Mathematical exercises for 1st grade on **numerical functions**.

These corrected math exercises on functions have been written by teachers of the national education.

You will find the following concepts:

- definition domain of a function;
- limit of a function;
- asymptote to a curve;
- canonical form;
- parity of a function;
- direction of variation of a function.

These exercises on numerical functions with their answers in Première S can be consulted online or downloaded in PDF format.

Exercise 1:

1)Copy and complete the following sentences.

a) “The parabola and the line d intersect at …”.

b)” The parabola is located strictly above the line d on …”.

“The parabola is located strictly below the line d on …”.

2)Deduce the solutions of the equations and inequations.

a) f(x) = g(x). b) f(x)>g(x). c) f(x) < g(x).

Exercise 2:

1)Study the relative position of the parabola and the line .

2)Deduce the solutions of the following equations and inequations:

a) f(x) = g(x). b) f(x)>g(x). c) f(x) < g(x).

Exercise 3:

Let f and g be two functions defined on whose curves and are represented

below in a reference frame of the plane.

1)Study the relative position of the curves and .

2) Deduce the solutions of the following equations and inequations.

a) f(x) = g(x). b) f(x)>g(x). c) f(x) < g(x).

Exercise 4:

Let f be a function defined and derivable on the interval [- 5 ; 6 ].

The representative curve of f is plotted below in a plane.

1)Describe the variations of f on [- 5 ; 6].

2) Derive the sign table of the derivative function f ‘ on [- 5 ; 6 ].

Exercise 5:

Let g be a differentiable function on and g ‘ its derivative.

The sign table for g ‘ is given.

Does the function g have a local extremum?

If so, is this a maximum?

Exercise 6:

Let g be a differentiable function on and g ‘ its derivative.

The sign table for g ‘ is given.

1)Does the function g have a local minimum ?

If yes, in what value?

2)Does the function g have a local maximum?

If yes, in what value?

Exercise 7:

Let f be the function defined on by .

1)Justify that f is derivable on and calculate f ‘ (x) for any real x.

2)Draw the sign table of f ‘ (x) on .

3)Deduce that f has a local extremum at a value to be determined.

Exercise 8:

We have the table of variations of a function g defined and derivable on [- 5 ; 8 ].

a)Give a frame for g(x) when .

b)Give a frame for g(x) when .

c) Give a frame for g(x) when .

d) Let a and b be two real numbers such that .

e) Let a and b be two real numbers such that .

Compare g(a) and g(b).

f) Let and be. Compare g(a) and g(b).

Exercise 9:

1. Study the relative position of the parabola and the line d.

2. Deduce the solutions of the equations and inequalities.

a) f(x) = g(x)

b) f(x) > g(x)

c) f(x) < g(x)

Exercise 10:

1. investigate the relative position of the parabola and the line d.

2.deduce the solutions of the following equations and inequations:

a) f(x) = g(x)

b) f(x) > g(x)

c) f(x) < g(x)

Exercise 11:

Let f and g be two functions defined on whose curves and are represented below in a plane reference frame.

1 . Study the relative position of the curves and .

2. Deduce the solutions of the following equations and inequations.

a) f(x) = g(x)

b) f(x) > g(x)

c) f(x)< g(x)

Exercise 12:

Let f and g be two functions defined on by and .

1. Show, that for any real x, .

2. Study, according to the values of x, the sign of f(x) – g(x).

3. Deduce the relative position of the curves and .

Exercise 13:

Let f be a function defined and derivable on the interval [- 5 ; 6].

The representative curve of f is plotted below in a plane.

Describe the variations of f on [- 5 ; 6].

2. deduce the sign table of the derivative function on [-5 ; 6].

Exercise 14:

Let f be a function defined and derivable on the interval [- 2 ; 10].

Its derivative is the function represented by the curve below in a plane reference frame.

1 . Read graphically the sign of according to the values of x in the interval [ – 2 ; 10].

And present your results in a table of signs.

2. Deduce the table of variations of the function f on the interval [ – 2 ; 10].

Exercise 15:

Let f be a differentiable function on and its derivative. The sign table of is given.

Does the function f have a local extremum? If so, is this a maximum or minimum?

Exercise 16:

Let g be a differentiable function on and its derivative.

The sign table of is given.

Does the function g have a local extremum? If so, is this a maximum or minimum?

Exercise 17:

Let f be a differentiable function on and its derivative.

The sign table of is given.

1. Does the function f have a local minimum ? If yes, in what value?

2. Does the function f have a local maximum ? If yes, in what value?

Exercise 18:

Let g be a differentiable function on and its derivative.

The sign table of is given.

1. Does the function g have a local minimum ? If yes, in what value?

Does the function g have a local maximum? If yes, in what value?

Exercise 19:

Let [AB] be a segment of length 10 and M a point of this segment.

On the same side of this segment, we construct two squares AMNP and MBCD.

Let AM = x and study the area of the domain formed by these two squares as a function of x.

1.to which interval I does the real x belong ?

2. Let f(x) be the area of the domain.

Show that, for any real x of l, we have

3. Justify that the function f is differentiable on I and determine for all x of I.

4. Deduce the variations of f on I and the value of x for which the area of the domain is minimal.

Exercise 20:

Let f be a function defined on by and g a function defined on by .

a) Show that, for any non-zero real x :

.

b) Study, according to the values of x, the sign of f(x) – g(x).

c) Deduce the relative position of the curves and .

Exercise 21:

We have the table of variations of a function g defined and derivable on [ – 5 ; 8] :

a) Give a frame for when .

b) Give a frame for when .

c) Give a frame for when .

d) Let a and b be two real numbers such that

Compare g(a) and g(b).

e) Let a and b be two real numbers such that

Compare g(a) and g(b).

f) Let and be.

Compare g(a) and g(b).

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