light = 3 × 10⁸ m/sSo, P = 2I/c
P = 2E²/2μ₀c ÷ c
P = E²/μ₀c²
Substituting the values of the variables intot he equation, we have
P = E²/μ₀c²
P = (175 V/m)²/[4π × 10⁻⁷ T/m × (3 × 10⁸ m/s)²]
P = 30625 V²/m²/[4π × 10⁻⁷ T/m × 9 × 10¹⁶ m²/s²]
P = 30625 V²/m²/[36π × 10⁹ Tm/s²]
P = 30625 V²/m²/113.097 × 10⁹ Tm/s²
P = 270.78 × 10⁻⁹ Pa
P = 2.7078 × 10⁻⁷ Pa
P ≅ 2.71 × 10⁻⁷ Pa
So, the radiation pressure directed at normal incidence onto a perfectly reflecting sheet is 2.71 × 10⁻⁷ Pa
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which names are correct for fm−→−? select each correct answer. fp−→− ray f p fd−→− ray f d mf−→− ray m f fw−→−
The correct answer is FM-MF
The crust of the Earth does not naturally contain ferromium. The heaviest element that can be created by neutron bombardment of lighter elements, fermium was first discovered in December 1952 by American scientists from the Argonne National Laboratory near Chicago, Illinois, the Los Alamos National Laboratory in Los Alamos, New Mexico, and The University of California Laboratory in Berkeley. Fermium is the heaviest element that can be synthesized in macroscopic quantities.
We can prove this by saying that if a point's distance from another point is six units, then the distance between those two points will also be six units. Therefore, FM-MF
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will anyone give me a free amazon card
Suppose we have a 600 kilogram great "yellow" shark swimming to the right at a speed of 3 meters traveled each second as it tries to get lunch. An unsuspecting 100 kilogram blue fin tuna is minding its own business swimming to the left at a speed of 0.5 meters traveled each second. GULP! After the great "yellow" shark "collides" with the blue fin tuna, what is the velocity of the well fed shark?
Answer:
2.64 m/s
Explanation:
Given that a 600 kilogram great "yellow" shark swimming to the right at a speed of 3 meters traveled each second as it tries to get lunch. An unsuspecting 100 kilogram blue fin tuna is minding its own business swimming to the left at a speed of 0.5 meters traveled each second. GULP! After the great "yellow" shark "collides" with the blue fin tuna
Momentum = MV
Momentum of the yellow shark before collision = 600 × 3 = 1800 kgm/s
Momentum of the tun final before collision = 100 × 0.5 = 50 kgm/s
Total momentum before collision = 1800 + 50 = 1850 kgm/s
Let's assume that they move together after collision. Then,
1850 = ( 600 + 100 ) V
1850 = 700V
V = 1850 / 700
V = 2.64285 m/s
Therefore, the momentum of the shark after collision is 2.64 m/ s approximately
According to the scientific method which do you have to do before you can form a hypothesis?
According to the scientific method, you have to make observations and ask questions before you can form a hypothesis.
What is the scientific method?The scientific method is a series of steps or procedures which scientists follow in their process of discovery and explanation of the natural world or natural phenomena.
The scientific method consists of the following steps:
observation/ asking of questionsformulation of hypothesisdesigning of experimentstesting the hypothesisanalyzing the data from the experimentconclusionsfurther experimentstheory/lawIn conclusion, observations and questions are important before formulation of hypothesis.
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would a larger, more heavier ball fall faster than a smaller and lighter ball?
Answer:
the larger-heavier would fall faster than lighter-smaller ball
Explanation:
please help me swear will mark brainiest
Answer:
I think that is d option 1.0Nm anticlockwise
what yall notice about this please help me out
Answer:
There is a blue sky with clouds
There is cracks in the ground and the ground is brown
There is stuff in the dictance
Explanation:
Hope this helps
If the total mechanical energy of the system is 100 J: Where does the pendulum have 100 J of kinetic energy? Where does the pendulum have 100 J of potential energy? (Label both points.)Where does the pendulum have 50 J of potential energy? (Label both
Answer:
See image and explanation
Explanation:
A simple pendulum consists of a string to which a bulb is attached and the string is tied to a support. The string swings from one end to the other as shown.
At the both ends, the energy of the pendulum is all potential. Hence we have 100J potential energy at both ends.
When the pendulum is in motion, the energy is all kinetic at the center of motion. Hence at that point, we have 100J kinetic energy.
In between both extremes the energy is both kinetic and potential as shown. The total energy is this a summation of the two.
Answer:
A simple pendulum consists of a string to which a bulb is attached and the string is tied to a support. The string swings from one end to the other as shown.
At the both ends, the energy of the pendulum is all potential. When the pendulum is in motion, the energy is all kinetic at the center of motion.
In between both extremes the energy is both kinetic and potential as shown. The total energy is this a summation of the two.
The reason that objects tend to rest in our daily lives is because there are
unbalanced forces acting on them.
true or false?
Answer:
True
Explanation:
An object at rest will stay at rest unless acted on by an unbalanced force.
Explain how inductive reasoning is inferior to deductive reasoning in establishing theory.
Answer: Created Fall 2016 by Ronald Wilson Inductive reasoning uses a set of specific observations to reach an overarching conclusion; it is the opposite of deductive reasoning. So, a few particular premises create a pattern which gives way to a broad idea that is likely true.
Explanation: No explanation lol. But I hope this is helpful!
write down the kirchhoff's laws in the adjoining circuit , find the current in each resistors
Answer:
You need a motor and a battery
Explanation:
The currents through 4 ohm, 2 ohm and 5 ohm are 0.42 amp, 0.15 amp and 0.26 amp respectively.
What are the Kirchhoff's laws ?
Kirchhoff's Current Law states that no charge is lost and that the total current flowing into a junction or node equals the charge leaving it.
Kirchhoff's Voltage Law states that for any closed network, the voltage around a loop is equal to the total of all voltage drops in that loop and is equal to zero.
Let the current through 4 ohm is I₁.
the current through 2 ohm is I₂.
Applying Kirchhoff's Current Law, current through 5 ohm is I = I₁ + I₂.
Applying Kirchhoff's Voltage Law in two loops we get:
4 I₁ + 5(I₁+I₂) = 3
2I₂ + 5(I₁+I₂) = 1
Hence, I₁ = 0.42 amp
I₂ = - 0.15 amp
I₃ = 0.26 amp
Hence, the currents through 4 ohm, 2 ohm and 5 ohm are 0.42 amp, 0.15 amp and 0.26 amp respectively.
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Mathew throws a ball straight up in the air. it rises for a period of time and then begins to drop. At which points in the balls journey will be gravity be the greatest force acting on the ball
Answer: The greatest force of gravity on the ball will occur at the point when the ball is near to hit the ground ... hope this helps have a good day :P
which of the following is not an electromagnet device
Answer:
[tex]\textbf {Engine}[/tex]
Explanation:
Electric motor in cars is an example of electromagnetic device.
If two people lift identical stacks of books the same distance and one person does the job twice as fast, which of the following has doubled?
-power
-work output
-work input
-efficiency
Answer:
Power
Explanation:
Power is defined as the rate of doing work with reference to the time spent and the formula is force multiplied by velocity.
In this case, if two people lift identical stacks of books the same distance and one person does the job twice as fast, then it means the velocity in the case is doubled which will also lead to an increase in the Power .
PowerExplanation:
The formula for power is Work divided by time taken, so P=W/T.
That case, involved power, since one of the people did work twice as fast, but, it was the same amount of work each person did, for the same distance (not exact distance, but usually the same)
Is Ampère's law valid for all closed paths surrounding a conductor? Why is it not useful for calculating →B for all such paths?
Although not always practical, Ampere's law is true for all closed channels around a conductor. The integral can be calculated along many different paths, although it is difficult to do so. Think about a path that is circular but not coaxial with a long, straight wire that carries current. If the current in a conductor has enough symmetry to allow the line integral to be reduced to the size of B times an integral, Ampere's law can be used to calculate B.
What applications does Ampere's Law have?Ampere's Law is applied to,
Find the magnetic induction caused by a long wire carrying current.Analyze the magnetic field a toroid has.Find the magnetic field produced by a long conducting cylinder that carries current.Ascertain the conductor's magnetic field.To learn more about Ampere's Law, visit:
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Sheila uses a 45N force on her bowling ball across a 15m lane. What work did she do on the bowling ball? Show your work.
Answer:
675J
Explanation:
Given parameters:
Force = 45N
Distance = 15m
Unknown:
Work done by Sheila = ?
Solution:
Work done by a body is the amount of force applied to make a body move through a distance;
Work done = Force x distance
Now;
Work done = 45 x 15 = 675J
Jennifer wears a black shirt on a sunny summer day. After spending one hour outdoors, Jennifer notices that the shirt is much warmer than it was originally. What role did sunlight play in warming the shirt
Explanation:
Black or Dark color clothes are good absorbers of heat. The heat from the sun is absorbed by the black shirt, thus making the shirt warmer. The sunlight here is source of heat. The amount of heat absorbed can be calculated according to the stephan's law.
PLEASE HELP ME
What is the net force acting on the car?
the Heat required to raise the temp. of 20 g water from 25 C to 36 C
Please help me I need it
Answer:
Parallel circuits :)
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Answer:
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Explanation:
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Answer:
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Explanation:
Show FORMULA AND SOLUTION:
What acceleration will result when a 12N net force applied to a 3kg object? A 6kg object?
Answer:
2m/s²
Explanation:
F = 12 N
m = 3kg
a = ?
F = ma
a = F/m = 12/3 = 4 m/s²
when m = 6kg
a = F/m = 12/6
a = 2 m/s²
the armature windings of a dc motor have a resistance of 6.5 ω . the motor is connected to a 120-v line, and when the motor reaches full speed against its normal load, the back emf is 108 v. the load is increased so it causes the motor to run at half speed.
The current in the load will be 10 A.
given,
resistance of the load (R) = 6.5Ω
voltage in the load (V) = 120V
Back emf in the load (E)= 108V
now to measure current we will use the formula
current (I)= {voltage (V)-Back emf (E)}/resistance(R)
now as the speed is halved so will the back electro motive force will be halved.
i.e.
back electro motive force will be = 108V/2
=54V
now,
substituting this value in the formula we get,
current (I) = (120-54)/6.5
current (I) = 66/6.5
current (I) =10A
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An over-caffeinated student stands on a table which weighs 600 newtons. The student has a mass of 50 kg. What is the weight of the student? What is the weight of the student AND the table combined? Now, what is the TOTAL normal force on the table by the floor?
Answer:50kg
Explanation: