The TangentLine(f, var1) command computes the line tangent to the curve f at the point specified by var1. The curve can be entered as a Vector, a Vector-valued procedure, a scalar expression, or a scalar-valued procedure. If f is entered as a procedure, then it must accept at least 1 argument; it will be evaluated as $f\left(\mathrm{lhs}\left(\mathrm{var}\right)\right)$ to obtain the expression for the curve and will be evaluated as $f\left(\mathrm{rhs}\left(\mathrm{var}\right)\right)$ to obtain the point of tangency.

A curve in two-dimensional space can be specified using any of the calling sequences.  A curve in three-dimensional space can only be specified using the calling sequence TangentLine(f, var1) where f is a 3-vector or a 3-vector-valued procedure.  The coordopt will be ignored in this case if the vector f has its own coordinate system attached, but the outvars option can be included.

The TangentLine(f, var1, var2) command computes the line tangent to the curve f at the point specified by the two var parameters, where f is defined implicitly by an equation, for example $x^2+y^2=1$.  No attempt is made to verify that the point specified by the var parameters is actually on the curve.

The TangentLine(f, pt) command computes the line tangent to the curve f at the point specified by the 2-element Vector pt parameters, where f is defined implicitly by an equation, for example $x^2+y^2=1$.  No attempt is made to verify that the point specified by the pt parameter is actually on the curve.

If f is a scalar expression, an equation, or a scalar-valued procedure, then the current coordinate system must be two-dimensional, e.g., cartesian (in two dimensions) or polar, or the coordopt option must be provided and specify a two-dimensional coordinate system.

If f is a scalar expression, then the second argument must be an equation in which the left-hand side is one of the coordinate names from the current coordinate system or from the coordopt option, if provided.  For example, if f is a scalar expression in $x$, the ambient coordinate system is ${\mathrm{cartesian}}_{x,y}$, and the second argument is $x=a$, respectively, then f is parameterized as the Vector <$x$, f>. The tangent line will be computed at the point $x=a$, $y=\genfrac{}{}{0ex}{}{f}{}|\genfrac{}{}{0ex}{}{}{x=a}$.

If f is a scalar-valued procedure, the second argument must be an equation in which the left-hand side is one of coordinate variable names from the current coordinate system or from the coordopt option. The procedure f will be evaluated at this variable, producing a scalar expression, and the previous paragraph applies.

If f is a Vector or Vector-valued procedure, var must be an equation whose left hand side is the curve parameter.

If f is a Vector or a scalar, the output is a position Vector. If f is a procedure, the output is a procedure that evaluates to a position Vector.  If f is described implicitly, as an equation, then the returned tangent line will be given as an equation.

If the input expression f is not a Vector with a coords attribute, it is interpreted in coordinate system specified by the coordopt option.  If this is not given, then the ambient coordinate system (as set, for example, by SetCoordinates) is assumed.

If neither the ambient coordinate system nor the coordinate system in which f is interpreted (see previous item) is cartesian of the appropriate dimension, then the outvars option must be used to provide the names of the output (cartesian) coordinate system variable names.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\&colon;$

$\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y\right]\right)$

${\mathrm{cartesian}}_{x,y}$

$\mathrm{TangentLine}\left(⟨t\&comma;t^2⟩\&comma;t=a\right)$

$\left[\begin{array}{c}x\\ -a\left(a-2x\right)\end{array}\right]$

$\mathrm{TangentLine}\left(x\mapsto x^2\&comma;x=a\right)$

$x\mapsto \mathrm{VectorCalculus}:-\mathrm{Vector}\left(\left[x\&comma;-a\cdot \left(a-2\cdot x\right)\right]\&comma;\mathrm{attributes}=\left[\mathrm{positionvector}\&comma;\mathrm{coords}={\mathrm{cartesian}}_{x,y}\right]\right)$

$\mathrm{TangentLine}\left(x^2\&comma;x=a\right)$

$\left[\begin{array}{c}x\\ -a\left(a-2x\right)\end{array}\right]$

$\mathrm{TangentLine}\left(t\mapsto ⟨1+t^2\&comma;t-t^2⟩\&comma;t=a\right)$

$t\mapsto \mathrm{VectorCalculus}:-\mathrm{Vector}\left(\left[x\&comma;\frac{a^2+\left(-2\cdot x+2\right)\cdot a+x-1}{2\cdot a}\right]\&comma;\mathrm{attributes}=\left[\mathrm{positionvector}\&comma;\mathrm{coords}={\mathrm{cartesian}}_{x,y}\right]\right)$

$\mathrm{SetCoordinates}\left('\mathrm{polar}'\left[r,\mathrm{\theta }\right]\right)$

${\mathrm{polar}}_{r,\mathrm{\theta }}$

$\mathrm{TangentLine}\left(⟨1\&comma;t⟩\&comma;t=\frac{\mathrm{\pi }}{4}\&comma;\mathrm{outnames}=\left[u\&comma;v\right]\right)$

$\left[\begin{array}{c}u\\ \sqrt{2}-u\end{array}\right]$

$\mathrm{TangentLine}\left(r=1\&comma;⟨1\&comma;\frac{\mathrm{\pi }}{4}⟩\&comma;\mathrm{outnames}=\left[u\&comma;v\right]\right)$

$\frac{\sqrt{2}\left(u-\frac{\sqrt{2}}{2}\right)}{2}+\frac{\sqrt{2}\left(v-\frac{\sqrt{2}}{2}\right)}{2}=0$

$\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y,z\right]\right)$

${\mathrm{cartesian}}_{x,y,z}$

$\mathrm{TangentLine}\left(⟨1\&comma;t\&comma;t^2⟩\&comma;t=2\right)$

$\left[\begin{array}{c}1\\ y\\ -4+4y\end{array}\right]$

$\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y\right]\right)$

${\mathrm{cartesian}}_{x,y}$

$\mathrm{`~`}\left[\mathrm{normal}\right]\left(\mathrm{TangentLine}\left({\mathrm{\theta }}^2\&comma;\mathrm{\theta }=\frac{\mathrm{\pi }}{4}\&comma;\mathrm{coords}=\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)\right)$

$\left[\begin{array}{c}x\\ \frac{{\mathrm{\pi }}^3\sqrt{2}-16\mathrm{\pi }x-128x}{16\left(-8+\mathrm{\pi }\right)}\end{array}\right]$

The Student[VectorCalculus][TangentLine] command was updated in Maple 17.

The var2, pt, outvars and coordopt parameters were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.