Exponential: terminal math exercises corrected in PDF.

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. $\frac{1}{e^3}$

b. $e^{-2}\times \,e^7$

Exercise #2:

f is a differentiable function on $\mathbb{R}$ such that $f'=f\,\,et\,\,f(0)=-\frac{1}{2}$ .

g is the function defined on $\mathbb{R}$ by .

Check that g is derivable on $\mathbb{R}$ 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.

a. $e^4\times \,e^6$

b. $e\times \,(e^5)^2$

c. $\frac{e^{30}\times \,e^{-10}}{e^{10}}$

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

$a)\,\,\frac{e^{2a}\times \,e^{-a}}{e^{5a}}\,\,;b)\,\,\frac{e^{2a}+1}{e^{1-a}}\,\,;\,\,c)(e^a)^3\times \,e$

Exercise #4:

f is the function defined on by .

In a reference frame, $\xi$ is the representative curve of the function f and is the tangent to $\xi$ 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 :

$T(t)=19,5e^{-7\times \,10^{-4}t}-10,5$ where $t\in%5B0;+\infty%5B$ 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 $+\infty$ and interpret the result obtained.

Exercise #6:

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

$1)e^{-7}\times \,e^3$

$2)e^{-1}\times \,e^{-5}$

$3)e^2,\times \,e$

$4)e\times \,e^{-1}$

$5)\frac{1}{e}$

$6)\frac{1}{e^{-1}}$

$7)\frac{1}{e^2}$

$8)\frac{1}{e^{-3}}$

$9)\frac{e^{-3}}{e^2}$

$10)\frac{e}{e^{-1}}$

$11)\frac{e^{-2}}{e}$

$12)\frac{e^2\times \,e^{-3}}{e^5}$

$15)(e^{-1})^6$

$16)e\times \,(e^{-1})^3$

Exercise #6:

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

$1)e^x,\times \,e^2$

$2)e^{-1}\times \,e^{-x}$

$3)e\times \,e^x$

$4)e^x\times \,e^x$

$5)e^x\times \,e^{-x}$

$6)e^{x-1}\times \,e^x$

$8)(e^{-x+1})^3$

$10)\frac{e^{5x}}{e^x}$

$11)\frac{e^{x+1}}{e}$

$12)\frac{e^3}{e^{2x-1}}$

Exercise #7:

We give the expression of three functions f,g and h defined and derivable on $\mathbb{R}$ .

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

$1)f(x)=e^{0,5x};g(x)=e^{3x};h(x)=e^{-x}$ .

$2)f(x)=e^{x+1};g(x)=e^{1-2x};h(x)=e^{-3x+1}$

$3)f(x)=2+e^{2x};g(x)=1-e^{-2x};h(x)=e^{-3x+1}$

$4)f(x)=2+e^{2x};g(x)=1-e^{-2x};h(x)=1+2e^{-x}$

Exercise #8:

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

$f(x)=17280e^{-0,024x}$

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 + $\infty$ .

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