Corrected math exercises for high school seniors on exponential functions.
These exercises involve the following concepts:
- definition of the exponential;
- direction of variation of the exponential function;
- derivative of the exponential function;
- limits of the exponential function;
- solve equations and inequalities;
- Gauss curve;
- 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
.
- Check that g is derivable on
and that g’ = g.
- Calculate g(0); derive the expression for g(x).
- 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
.
- 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.
- Conjecture the limits of the functions f and g at the limits of their definition set.
- Prove these conjectures.
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