Derivative of a function : 11th grade math lesson in PDF.

11th grade math lessons

Sommaire de cette fiche

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
- 5.1 1. special case where f is derivable on an open interval.

The derivative of a function in a 11th grade math lesson where we will find the derivative at a point and the concrete meaning of the derivative number and the equation of the tangent at a point.
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.

Definition:

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 $(O,\vec{i},\vec{j})$ . M and N are two points of (C) with respective abscissas $a\in\,I$ and $x=a+h\in\,I$ where $h\in\,\mathbb{R}^*$ .

M and N therefore have the following coordinates: and , i.e.: .

So we have: $\vec{MN}\,(\,x-a;f(x)-f(a)\,)$ or $\vec{MN}\,(\,h;f(a+h)-f(a)\,)$

The directrix of the line (MN) secant to (C) is therefore

$m=\frac{f(x)-f(a)}{x-a}=\frac{f(a+h)-f(a)}{h}$ .

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:

$\lim_{x\to\,a}\frac{f(x)-f(a)}{x-a}=d$ i.e.: $\lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d$ .

So we have: $\lim_{h\to\,0}\,%5B\frac{f(a+h)-f(a)}{h}-d\,\,%5D=0$ .

If we call $\Phi$ , the function defined for $h\in\,\mathbb{R}^*$ and $a+h\in\,I$ by :

$\Phi\,(h)=\frac{f(a+h)-f(a)}{h}-d$ .

we have :

$\lim_{h\to\,0}\Phi\,(h)=0$ and $h\Phi\,(h)=\,f(a+h)-f(a)-dh$ , which is also written: $f(a+h)=\,f(a)+dh+h\Phi\,(h)$ .

Conversely, if there exists a real d and a function $\Phi$ such that, for all $h\in\,\mathbb{R}^*$ and $a+h\in\,I$ , we have: $f(a+h)=\,f(a)+dh+h\Phi\,(h)$ with $\lim_{h\to\,0}\Phi\,(h)=0$ ,

we deduce that: $\Phi\,(h)=\frac{f(a+h)-f(a)}{h}-d$ and therefore that: $\lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d$ .

This allows us to give the three equivalent definitions:

Definition 1:

If f is a function defined on an interval and if $a\in\,I$ .

When there is a real number d such that, for any real h close to 0, we have

$\lim_{h\to\,0}\frac{f(a+h)-f(a)}{h}=d$

We say that the function f is derivable in a and that d=f'(a) is the derivative of f in a.

Definition 2:

If f is a function defined on an interval I and if $a\in\,I$ .

When there is a real number d such that, for any real $x\in\,I$ and close to a, we have:

$\lim_{x\to\,a}\frac{f(x)-f(a)}{x-a}=d$

We say that the function f is derivable in a and that d=f'(a) is the derivative of f in a.

II. Derivative function on an interval I. Derivative function of a function derivable on I

Definition:

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 $x\in\,I$ the derivative number of f at x is called the derivative function of f on I. This function is noted .

Graphical interpretation of the derivative number.

If f is a function defined on an interval I. If $a\in\,I$ and if f is derivable at x=a , then :The representative curve of f has a tangent at the point and the directing coefficient of this tangent is the derivative number f'(a) 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

Definition:

If function f is derivable at a, the tangent (MP) to the curve (C) at M of abscissa x=a exists. It has the directing coefficient m=f'(a) .

Its equation is therefore of the form: y=mx+p , where m=f'(a) and its intercept p can be calculated.

It is sufficient to write that (MP) passes through m=f'(a) .

So we have: $f(a)=f'(a)\times \,a+p$ . This gives: $p=f(a)-a\times \,f'(a)$ .

So: which is often written in one of the forms, easier to remember:

Equation of the tangent at the point :

Definition:

or .

IV. Sign of the derivative and direction of variation of a function

We admit without proof the following theorems:

Theorem 1:

Let f be a differentiable function on an interval I.

If f is increasing on I, then for all $x\in\,I$ , we have $f'(x)\geq\,\,0$
If f is decreasing on I, then for all $x\in\,I$ , we have: $f'(x)\leq\,\,0$ .
If f is constant on I, then for all $x\in\,I$ , we have: $f'(x)=\,0$ .

Theorem 2:

Let f be a differentiable function on an interval I.

If, for all $x\in\,I$ , we have: $f'(x)\geq\,\,0$ , then f is increasing on I.
If, for all $x\in\,I$ , we have: $f'(x)\leq\,\,0$ , then f is decreasing on I.
If, for all $x\in\,I$ , we have: $f'(x)=\,0$ , then f is constant on I.

Theorem 3:

Let f be a differentiable function on an interval I.

If, for all $x\in\,I$ , we have: $f'(x)>\,0$ ( except perhaps at isolated points where $f'(x)=\,0$ ), then f is strictly increasing on I.
If, for all $x\in\,I$ , we have: $f'(x)<\,0$ ( except perhaps at isolated points where $f'(x)=\,0$ ), then f is strictly decreasing on I.

In particular:

Let f be a function derivable on an interval .

Ownership:

If, for any $x\in%5Ba;b%5D$ , we have $f'(x)>\,0$ , then f is strictly increasing on .
If, for any $x\in%5Ba;b%5D$ , we have $f'(x)<\,0$ , then f is strictly decreasing on .

Examples:

1) Let the function f be defined on $\mathbb{R}$ by .

f is derivable on $\mathbb{R}$ and for all $x\in\,\mathbb{R}$ .

– For all $x\in\,%5D-\infty;0%5D$ , we have $f'(x)\leq\,\,0$ , so f is decreasing on $%5D-\infty;0%5D$ .

– For all $x\in\,%5B0;+\infty%5B$ , we have $f'(x)\,\geq\,\,0$ , so f is increasing on $\,%5B0;+\infty%5B$ .

Although $f'(0)\,=\,0$ , we have more precise:

– For all $x\in\,%5D-\infty;0%5B$ , we have $f'(x)\,<\,0$ , so f is strictly decreasing on $%5D-\infty;0%5D$ .

– For all $x\in\,%5D0;+\infty%5B$ , we have $f'(x)\,>\,0$ , so f is strictly increasing on $%5D0;+\infty%5B$ .

V. Change of sign of the derivative and extremum of a function

We admit without proof the following theorems:

Theorem:

If f is a differentiable function on an interval I,

And if f admits a local maximum or a local minimum at x=a different from the extremities of the interval I,

So: f'(a)=0 .

1. special case where f is derivable on an open interval.

Ownership:

If f is a differentiable function on an open interval I,

And if f admits a local maximum or a local minimum in $a\in\,I$ ,

So: f'(a)=0 .

Ownership:

If f is a differentiable function on an open interval I,

and if $a\in\,I$ and if f'(x) cancels for x=a by changing sign,

Then f(a) is a local extremum of f on I.

Examples:

1) Let the function f be defined on $\mathbb{R}$ by . f is derivable on $\mathbb{R}$ with $f'(x)\,=6x^2-6x-12=6(x+1)(x-2)$ .

cancels out at and by changing sign, because :

for x belonging to $-%5D\infty;-1%5B$ , we have: so f is strictly increasing on $-%5D\infty;-1%5B$
.
for x belonging to , we have : so f is strictly decreasing on.
for x belonging to $%5D2;+\infty%5B$ , we have : $f'(x)>0$ so f is strictly increasing on $%5D2;+\infty%5B$ .

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 :

Cette publication est également disponible en : Français (French) Español (Spanish) العربية (Arabic)

Download and print this document in PDF for free

You have the possibility to download then print this document for free «derivative of a function : 11th grade math lesson to download in PDF» in PDF format.

Other forms similar to derivative of a function : 11th grade math lesson to download in PDF.

Derivative of a function: course in Première S
99
A mathematical course on the derivation of a function, including the derivative at a point and the concrete meaning of the derivative number and the equation of the tangent at a point, the derivative of a sum, a product and a quotient, the derivative and the direction of variation of…
Generalities on numerical functions: 11th grade math lesson
94
Generalities about numerical functions in a 11th grade math lesson that involves the tables of variation of a function as well as its graphical representation. In this eleventh grade lesson, we will study the square root and absolute value functions and the direction of variation of the functions u+k. I.…
Probability : 11th grade math lesson to download in PDF.
85
Probability in a 11th grade math lesson where we will study the law of large numbers and the law of probabilities. In this first grade lesson, we will also cover the concept of equiprobability, the expectation and standard deviation and the variance of a random variable. I. Random variables and…

Les dernières fiches mises à jour.

Voici les dernières ressources similaires à derivative of a function : 11th grade math lesson to download in PDF mis à jour sur Mathovore (des cours, exercices, des contrôles et autres), rédigées par notre équipe d'enseignants.

On Mathovore, there is 13 703 392 math lessons and exercises downloaded in PDF.

Derivative of a function : 11th grade math lesson to download in PDF

I. Derivative number – derivative function – tangent to a curve.

II. Derivative function on an interval I. Derivative function of a function derivable on I

III. Equation of the tangent to a curve

IV. Sign of the derivative and direction of variation of a function

V. Change of sign of the derivative and extremum of a function

1. special case where f is derivable on an open interval.

Other documents in the category 11th grade math lessons

Other forms similar to derivative of a function : 11th grade math lesson to download in PDF.