Examples of Transcendental
Expressions. Transcendental
(also called non-algebraic) expressions are expressions that
cannot be constructed solely from polynomials by adding, subtracting,
multiplying, dividing, raising to an integer power, or taking
roots. The list includes some algebraic-looking
operations such as: powers of x that are irrational
numbers like Examples of Transcendental
Functions. The list includes
special function names such as: Periodic Transcendental
Functions. A periodic function is one whose graph repeats
over fixed interval lengths. The six basic trigonometric functions— Non-periodic Transcendental
functions. Among the functions studied in calculus,
only the trigonometric functions are periodic. So, in addition
to irrational powers and tower functions, the list of basic,
transcendental functions includes exponential Complete Graphs for NonPeriodic Functions. A complete graph is one that includes all the important mathematical features. For nonperiodic transcendental functions, this means the viewing window should be just large enough to display all x- and y-intercepts, all maxima and minima, all points of inflections, all holes, all jumps (1- and 2-hole), all vertical asymptotes, all horizontal asymptotes, all vertical tangents, all vertical cusps, all corners, and all isolated points. When you enter a nonperiodic transcendental function and click the GraphIt button, WebGraphing.com automatically displays a complete graph. Armed with this knowledge, you can select different viewing windows to zoom in on particular aspects of the graph or refine it more to your liking. Fixed Graph Intervals for
Periodic Functions. Periodic functions have infinitely-many
points of mathematical interest. All the basic trig functions
have infinitely-many x-intercepts, all but Zoom In/Out (Resizing the Viewing Window). If you wish to graph again with a different display window, you can do this in two ways: (1) you can enter the lower and upper x-bounds only and let WebGraphing.com determine optimal y-bounds (Smart Zooming) that include all the important mathematical features contained within your x-bounds, or (2) you can enter both the lower and upper x- and y-bounds to choose the precise viewing window. The first option is especially useful if you are zooming in on a peak or valley or a change in concavity, since in that case WebGraphing.com chooses an optimal range of y-values. The choice of an appropriately-chosen display can be important to visually verify, beyond the indicated color-coding, underlying features like maxima, minima, and points of inflection. Domain.An
irrational number power, like Horizontal and Vertical Asymptotes Horizontal
Asymptotes. A horizontal
asymptote is a horizontal line, Vertical
Asymptotes. A vertical asymptote is a vertical
line, Holes and Jump Discontinuities Holes. With transcendental functions, a hole can occur at a point along with a vertical asymptote, as happens with the exponential function e^{(1/x)}. In this example, the function is nearly continuous but only from one side.
Jump Discontinuities. If, at a given point, a function has limits from the left side and right side that are finite but unequal, the function is said to have a jump discontinuity at that point. Here is an example of a transcendental function that, at x=0, has a left hand limit of – π/2 and a right hand limit of π/2
Vertical Tangent Lines, Vertical Cusps, and Corners
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