Function analysis

The app analyzes a function in the usual order: domain, even or odd, intercepts, the sign of the function, limits at the ends of the domain and asymptotes, the first derivative – monotonicity and extrema (also by higher derivatives), the second derivative – concavity and inflection points, and finally the range. It sums everything up in an overview and draws a graph with the points marked.

For a trigonometric function the app finds the period, examines the function on one period and writes the results with the period (a maximum at x = π/2 + 2kπ); if only the derivative is periodic (x + 2 sin x), the extrema and monotonicity repeat with the period. Absolute values are removed by the signs on the intervals and the one-sided derivatives compared at the corners; where the derivative is infinite, a vertical tangent is shown. Nested absolute values and sums of absolute values are analyzed by pieces; where the function is constant on an interval, this is stated with its value (a minimum at every point of the interval).

How to enter it

`analysis y = x^3 - 3x`	With the word “analysis” before the function.
`x^3 - 3x`	A function with the “Analyze function” chip.

Examples

A polynomial – extrema and an inflection point.

Input: x^3 - 3x Analyze function

A rational function – asymptotes.

Input: x^2/(x - 1) Analyze function

A periodic function – everything on one period, the results with the period.

Input: sin 2x + 2cos x Analyze function

Tangent – vertical asymptotes, range ℝ.

Input: tg x Analyze function

Hyperbolic tangent – horizontal asymptotes y = ±1.

Input: tanh x Analyze function

Only the derivative is periodic – the extrema values grow with k.

Input: x + 2sin x Analyze function

Absolute value – corners, one-sided derivatives.

Input: |x^2 - 1| Analyze function

Cube root – a vertical tangent at zero.

Input: x^(1/3) Analyze function

A nested absolute value – by pieces, an even function.

Input: |-x^2 + 2|x| + 3| Analyze function

A sum of absolute values – a constant piece and two oblique asymptotes.

Input: |x - 2| + |x + 3| Analyze function

Known limitations

A function without even a periodic derivative (x·sin x) is analyzed only numerically on ⟨−2π; 2π⟩, without the period; inverse trigonometric functions (arcsin, arctan) are not supported yet.