audio audioduration (s) 0.83 21 | transcription stringlengths 3 127 | LaTeX stringlengths 7 142 | Source stringclasses 89
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the limit of 1 minus cosine x divided by x x goes to 0 is 0 | $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ | https://www.youtube.com/watch?v=ryLdyDrBfvI | |
limit as x goes to x0 of f of x is equal to f of x0 | $\lim_{x\to x_{0}}f(x)=f(x_{0})$ | https://www.youtube.com/watch?v=ryLdyDrBfvI | |
sine a cos b plus cos a sin b | $\sin(a)\cos(b)+\cos(a)\sin(b)$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
cosine delta x minus 1 divided by delta x | $\cos\Delta x - 1 \over \Delta x$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
minus cosine x divided by delta x | $-\frac{\cos x}{\Delta x}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
sine delta x divided by delta x | $\frac{\sin\Delta x}{\Delta x}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
cosine delta x minus 1 divided by delta x | $\frac{\cos\Delta x - 1}{\Delta x}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
the limit as delta x goes to 0 of sine delta x over delta x | $\lim_{\Delta x\to 0} \frac{\sin \Delta x}{\Delta x}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
delta y over delta theta | $\frac{\Delta y}{\Delta \theta}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
u prime v minus u v prime | $u^{\prime}v-uv^{\prime}$ | https://www.youtube.com/watch?v=kCPVBl953eY | |
n x to the n minus 1 | $nx^{n-1}$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
x to the n times sine of x | $x^{n}\sin(x)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
u times v of x plus delta x | $uv(x + \Delta x)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
delta of uv divided by delta x | $\frac{\Delta (uv)}{\Delta x}$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
delta u divided by delta x times v of x plus delta x | $\frac{\Delta u}{\Delta x}\cdot v(x+\Delta x)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
u of x plus delta x times v of x plus delta x | $u(x+\Delta x)v(x+\Delta x)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
u prime v minus uv prime divided by v squared | $u^{\prime}v - uv^{\prime} \over v^{2}$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
v plus delta v | $v+\Delta v$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
v to the minus 2 v prime minus v prime divided by v squared | $v^{-2}v^{\prime} - \frac{v^{\prime}}{v^2}$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
10 times sine of t raised to the 9th power | $10 \sin^{9}(t)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
sine to the 9th of t cosine of t | $\sin^{9}(t)\cos(t)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
D cubed x to the n | $D^{3}x^{n}$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
n times n minus 1 | $n\times (n-1)$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
D to the n plus 1 applied to x to the n | $\mathcal{D}^{n+1}(x^{n})$ | https://www.youtube.com/watch?v=4sTKcvYMNxk | |
length A squared equals length B squared plus one | ${a}^{2}={b}^{2} + 1$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
length of A is square root of length B squared plus one | $\mathrm{|A|} = \sqrt{\mathrm{|B|^{2} + 1}}$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
square root of 13 plus one is square root of 14 | $\sqrt{{13}+1}=\sqrt{14}$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
the length of a is the square root of a one squared plus a two squared plus a three squared | $|a|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
a dot b minus b dot a plus b dot b | $a\cdot b-b\cdot a+b\cdot b$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
a dot a is length a squared | $a\cdot a = |a|^2$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
cosine theta is PQ dot PR over length PQ length PR | $\cos(\theta) = \frac{{\mathsf{PQ} \cdot \mathsf{PR}}}{{|\mathsf{PQ}| \cdot |\mathsf{PR}|}}$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
minus one squared plus one squared plus zero squared square root | $\sqrt{(-1)^{2} + 1^{2} + 0^{2}}$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
x plus two y plus three z equals zero | $x+2y+3z=0$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
OP dot A equals zero | $\mathsf{OP}\cdot \mathsf{A}=0$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
x times one plus y times two plus z times three | $x\cdot 1 + y\cdot 2 + z\cdot 3$ | https://www.youtube.com/watch?v=PxCxlsl_YwY | |
x equals r cosine theta | $x=r\cos \theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
y equals r sine theta | $y=r\sin \theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
one minus x squared minus y squared | $1-x^{2}-y^{2}$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
r squared cosine squared theta | $r^{2}\cos^{2}\theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
r squared sine squared theta | $r^{2}\sin^{2}\theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
the integral from zero to pi over two integral from zero to one of one minus r squared r dr d theta | $\displaystyle\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}(1-r^{2})r dr d\theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
r minus r cubed | $(r-r^{3})$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
r to the four over four | $\frac{r^4}{4}$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
one-half m v squared | $\frac{1}{2}mv^{2}$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
one-half m r squared omega squared | $\frac{1}{2}mr^{2}\omega^{2}$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
x squared plus y squared | $x^{2}+y^{2}$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
pi a to the four over two | $\pi a^{4} \over 2$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
four a to the four cosine to the four theta | $4a^4\cos^{4}\theta$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
three halves of pi a to the fourth | $\frac{3}{2}(\pi a^{4})$ | https://www.youtube.com/watch?v=60e4hdCi1D4 | |
y of t minus two equals t | $y(t)-2=t$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
z of t minus two equals minus 3t | $z(t)-2=-3t$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
minus one plus two t | $\displaystyle -1+2t$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
y of t equals two plus t | $y(t)=2+t$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
z of t equals two minus 3t | $z(t)=2-3t$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
Q of t is Q0 plus t times the vector Q0 Q1 | $Q(t)=Q_{0}+t\vec{Q_{0},Q_{1}}$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
x plus 2y plus 4z equals seven | $x+2y+4z=7$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
minus one plus 2t plus twice two plus t plus four times two minus 3t | $(-1+2t)+2(2+t)+4(2-3t)$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
minus one plus four plus eight is eleven | $-1+4+8=11$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
one times two plus two times one plus four times minus three | $1\times2+2\times1+4\times(-3)$. | https://www.youtube.com/watch?v=57jzPlxf4fk | |
a theta minus a sine theta | $a\theta-a\sin \theta $ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
x of theta is theta minus sine theta | $x(\theta) = \theta-\sin \theta$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
one minus theta squared over two | $1-\theta^{2}\over2$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
theta minus theta cubed over six | $\theta-\frac{\theta^{3}}{6}$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
theta squared over two | $\frac{\theta^2}{2}$ | https://www.youtube.com/watch?v=57jzPlxf4fk | |
square root of one minus cosine t squared plus sine squared t | $\sqrt{1-\cos^{2}t+\sin^{2}t}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
square root of one minus two cosine t plus cosine squared t plus sine squared t | $\displaystyle\sqrt{1-2\cos t+\cos ^{2}t+\sin ^{2}t}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
cosine squared plus sine squared | $\cos ^{2}+\sin ^{2}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
square root of two minus two cosine t | $\sqrt{2-2\cos t}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
d by dt of r cross v is the zero vector | $\displaystyle\frac{d}{dt}(\mathbf{r}\times\mathbf{v})=\mathbf{0}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
uv prime equals u prime v plus u v prime | $(uv)^{\prime}=u^{\prime}v+uv^{\prime}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
dr dt cross v plus r cross dv dt | $\displaystyle\frac{d\mathbf{r}}{dt}\times \mathbf{v}+\mathbf{r}\times \frac{d\mathbf{v}}{dt}$ | https://www.youtube.com/watch?v=0D4BbCa4gHo | |
the antiderivative of sine x cosine x | $\int{\sin x \cos x} dx$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
C1 minus C2 plus a half is equal to zero | $\mathcal{C}_{1}-\mathcal{C}_{2}+\frac{1}{2}=0$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
minus x squared over two | $-\frac{x^{2}}{2}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
A e to the minus x squared over two | $Ae^{-\frac{x^{2}}{2}}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
a times d by dx of e to the minus x squared over two | $\displaystyle a\frac{d}{dx}(e^{-\frac{x^{2}}{2}})$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
h inverse of f of x plus c | $h^{-1}(f(x))+c$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
natural log of absolute y is equal to minus x squared over 2 plus c | $\ln(|y|)=-\frac{x^{2}}{2}+c$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
plus or minus a e to the minus x squared over two | $\pm ae^{-x^{2}/2}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
log y plus c one is equal to minus x squared over two plus c two | $\log y+c_{1}=-\frac{x^{2}}{2}+c_2$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
log y is equal to minus x squared over 2 plus c2 minus c1 | $\log y=-\frac{x^{2}}{2}+c_{2}-c_{1}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
a times e to the minus 0 squared over 2 is equal to 3 | $ae^{-\frac{0^{2}}{2}}=3$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
y is equal to 3e to the minus x squared over 2 | $y = 3e^{-x^{2}/2}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
twice the logarithm of x plus a constant | $2\log(x)+\text{c}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
e to the log y is equal to e to the 2 log x plus c | $e^{\log y} = e^{2\log(x)+c}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
e to the log x squared | $(e^{\log x})^2$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
minus 1 divided by 2 y divided by x | $-\frac{1}{2(y/x)}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
dy dx is equal to minus x over 2y | $\frac{dy}{dx} = -\frac{x}{2y}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
x squared over 2 plus y squared | $\frac{x^{2}}{2}+y^{2}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
y is equal to minus the square root of a squared minus x squared over two | $y=-\sqrt{a^{2}-\frac{x^{2}}{2}}$ | https://www.youtube.com/watch?v=60VGKnYBpbg | |
x to the 1 half times 2 | $x^{\frac{1}{2}} \times (2)$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
dx over x to the p | $\frac{dx}{x^{p}}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
1 over 1 minus p | $\frac{1}{1-p}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
the integral from 1 to infinity dx over x | $\int_{1}^{\infty} \frac{dx}{x}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
y equals 1 over the square root of x | $y=1/\sqrt{x}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
y equals 1 over x minus 3 squared | $y=\frac{1}{(x-3)^{2}}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
1 plus a plus a squared | $1+a+a^{2}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
one over one minus a | $\frac{1}{1-a}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
sum 1 over n squared n equals 1 to infinity | $\sum_{n=1}^{\infty} 1/n^{2}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA | |
pi squared over 6 | $\frac{\pi^{2}}{6}$ | https://www.youtube.com/watch?v=MK_0QHbUnIA |
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