Functions: math exercises in 1st grade corrected in PDF.

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 $\mathbb{R}$ 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 $%5D0;+\infty%5B$ 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 $\mathbb{R}$ 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 $\mathbb{R}$ by $f(x)=\frac{3}{4}x^2-15x+100$ .

1)Justify that f is derivable on $\mathbb{R}$ and calculate f ‘ (x) for any real x.

2)Draw the sign table of f ‘ (x) on $\mathbb{R}$ .

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 $3\leq\,\,x\leq\,\,8$ .

b)Give a frame for g(x) when $-2\leq\,\,x\leq\,\,3$ .

c) Give a frame for g(x) when $-5\leq\,\,x\leq\,\,3$ .

d) Let a and b be two real numbers such that $-5\leq\,\,a<b\leq\,\,-2$ .

e) Let a and b be two real numbers such that $-2\leq\,\,a<b\leq\,\,3$ .

Compare g(a) and g(b).

f) Let $a\in%5B-5;-2%5D$ and $b\in%5B3;8%5D$ 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 $\mathbb{R}$ 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 $\mathbb{R}$ by $f(x)\,=\,x^2-\,3x+\,7$ and $g(x)\,=\,5x-\,9$ .
1. Show, that for any real x, $f(x)-\,g(x)\,=\,x^2\,-\,8x+\,16$ .
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 $f\,'$ on [-5 ; 6].

Exercise 14:

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

Its derivative is the function $f\,'$ represented by the curve below in a plane reference frame.
1 . Read graphically the sign of $f\,'(x)$ 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 $\mathbb{R}$ and $f\,'$ its derivative. The sign table of $f\,'$ 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 $%5D0;+\infty%5B$ 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 $\mathbb{R}$ and $f\,'$ its derivative.

The sign table of $f\,'$ 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 $\mathbb{R}$ 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
$f(x)\,=2x^2\,-\,20x+\,100.$
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 $\mathbb{R}$ by $f(x)=\frac{1}{4}x-5$ and g a function defined on $\mathbb{R}^*$ by $g(x)=\,-\frac{7}{x}$ .
a) Show that, for any non-zero real x :
$f(x)-g(x)=\frac{\frac{1}{4}x^2-5x+7}{x}$ .
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 $3\,\leq\,\,x\leq\,\,8$ .
b) Give a frame for when $-2\leq\,\,x\leq\,\,3$ .
c) Give a frame for when $-5\leq\,\,x\leq\,\,3$ .
d) Let a and b be two real numbers such that $-5\,\leq\,\,a\,<\,b\leq\,\,-2.$
Compare g(a) and g(b).
e) Let a and b be two real numbers such that $-2\,\leq\,\,a\,<\,b\leq\,\,3.$