Study of function

This article relates to the real functions of only one Variable.

The study of a function ƒ is the determination of the behavior of the Chart of this function. The chart is the whole of the points of the plan checking

there = ƒ ( X )

By abuse language, one speaks sometimes about Graphe of a function .

Method of study

The study consists in determining the particular points and directions and the behavior in extreme cases of the interval of definition (which can be finished or ±∞). That passes by the calculation of its Dérivée and its Dérivée second:

discontinuity;
direction of variation, defined by the sign of the Derived ;
Not of inflection;
Not of graining;
intersection with the axes;
tangent Horizontal E;
Asymptotic.

After having traced and having graduated the axes, one places the particular points, one plots the straight lines of asymptote and the tangents remarkable, then by a show of hands, one plots a smooth curve while passing by the point given and respecting the directions.

One can also calculate a certain number of points (for example ten) judiciously distributed to facilitate the layout. These points are represented in the form of a right cross (+).

Examples

Example of a polynomial

Let us consider the Polynôme

p ( X ) = X + 3 X + 1

One knows that it tends towards +∞ in +∞ and - ∞ ( p ( X ) ≈ X in + and - ∞);
it cuts the axis of the there into +1 ( p (0) = 1);
the resolution of the quadratic equation indicates to us that it cuts the axis of the X in $r_1= \ frac {- 3 - \ sqrt {5}} {2}$ and in $r_2= \ frac {- 3 + \ sqrt {5}} {2}$
its derivative vaut
p ' ( X ) = 2 X +3
it is cancelled in X = -3/2, there is thus a horizontal tangent in this point, the curve is decreasing before ( p '<0), increasing after ( p '>0);
its derivative second vaut
p ( X ) = 2
the curve is thus convex, it does not have there a point of inflection

There is thus the table of variation according to:

One chooses an interval of X giving “representable” values, a readable graph, for example; on this interval, the polynomial will take values between -1,25 (- 5/4) and 19, one thus traces the axes. The remarkable points are placed, (first root), (- 1,5; - 1,25) with the end of horizontal tangent, (second root), and. Then, one plots the free-hand curve.

Example of the tangent function

The tangent function is defined by

\ tan {X} = \ frac {\ sin {X}} {\ cos {X}}

The functions sine and cosine being periodic, it is also a Periodic function, it is thus enough to study it on an interval whose width is the period. One initially does not know the period of the tangent, one thus starts by taking an interval of 2 π, period of the sine and the cosine; let us take for example.

The cosine cancels himself for values π/2 + K ·π, and in these values, the sine is nonnull (it is worth ±1), therefore in these values, the function tends towards ±∞.

The sine is cancelled for values K ·π, and for these values, the cosine is nonnull (it is worth ±1), therefore the function is cancelled for these values.

We thus determined vertical asymptotes π/2 + K ·π, and of the simple points of passage in K ·π.

The derivative is worth, according to the law of composition (( has / B ) “= ( a' B - ab” ) /b ²):

\ tan' X = \ frac {\ sin^2 X + \ cos^2 X} {\ cos^2 X} = \ frac {1} {cos^2 X}

it is thus seen that the function is always increasing, since its derivative is always positive, and that its slope tends towards +∞ for values of the π/2 type + K ·π, which corresponds to the vertical asymtotes.

The derivative second is worth (with 1 b' = - b' / B ² and ( C ²) '= 2 cc' )

\ tan

X = - \ frac {- 2 \ cos X \ cdot \ sin X} {\ cos^4 X} = \ frac {2 \ sin X} {\ cos^3 X}

it is seen that the derivative second is cancelled for the values K ·π, there are thus points of inflection; in these points, the derivative is worth 1.
There is thus the table of variation according to:

Within sight of this table, the function seems to present a periodicity of π. One can check it simply:

$\ tan (x+ \ pi) = \ frac {\ sin (x+ \ pi)}{\ cos (x+ \ pi)} = \ frac {- \ sin (X)}{- \ cos (X)} = \ tan (X)$
One can thus restrict the interval of layout at. One thus traces the vertical asymptotes X = π/2 + K ·π, the tangent of slope 1 at the points of inflection ( K ·π, 0), then one traces the function by a show of hands.

See too

External bonds

Function analyzes: Explanations - Example - Exercises

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