fouriestseries:

Taylor Series Approximations
A Taylor series is a way to represent a function in terms of polynomials. Since polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.
There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].
The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.
Mathematica code:
f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]

fouriestseries:

Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomialsSince polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.

There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].

The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.

Mathematica code:

f[x_] := Sin[x]
ts[x_, a_, nmax_] := 
    Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, 
    PlotRange -> {-1.45, 1.45}, 
    PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}}, 
    AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]], 
    {nmax, 1, 30, 1}]

(via likeaphysicist)

molecularlifesciences:

popmech:

10 Scientific Images That Changed How We Look at the World

Any conversation about progress in science will include visualization.

(via likeaphysicist)

space-wallpapers:

The Helix Nebula a.k.a. Eye of God

space-wallpapers:

The Helix Nebula a.k.a. Eye of God

(via likeaphysicist)

isaacgoodhart:

Guardians is finally out!! Her’e a fan art I made after the first trailer came out and melted my mind. 

isaacgoodhart:

Guardians is finally out!! Her’e a fan art I made after the first trailer came out and melted my mind. 

(via bestnatesmithever)

canadian-space-agency:

ESA Astronaut Alexander Gerst aboard the ISS: “At sunset, our solar panels look like they are made from pure gold.”
Credit: Alexander Gerst/ESA

canadian-space-agency:

ESA Astronaut Alexander Gerst aboard the ISS: “At sunset, our solar panels look like they are made from pure gold.”

Credit: Alexander Gerst/ESA

(via bewbin)

lizclimo:

happy first birthday, marlow. i love you more than i can count on all of my fingers and toes (and yours, too)

lizclimo:

happy first birthday, marlow. i love you more than i can count on all of my fingers and toes (and yours, too)

(via bewbin)

thatscienceguy:

The glass tank is filled with a gas called Sulfur Hexafluoride which is a one of the more dense gases, and it also happens to be as transparent as air so it gives the illusion of something floating in mid air, when it is really just acting like a boat in water.

thatscienceguy:

The glass tank is filled with a gas called Sulfur Hexafluoride which is a one of the more dense gases, and it also happens to be as transparent as air so it gives the illusion of something floating in mid air, when it is really just acting like a boat in water.

(via thatscienceguy)

carmenbass:

 

To find the universal elements enough; to find the air and the water exhilarating; to be refreshed by a morning walk or an evening saunter… to be thrilled by the stars at night; to be elated over a bird’s nest or a wildflower in spring - these are some of the rewards of the simple life.

carmenbass:

 

To find the universal elements enough; to find the air and the water exhilarating; to be refreshed by a morning walk or an evening saunter… to be thrilled by the stars at night; to be elated over a bird’s nest or a wildflower in spring - these are some of the rewards of the simple life.

(via nihilsupernum)

enjolradz:

friendly reminder that if we’re mutuals and you wanna exchange snapchat names or instagram or something you’re more than welcome (encouraged) to shoot me an ask

(via haveanadventure)

generalelectric:

Detailed views of the blades on GE gas turbines at our Greenville, SC, facility. Footage captured by @seenewphoto