$\mathrm{TaylorApproximationTutor}\left(\right)$

$\mathrm{TaylorApproximationTutor}\left(\sin \left(x\+y\right)\,\left[x\,y\right]\=\left[1\,2\right]\,8\right)$

$\mathrm{TaylorApproximation}\left(\sin \left(x\+y\right)\,\left[x\,y\right]\=\left[1\,0\right]\,5\right)$

$\frac{\cos \left(1\right)y^5}{120}+\frac{\sin \left(1\right)y^4}{24}+\frac{\cos \left(1\right)\left(x-1\right)y^4}{24}-\frac{\cos \left(1\right)y^3}{6}+\frac{\sin \left(1\right)\left(x-1\right)y^3}{6}+\frac{\cos \left(1\right){\left(x-1\right)}^2{y}^3}{12}-\frac{\sin \left(1\right)y^2}{2}-\frac{\cos \left(1\right)\left(x-1\right)y^2}{2}+\frac{\sin \left(1\right){\left(x-1\right)}^2{y}^2}{4}+\frac{\cos \left(1\right){\left(x-1\right)}^3{y}^2}{12}+\cos \left(1\right)y-\sin \left(1\right)\left(x-1\right)y-\frac{\cos \left(1\right){\left(x-1\right)}^{2}y}{2}+\frac{\sin \left(1\right){\left(x-1\right)}^{3}y}{6}+\frac{\cos \left(1\right){\left(x-1\right)}^{4}y}{24}+\sin \left(1\right)+\cos \left(1\right)\left(x-1\right)-\frac{\sin \left(1\right){\left(x-1\right)}^2}{2}-\frac{\cos \left(1\right){\left(x-1\right)}^3}{6}+\frac{\sin \left(1\right){\left(x-1\right)}^4}{24}+\frac{\cos \left(1\right){\left(x-1\right)}^5}{120}$

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$\mathrm{TaylorApproximation}\left(\sin \left(x\+y\right)\,\left[x\,y\right]\=\left[1\,0\right]\,5\,\mathrm{output}\=\mathrm{animation}\right)$

$\mathrm{CrossSectionTutor}\left(\right)$

$\mathrm{CrossSectionTutor}\left(x^2\+y^2\+z^2\=4\,x\+z\=\left[2\,1\,0\right]\,x\=-2..2\,y\=-2..2\,z\=-2..2\right)$

$\mathrm{CrossSection}\left(x^2\+y^2\+z^2\=4\,x\+y\=\left[1.1\,0.1\right]\,x\=-2..2\,y\=-2..2\,z\=-2..2\,\mathrm{showfunction}\=\mathrm{false}\,\mathrm{title}\=''Sphere''\right)$

$\mathrm{CrossSection}\left(x^2\+y^2\,z\=0..24\,x\=-4..4\,y\=-4..4\,z\=0..25\,\mathrm{planes}\=10\,\mathrm{output}\=\mathrm{animation}\right)$

$\mathrm{DirectionalDerivativeTutor}\left(\right)$

$\mathrm{DirectionalDerivativeTutor}\left(x^2\+3y^{\frac{1}{2}}\,\left[x\,y\right]\=\left[1\,2\right]\,\left[2\,3\right]\right)$

$\mathrm{GradientTutor}\left(\right)$

$\mathrm{GradientTutor}\left(y{x}^2\+y^2\,\left[x\,y\right]\=\left[\left[1\,2\right]\,\left[3\,5\right]\right]\right)$

$\mathrm{DirectionalDerivative}\left(x^2\+y^2\,\left[x\,y\right]\=\left[1\,2\right]\,\left[3\,4\right]\right)$

$\frac{22}{5}$

$\mathrm{DirectionalDerivative}\left(x^2\+3y^{\frac{1}{2}}\,\left[x\,y\right]\=\left[1\,1\right]\,\left[1\,-1\right]\,\mathrm{output}\=\mathrm{animation}\,\mathrm{frames}\=15\right)$

$\mathrm{Gradient}\left(x^2\+y^2\, \left[x\,y\right]\=\left[0\,1\right]\right)\;$

$\left[\left[\begin{array}{r}0\\ 2\end{array}\right]\right]$

$\mathrm{Gradient}\left(x^2\+y^2\, \left[x\,y\right]\=\left[\left[0\,1\right]\,\left[2\,1\right]\right]\, \mathrm{output}\=\mathrm{plot}\right)\;$

$\mathrm{Gradient}\left(x^2\+y^2\, \left[x\,y\right] \= \left[\left[0\,1\right]\,\left[2\,1\right]\right]\, \mathrm{output} \= \mathrm{gradplot}\right)\;$

$\mathrm{SecondDerivativeTest}\left(x^2\+y^2\,\left[x\,y\right]\=\left[0\,0\right]\,\mathrm{output}\=\mathrm{hessian}\right)$

$\left[\begin{array}{rr}2& 0\\ 0& 2\end{array}\right]$

$\mathrm{SecondDerivativeTest}\left(x^2-y^2\,\left[x\,y\right]\=\left[0\,0\right]\right)$

$\mathrm{LocalMin}=\left[\right],\mathrm{LocalMax}=\left[\right],\mathrm{Saddle}=\left[\left[0\,0\right]\right]$

$\mathrm{SecondDerivativeTest}\left(x^2\+y^2\+z^2\,\left[x\,y\,z\right]\=\left[\left[1\,2\,1\right]\,\left[2\,3\,1\right]\,\left[0\,0\,0\right]\right]\,\mathrm{output}\=\mathrm{hessian}\right)$

$\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right],\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right],\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$

$\mathrm{SecondDerivativeTest}\left(x^2\+y^2\+z^2\,\left[x\,y\,z\right]\=\left[\left[1\,2\,1\right]\,\left[2\,3\,1\right]\,\left[0\,0\,0\right]\right]\right)$

$\mathrm{LocalMin}=\left[\left[0\,0\,0\right]\right],\mathrm{LocalMax}=\left[\right],\mathrm{Saddle}=\left[\right]$

$\mathrm{MultiInt}\left(x^{2}y\,y\=x-1..x\+1\,x\=0..4\,\mathrm{output}\=\mathrm{integral}\right)$

$\mathrm{Int}\left(x^{2}y\,\left[y\=x-1..x\+1\,x\=0..4\right]\right)$

$\mathrm{MultiInt}\left(x^{2}y\,y\=x-1..x\+1\,x\=0..4\right)$

$128$

$\mathrm{MultiInt}\left(x^2\+y^2\+z\,z\=-2..4\+y^2\,y\=x-1..x\+6\,x\=2..4\,\mathrm{output}\=\mathrm{integral}\right)$

$\mathrm{Int}\left(x^2\+y^2\+z\,\left[z\=-2..y^2\+4\,y\=x-1..x\+6\,x\=2..4\right]\right)$

$\mathrm{MultiInt}\left(x^2\+y^2\+z\,z\=-2..4\+y^2\,y\=x-1..x\+6\,x\=2..4\right)$

$\frac{2133754}{45}$

$\mathrm{ApproximateIntTutor}\left(\right)$

$\mathrm{ApproximateIntTutor}\left(x^3-y^2\+xy\,y\=-4..4\right)$

$\mathrm{ApproximateIntTutor}\left(r\,r\=0..2\,\mathrm{\θ}\=0..\frac{\mathrm{\π}}{2}\,\mathrm{coordinates}\=\mathrm{polar}\right)$

$\mathrm{ApproximateInt}\left(r\,r\=0..5\,t\=0..\frac{\mathrm{\π}}{2}\,\mathrm{coordinates}\=\mathrm{polar}\,\mathrm{output}\=\mathrm{plot}\,\mathrm{showfunction}\=\mathrm{false}\,\mathrm{prismoptions}\=\left[\mathrm{color}\=\mathrm{green}\right]\,\mathrm{method}\=\mathrm{upper}\right)$

$\mathrm{ApproximateInt}\left(x^2\+y^2\,x\=-4..4\,y\=-6..6\,\mathrm{output}\=\mathrm{animation}\,\mathrm{partition}\=\left[6\,6\right]\,\mathrm{frames}\=2..7\right)$

$\mathrm{ApproximateInt}\left(x^2\+y^2\,x\=-4..4\,y\=-6..6\,\mathrm{output}\=\mathrm{value}\,\mathrm{partition}\=\left[6\,6\right]\right)$

$1617.777776$

$\mathrm{Jacobian}\left(\left[zx-4yz\,x\+y-2z\,z^3\right]\,\left[x\,y\,z\right]\=\left[2\,1\,C\right]\right)$

$\left[\begin{array}{ccc}C& -4C& -2\\ 1& 1& -2\\ 0& 0& 3C^2\end{array}\right]$

$\mathrm{Jacobian}\left(\left[zx-4yz\,x\+y-2z\,z^3\right]\,\left[x\,y\,z\right]\=\left[2\,1\,C\right]\,\mathrm{output}\=\mathrm{determinant}\right)$

$15C^3$

$a:={\∫}_0^2{\∫}_0^4{x}^{2}y\ⅆx\ⅆy$

$\mathrm{Int}\left(x^{2}y\,\left[x\=0..4\,y\=0..2\right]\right)$

$b:=\mathrm{ChangeOfVariables}\left(a\,\left[{\mathrm{cartesian}}_{x\,y}\,{\mathrm{polar}}_{r\,\mathrm{\θ}}\right]\right)$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{ChangeOfVariablesObject}\left(\mathrm{Int}\left(\mathrm{Int}\left(r^4\right)\right)\right)$