Calculus 1: Tangents, Inverses, and Sampling

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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.

The following sections show how the routines work.

Tangents

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

(1.1)

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

(1.2)

You can also learn about tangents using the TangentTutor command.

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.

InversePlot((3*x^3+x+1)/(x^2+1), x = -3 .. 3)

You can also plot the inverse of a function using the InverseTutor command.

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 at the points x = , and 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.

In the following example, the global cubic behavior is very well approximated by the sampling, but the asymptote at is missed.

In other cases, some of the behavior of the expression occurs outside the sampling region. The following misses that the expression goes to , and not as the plot suggests.