In mathematics, the exponential function is the function
f
(
x
)
=
e
x
,
{\displaystyle f(x)=e^{x},}
where e = 2.71828... is Euler's constant.
More generally, an exponential function is a function of the form
f
(
x
)
=
a
b
x
,
{\displaystyle f(x)=ab^{x},}
where b is a positive real number, and the argument x occurs as an exponent. For real numbers c and d, a function of the form
f
(
x
)
=
a
b
c
x
+
d
{\displaystyle f(x)=ab^{cx+d}}
is also an exponential function, since it can be rewritten as
a
b
c
x
+
d
=
(
a
b
d
)
(
b
c
)
x
.
{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}
The exponential function
f
(
x
)
=
e
x
{\displaystyle f(x)=e^{x}}
is sometimes called the natural exponential function for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since
a
b
x
=
a
e
x
ln
b
{\displaystyle ab^{x}=ae^{x\ln b}}
As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
d
d
x
b
x
=
b
x
log
e
b
.
{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}
For b > 1, the function
b
x
{\displaystyle b^{x}}
is increasing (as depicted for b = e and b = 2), because
log
e
b
>
0
{\displaystyle \log _{e}b>0}
makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1/2); and for b = 1 the function is constant.
The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:
This function, also denoted as exp x, is called the "natural exponential function", or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as
b
x
=
e
x
log
e
b
{\displaystyle b^{x}=e^{x\log _{e}b}}
, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression. The graph of
y
=
e
x
{\displaystyle y=e^{x}}
is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation
d
d
x
e
x
=
e
x
{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}
means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point. Its inverse function is the natural logarithm, denoted
log
,
{\displaystyle \log ,}
ln
,
{\displaystyle \ln ,}
or
log
e
;
{\displaystyle \log _{e};}
because of this, some old texts refer to the exponential function as the antilogarithm.
The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well):
It can be shown that every continuous, nonzero solution of the functional equation
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
{\displaystyle f(x+y)=f(x)f(y)}
is an exponential function,
f
:
R
→
R
,
x
↦
b
x
,
{\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},}
with
b
≠
0.
{\displaystyle b\neq 0.}
The multiplicative identity, along with the definition
e
=
e
1
{\displaystyle e=e^{1}}
, shows that
e
n
=
e
×
⋯
×
e
⏟
n
factors
{\displaystyle e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ factors}}}}
for positive integers n, relating the exponential function to the elementary notion of exponentiation.
The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (e.g., matrix).
The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
Find the Taylor polynomial of degree 9 of
f(x) = e^x
about x=0 and hence approximate the value of e. Estimate the error in the approximation.
I have written the taylor polynomial and evaluated for x=1 to give an approximation of e.
Its just the error that is confusing me. I have:
R_n(x) =...
[SOLVED] Re: Integral involving square root of e^x
Homework Statement
\int \sqrt{1-e^{-x}}
Homework Equations
Sub rule.
The Attempt at a Solution
I realized that it's fairly obvious I can use u=e^-x/2 to give \sqrt {1-u^2}
but I'm kind of looking at the answers and I'm not seeing how I...
Homework Statement
Integrate from e^x to e^2x: (sin^2(t) + cos^2(t) -1)dt
Homework Equations
just standard integral equations
The Attempt at a Solution
I know how to do most of it, my only question is: is (sin^2(e^2x) + cos^2(e^x) -1) a special trig identity? or would i just...
the quesion is below, show that
show that ( 1+ \frac{x}{n} ) ^n < e^x
at the first, i take log on both sides,.. but i couldn't go further.
can someboday help me?
thx
please explain in more detail on how we come up with the answers below. Thanks in advance!
(formulas much appreciated)
Differentiate:
1.
y=e^x
=e^x
2.
y=lnx
=1/x
How can I solve:
(x^2)(e^x) - e^x = 0
and
2e^(2+x)=6
For the first one, tell me if this is right:
e^x(x^2-1)=0 ->
e^x = 0 and x^2 - 1 = 0
so x = 1 and 0? but 0 doesn't work when I plug it back in. so is 1 the solution for x?
Homework Statement
For which real numbers c is (e^x+e^{-x})/2 \leq e^{cx^2} for all real x?
Homework Equations
The Attempt at a Solution
I think you can expand both sides into series and term by term compare them. The left side is
\sum_{n=0}^{\infty}\frac{x^{2n}}{2n!}
Can...
Is it possible to intergrate
e^x (cosx)
i wondered because i tried to intergrate it by parts, but ended up going round in circles.
I wondered because i had this question and I am stuck on how to do it :)
http://img505.imageshack.us/img505/320/frfbc8.png
Homework Statement
I want to solve for the derivative of e^x using the limit definition.
Homework Equations
http://www.math.hmc.edu/calculus/tutorials/limit_definition/img10.png
The Attempt at a Solution
obviously the derivative of e^x is itself, so i konw the answer. i just...
I am a little stuck how to solve this equation
e^x = 5-2x?
I did ln e^x = ln (5-2x)
x = ln(5-2x) / ln e
but iam not sure how to bring the other x around to the side with the x to solve the equation?
How does one integrate \int_{}^{} \frac{e^x}{x}dx
I could expand it using a Laurent series and than integrating term by term but are there more elementary methods?
i need to prove that:
1+x+x^2/2!+...+x^n/n!<=e^x<=1+x+x^2/2!+...+x^n/n!+(e^x)x^(n+1)/(n+1)! for x>=0
without using the power sum of e^x.
the textbook hints that i should evaluate the integral \int_{0}^{x}e^udu and then i should integrate over and over n times,
and obtain the upper and...
Hi I'm having some trouble with evaluating these limits. I can't figure out what to do. I guess i forgot some calc one. I don't have much work but All I'm asking for for now is a couple hints.
\lim_{x\rightarrow\infty} \frac{e^{3x} -e^{-3x}}{e^{3x} + e^{-3x}}
I tried dividing numerator and...
Hi,
I'm trying to solve this:
Find all general maximum solutions of this equation
y'(2-e^{x}) = -3e^{x}\sin y\cos y
First, there are some singular solutions:
If
y \equiv k\frac{\pi}{2}
Then right side is zeroed and so is the left.
To convert it to the separate form, I need to divide...
just starting up the school year again and my brain is not there yet.
Is e^x an even or odd function.
also what about
e^x + e^-x
and
e^x - e^-x
thanks for the help.
Hey, everyone
I am working on a calc problem, and I have no idea where to start. The integral is
e^x
------------- [division problem]
(25+e^2x)^4
Do I let my u equal to the 25+e^2x? or what...
then after that what do i do.
Thanks for all the help in advance.
-Eiano
hi ,
1.)how do I find the limit of (x! e^x) / (x^x *x^1/2) as x tends to infinity ?
2.)and is f(x)= x! a function ? if so, how do I find the derivative ?
thanks for any help
Roger
Hi all,
I've been having little problems getting Fourier series of e^x.
I have given
f(x) = e^{x}, x \in [-\pi, \pi)
Then
a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\ dx = \frac{2\sinh \pi}{\pi}
a_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} e^{x}\cos (nx)\ dx =...
Ok, here is the integral i seem to be having some issues with. I know there's a very simple step I am missing.
\int_{}^{} e^x \sin(\pi x) dx
i attempted to do this using by parts integration.
I tried u = \sin(\pi x) so du= \pi \cos(\pi x) dx
so then dv= e^x dx and v= e^x...
The text wants me to find x using a graph, but since I don't like taking my sweet time building one so accurate to find an answer to one decimal place, I rather find x without the graph.
This is it:
e^x = x^10
This isn't important or anything, but I figured that I can use some practice...
I had a question on a math test which said that you should find an approximation for e^x which is very good for x \approx 0 . First I declared the function f(x) = e^x . We have the interesting thing that f(x) = f'(x) = f''(x) = f'''(x) \ldots \ \forall x . And because of this we have f(0)...
I'm revising over my maths for my exams and I just came across something I didn't understand. How do we know that:
\frac{d}{dx} \left( e^x \right) = e^x
I've seen the infinite series for e^x but in our maths class we derived it by assuming the above statement :confused:. Preemptive thanks...