 Calculus1 Tangents - Maple Help

Calculus 1: Tangents, Inverses, and Sampling

The Student[Calculus1] package contains three routines that can be used to both work with and visualize the concepts of tangents, the inverses of functions, and the errors of plotting a function by sampling.  This worksheet demonstrates this functionality.

For further information about any command in the Calculus1 package, see the corresponding help page.  For a general overview, see Calculus1.

Getting Started

While any command in the package can be referred to using the long form, for example, Student[Calculus1][Tangent],  it is easier, and often clearer, to load the package, and then use the short form command names.

 > $\mathrm{restart}$
 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$

The following sections show how the routines work. Tangents

The Tangent routine returns the tangent to a curve at a given point.

 > $\mathrm{Tangent}\left(\mathrm{sin}\left(x\right),x=1,\mathrm{output}=\mathrm{line}\right)$
 ${x}{\mathrm{cos}}{}\left({1}\right){+}{\mathrm{sin}}{}\left({1}\right){-}{\mathrm{cos}}{}\left({1}\right)$ (1.1)

Where the tangent is vertical, an equation form is returned.

 >
 ${x}{=}{0}$ (1.2)
 > $\mathrm{Tangent}\left(\mathrm{sin}\left(x\right),x=1,\mathrm{output}=\mathrm{plot}\right)$ > You can also learn about tangents using the TangentTutor command.

 > $\mathrm{TangentTutor}\left(\right)$  Inverse of a Function

The inverse of a function can be plotted using the InversePlot routine.  The default plot domain and range are chosen to the display reasonable portions of the function and its inverse.

 > $\mathrm{InversePlot}\left(\mathrm{sin}\left(x\right),x=0..4\mathrm{π}\right)$ > $\mathrm{InversePlot}\left(\mathrm{tan}\left(x\right),x=0..\mathrm{π}\right)$ > $\mathrm{InversePlot}\left(\frac{3{x}^{3}+x+1}{{x}^{2}+1},x=-3..3\right)$ You can also plot the inverse of a function using the InverseTutor command.

 > $\mathrm{InverseTutor}\left(\right)$  The Failures of Approximating by Sampling

One reason for studying derivatives is to get qualitative information about a function.  The easiest way to sketch a function is to sample it at a number of points and connect the dots.  For example, sampling the function $\mathrm{sin}\left(12x\right)$ at the points x = $0,1,2,3,4$, and $5$ suggests the following approximation (shown in blue). Knowing that the sine function oscillates, you may be satisfied with this result.  The actual expression is plotted in red.

 > $\mathrm{PointInterpolation}\left(\mathrm{sin}\left(12x\right),x=0..5\right)$ In the following example, the global cubic behavior is very well approximated by the sampling, but the asymptote at $x=-1$ is missed.

 > $\mathrm{PointInterpolation}\left(\frac{{x}^{4}-2{x}^{3}-3{x}^{2}+3x+1}{x+1},x=-6..6\right)$ In other cases, some of the behavior of the expression occurs outside the sampling region. The following misses that the expression goes to $\infty$, and not $-\infty$ as the plot suggests.

 > $\mathrm{PointInterpolation}\left({x}^{4}-3{x}^{3}-x+3,x=-2..2\right)$ > 

Main: Visualization

Next: Derivatives