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- 1 I. Derivative number – derivative function – tangent to a curve.
- 2 II. Derivative function on an interval I. Derivative function of a function derivable on I
- 3 III. Equation of the tangent to a curve
- 4 IV. Sign of the derivative and direction of variation of a function
- 5 V. Change of sign of the derivative and extremum of a function
In this eleventh grade lesson, we will discuss the derivative of a sum, product and quotient. The derivative and the direction of variation of a function, as well as the derivatives of usual functions.
I. Derivative number – derivative function – tangent to a curve.
Let f be a function defined on an interval I. The curve (C) below is the graphical representation of f in an orthonormal frame of reference . M and N are two points of (C) with respective abscissas
and
where
.
M and N therefore have the following coordinates: and
, i.e.:
.
So we have: or
The directrix of the line (MN) secant to (C) is therefore
.
If the curve (C) has at M a tangent of directing coefficient d, then when the point N approaches M, that is to say when x tends to a, or, which is the same, when h tends to 0,
the secants (MN) will reach a limit position which is that of the tangent(MP) in M to (C).
This can then be translated using the governing coefficients as:
i.e.:
.
So we have: .
If we call , the function defined for
and
by :
.
we have :
and
, which is also written:
.
Conversely, if there exists a real d and a function such that, for all
and
, we have:
with
,
we deduce that: and therefore that:
.
This allows us to give the three equivalent definitions:
If f is a function defined on an interval and if
.
When there is a real number d such that, for any real h close to 0, we have
We say that the function f is derivable in a and that is the derivative of f in a.
If f is a function defined on an interval I and if .
When there is a real number d such that, for any real and close to a, we have:
We say that the function f is derivable in a and that is the derivative of f in a.
II. Derivative function on an interval I. Derivative function of a function derivable on I
We say that f is derivable on an interval I when it is derivable at any point of I.
When f is derivable on an interval I, the function which associates to any the derivative number of f at x is called the derivative function of f on I. This function is noted
.
If f is a function defined on an interval I. If and if f is derivable at
, then :The representative curve of f has a tangent at the point
and the directing coefficient of this tangent is the derivative number
of the function f at
.
Remarks:
If the graph of f does not have a tangent at the point M with abscissa , then the function f is not derivable at a. This is the case for the absolute value function in
.
The graph of a function may well have a tangent at a point without the function being derivable at that point: it is sufficient that the directing coefficient of this tangent does not exist (tangent parallel to the ordinate axis).
This is the case for the square root function in .
III. Equation of the tangent to a curve
If function f is derivable at a, the tangent (MP) to the curve (C) at M of abscissa exists. It has the directing coefficient
.
Its equation is therefore of the form: , where
and its intercept p can be calculated.
It is sufficient to write that (MP) passes through .
So we have: . This gives:
.
So: which is often written in one of the forms, easier to remember:
Equation of the tangent at the point :
or
.
IV. Sign of the derivative and direction of variation of a function
We admit without proof the following theorems:
Let f be a differentiable function on an interval I.
- If f is increasing on I, then for all
, we have
- If f is decreasing on I, then for all
, we have:
.
- If f is constant on I, then for all
, we have:
.
Let f be a differentiable function on an interval I.
- If, for all
, we have:
, then f is increasing on I.
- If, for all
, we have:
, then f is decreasing on I.
- If, for all
, we have:
, then f is constant on I.
Let f be a differentiable function on an interval I.
- If, for all
, we have:
( except perhaps at isolated points where
), then f is strictly increasing on I.
- If, for all
, we have:
( except perhaps at isolated points where
), then f is strictly decreasing on I.
In particular:
Let f be a function derivable on an interval .
- If, for any
, we have
, then f is strictly increasing on
.
- If, for any
, we have
, then f is strictly decreasing on
.
Examples:
1) Let the function f be defined on by
.
f is derivable on and
for all
.
– For all , we have
, so f is decreasing on
.
– For all, we have
, so f is increasing on
.
Although , we have more precise:
– For all , we have
, so f is strictly decreasing on
.
– For all , we have
, so f is strictly increasing on
.
V. Change of sign of the derivative and extremum of a function
We admit without proof the following theorems:
If f is a differentiable function on an interval I,
And if f admits a local maximum or a local minimum at different from the extremities of the interval I,
So: .
1. special case where f is derivable on an open interval.
If f is a differentiable function on an open interval I,
And if f admits a local maximum or a local minimum in ,
So: .
If f is a differentiable function on an open interval I,
and if and if
cancels for
by changing sign,
Then f(a) is a local extremum of f on I.
Examples:
1) Let the function f be defined on by
. f is derivable on
with
.
cancels out at
and
by changing sign, because :
- for x belonging to
, we have:
so f is strictly increasing on
- .
- for x belonging to
, we have :
so f is strictly decreasing on
.
- for x belonging to
, we have :
so f is strictly increasing on
.
The function thus has a local maximum at
and a local minimum at
.
All this study can be summarized in the table below:
Here is a piece of the graphical representations of f and :
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