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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.
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![with(Student[Calculus1])](/support/helpjp/helpview.aspx?si=5022/file03813/math33.png)
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The following sections show how the routines work.
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Tangents
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The Tangent routine returns the tangent to a curve at a given point.
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| (1.1) |
Where the tangent is vertical, an equation form is returned.
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| (1.2) |
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You can also learn about tangents using the TangentTutor command.
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Inverse of a Function
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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.
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You can also plot the inverse of a function using the InverseTutor command.
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The Failures of Approximating by Sampling
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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.
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In the following example, the global cubic behavior is very well approximated by the sampling, but the asymptote at  is missed.
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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.
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Main: Visualization
Next: Derivatives
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