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.
restart
withStudentCalculus1:
The following sections show how the routines work.
Tangents
The Tangent routine returns the tangent to a curve at a given point.
Tangentsinx,x=1,output=line
xcos1+sin1−cos1
Where the tangent is vertical, an equation form is returned.
Tangent surdx,3, x =0, output = line
x=0
Tangentsinx,x=1,output=plot
Tangent surdx − 1,3, x=1, output = plot
You can also learn about tangents using the TangentTutor command.
TangentTutor
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.
InversePlotsinx,x=0..4π
InversePlottanx,x=0..π
InversePlot3x3+x+1x2+1,x=−3..3
You can also plot the inverse of a function using the InverseTutor command.
InverseTutor
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 sin12x 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.
PointInterpolationsin12x,x=0..5
In the following example, the global cubic behavior is very well approximated by the sampling, but the asymptote at x=−1 is missed.
PointInterpolationx4−2x3−3x2+3x+1x+1,x=−6..6
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.
PointInterpolationx4−3x3−x+3,x=−2..2
Main: Visualization
Next: Derivatives
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