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