smilesandsunrays:

brawnbrainybombshell:

I don’t need any more caffeine but I need more caffeine.

Was planning on not drinking coffee for two weeks, gave up this morning on day 10.



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}]

Q
Why not fall in love?
Anonymous
A

brianashanee:

I got shit to do


crossfitters:

Libby DiBiase

crossfitters:

Libby DiBiase

(via my-trainer-tiffany)



I am the girl who prefers to spend her Friday night curled up with her pillow, reading a good novel, and I am also the girl who likes to go out on a Saturday night and dance until the DJ plays his last song. I am the girl who wants to wear beat up converses and an oversized sweatshirt, and I am also the girl who who owns over sixty dresses and too many shoes to count. Why did it become okay to say one is better than the other? Because I am all of that.
Ming D. Liu, What is “better?” (via dancebeforethelord)

(via my-trainer-tiffany)