# Exponential: senior math exercises corrected in PDF.

Corrected math exercises for high school seniors on exponential functions.

These exercises involve the following concepts:

1. definition of the exponential;
2. direction of variation of the exponential function;
3. derivative of the exponential function;
4. limits of the exponential function;
5. solve equations and inequalities;
6. Gauss curve;
7. simplify exponentials using algebraic formulas.

Exercise #1:

Write using a single exponential :

a.

b.

Exercise #2:

f is a differentiable function on such that .

g is the function defined on by .

1. Check that g is derivable on and that g’ = g.
2. Calculate g(0); derive the expression for g(x).
3. Deduce the expression of f(x).

Exercise #3:

In each case, write the expression with a single exponential.

1.

a.

b.

c.

2. a is a real number, simplify the writing of each expression :

Exercise #4:

f is the function defined on by .

In a reference frame, is the representative curve of the function f and is the tangent to at the point A of abscissa a with .

1. give an equation of .

2. Prove that there are two values of a for which passes through the origin of the frame of reference.

Exercise #5:

We model the average temperature T inside a freezer by posing :

where is the elapsed time, expressed in minutes

since it was switched on and T(t) its temperature in °C.

1. Give the average temperature inside the freezer:

a. before starting it up;

b. after one day of operation.

2. Study the limit of T in and interpret the result obtained.

Exercise #6:

Write the given reals in exponential form where k is an integer.

Exercise #6:

Write the given expression in the form where A is an expression.

Exercise #7:

We give the expression of three functions f,g and h defined and derivable on .

Calculate the derivative of the functions f, g and h.

.

Exercise #8:

It is estimated that future oil discoveries in the world can be modeled,
from 2015, by the function f defined on [15 ; +[ by:

where f(x) is the estimated quantity in millions of barrels
of oil that will be discovered in the year 2000 + x.
1. Determine the limit of the function f in +.

2. Calculate f ‘ (x) and deduce the direction of variation of the function f on the interval [15 ; + [.
3. Interpret the results of questions 1 and 2.

Exercise #9:

Let f be the function defined on by .

1. Express as a function of x.

2) Justify that, for any real x in the interval , .

3) Deduce the variations of the function f on .

Exercise #10:

Write the following expressions in exponential form , where A is an expression.

Exercise #11:

Prove the following equalities:

For any real x, .

For any real x, .

For any real x,

Exercise #12:

1)Show that the equation is equivalent to the equation .

2)Solve in the equation .

Exercise #13:

1)Solve in the inequation .

2)Deduce the sign of on .

Exercise #14:

Let f be the function defined on by

and g the function defined on by .

The representative curves and of the functions f and g are given below.

1. Conjecture the limits of the functions f and g at the limits of their definition set.
2. Prove these conjectures.

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