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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