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.
If $f^{-1}(x)$ is the inverse of $f(x)=e^{2x}$, then $f^{-1}(x)=$$a. \ln\dfrac{2}{x}$
$b. \ln \dfrac{x}{2}$
$c. \dfrac{1}{2}\ln x$
$d. \sqrt{\ln x}$
$e. \ln(2-x)$
ok, it looks slam dunk but also kinda ?
my initial step was
$y=e^x$ inverse $\displaystyle x=e^y$
isolate
$\ln{x} = y$
the...
$\textsf{ Find the solution of the given initial value problem.}$
$$xy^\prime+y=e^x, \qquad y(1)=1$$
$$\begin{array}{lrll}
\textit{Divide thru with x}\\
&\displaystyle y' +\frac{1}{x}y
&\displaystyle=\,\frac{e^x}{x} &_{(1)}\\
\textit {Find u(x)}\\...
(Not homework, just curious)
Homework Statement
Is there an explicit way to write the x which solves nx = e^x for constant n?
Homework Equations
nx = e^x
Or equivalently,
x = ln(x) + m
The Attempt at a Solution
For any n we could just find x numerically, but is there an explicit expression...
Homework Statement
[/B]
Determine the value that A (assumed real) must have if the wavefunction is to be correctly normalised, i.e. the volume integral of |Ψ|2 over all space is equal to unity.
Homework Equations
Integration by parts
(I think?)
The Attempt at a Solution
So, I've managed...
Homework Statement
[/B]
$$\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx$$
or
http://www.HostMath.com/Show.aspx?Code=\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx
Homework Equations
the sinh identity, which is (e^x-e^-x)/2
The Attempt at a Solution
Tried...
Hello all,
I was just experimenting around with some derivatives and ended up coming across a function whose derivative is the same as the original function. I know that the derivative of e^x is e^x, but I didn't know it was possible for other functions to also follow this. I'm just wondering...
I learned that differential of e^x is same but what's so special about it? What makes is so special as it seems like a normal function to me other than the fact that e= sum of series of reciprocal of factorial numbers. What i want to ask is if e^x differential is e^x then do this rule apply to...
Hi. I derived a formula for the length of the graph of e^x in the first quadrant between x=a and x=b. It is:
sqrt(1+e^(2b)) - sqrt(1+e^(2a)) + (a-b) + log [(sqrt(1+e^(2b)) - 1) / (sqrt(1+e^(2a)) - 1)]
I think it works because it gave a value of approx 2.003 units for a=0 and b=1. For a=0 and...
lim_(h->0^-) (e^(x+h)/((x+h)^2-1)-e^(x+h)/(x^2-1))/h = -(2 e^x x)/(x^2-1)^2
I know how to differentiate the expression using the quotient rule; however, I want to use the limit definition of a derivative to practice it more.This desire to practice led me into a trap! Now I just can't simplify...
Greetings,
I have some questions about ln(x) and e^x graphs , with figuring out Domain , range and line of asymptote.
Q1) How can I know if this graph is ln(x) or e^x
(I thought it was e^x graph since there's no x-axis intercept , however the answer in marking scheme is:
Domain : xεR , x>-3...
Homework Statement
1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1)
Homework Equations
(dy/dx)(e^x) = e^x
(dy/dx)ln(e^-x) = -x ?
The Attempt at a Solution
First I believe I put ln on each set of terms (Though I don't know...
The problem I have is
∫(2e^x)/(e^x+e^-x)dx
I cannot seem to get to the correct result, ln(e^2x + 1). I always have (e^2x)ln(e^2x + 1). What do I need to do, to get rid of the e^2x.
Any help would be greatly appreciated.
Differentiation by first principles is as followed:
$$y'=\lim_{h\rightarrow 0}\dfrac {f\left( x+h\right) -f\left( x\right) }{h}$$
So, assuming that ##y= e^{x},## can we prove, using first principle, that:
$$\dfrac{dy}{dx}\left( e^{x}\right) =e^x$$
Or is there other methods that are...
Reflect e^x at y=2
If you reflect e^x at y=0. You just turn e^x negative and it reflects
On my homework it says the correct answer is 4 - e^x
But it makes more sense for me to say its 2 - e^x.
Can someone explain?
Hey Guys!
I'm stick on this question,
I know that the summation of n=0 to infinity for x^n/n! equals e^x
In the question it wants me to come up with a corresponding summation for the function x^2(e^(3x^2) - 1) … I don't know how to manipulate it to get the -1. I know i can substitute x for...
Hi, I need this for part of a math project I'm working on. So as most of us know, e^x, when differentiated, is equal to e^x, hence it is its own derivative.
Now, I could prove this quite easily with the Chain Rule, but my teacher said that this is predicated on too many assumptions (not sure...
Homework Statement
∫e^x lnx dx
I don't really know how to solve it.
The Attempt at a Solution
This is what i have:
∫e^x lnxdx = lnx e^x - ∫e^x/x dx
And my prof says its wrong, that i can go further with it with some method they discussed about ( i missed it :(
I'm wondering how to integrate something of the form 2^x * e^x.
I tried using integration by parts, ∫u dv = v*u - ∫v du. The problem is no matter which term I choose to integrate and which to differentiate, the final solution still has an integral with a product of terms in the form (2^x)...
Homework Statement
Consider the graph y = e^x
Homework Equations
Make an equation that results in reflecting about the line y = 2
The Attempt at a Solution
I came up with 2 - e^x
And when I put it in my calculator it makes sense.
Just -e^x is reflected among y = 0
so if you...
I don't need an answer, I'm just stumped as to how to properly turn these into point slope form to find the tangent, can anyone guide me through this? Help would be greatly appreciated.
I believe I understand the formulas that are used to solve problems such as these.
It starts by finding the...
Hi folks,
If $e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}$
what do I evaluate $x$ at?
How does the sigma notation tell me what to do with $x$?
$$e^x= \Sigma_{k=0}^\infty \frac{x^k}{k!}\ = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} ... \text {ad infinitum}$$
Sorry, I just realized my error...
1.
a. Find Taylor series generated by ex2 centered at 0.
b. Express ∫ex2dx as a Taylor series.
2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series:
ex: 1 + x + x2/2! + x3/3! + ...
ex2: 1 + x2 + x4/2! + x6/3! + ...
For part b...
Homework Statement
Explain why ∫(1+(1/x2)1/2dx over [1,e] = ∫(1+e2x)1/2dx over [0,1]
The Attempt at a Solution
The two original functions are ln(x) and ex and are both symmetrical about the line y = x. If I take either of the functions and translate it over the line y = x the two...
Homework Statement
The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation:
f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)}
where h is Planck's constant, c is the speed of light, and...
Homework Statement
Find the derivative of these e^x functions [paraphrased]
Homework Equations
x^{2} e^{x}
The Attempt at a Solution
2x e^{x}
This is how I believe it to be correct from my understanding. It does feel wrong, with the answers agreeing with the feeling.
Homework Statement
Would you give me a clue as to how, limit as z approaches infinity,
[[1 + (1/z)]^z]^(1/3) = e^(1/3)Homework Equations
The Attempt at a Solution
May sound trivial but I don't find it trivial from the things I am given as a definition of the exponential function:
The exponential function satisfies f(a+b)=f(a)f(b), is the inverse to the ln and is restricted by the condition:
exp(x)≥1+x
I can't see how I can from this only proove...
Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :)
1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum.
(Please ignore the __ behind the n's. The format is not kept without it...)
_____n
2. lim Ʃ f(xi)(Δx)dx
x→∞ i=1
Δx= (b-a)/n = 8-1/n = 7/n
xi= 1 + 7i/n
____n
lim Ʃ...
Homework Statement
What is the limit of e^x when x approaches zero from negative side
Homework Equations
The Attempt at a Solution
Taylor series? Then the answer is put all x= 0 , and the answer is 1, but why the question ask from negative side??
Thank you very much
I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course:
\sum_{n=0}^\infty \frac{x^n}{n!}
would change the equation.
How would changing this to:
\sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)}
change the equation? Would it still be...
Homework Statement
Let g(x) = (e^x - e^-x)/2. Find g^-1(x) and show (by manual computation) that g(g^-1(x)) = x.
Homework Equations
g(x) = (e^x - e^-x)/2
The Attempt at a Solution
I get the inverse = ln[ (2x + sqrt(4x^2 + 4) ) / 2 ]
How do I proceed?
Homework Statement
Integrate
\int e2xsin(ln(x)) dx
Homework Equations
Well, I'm not exactly sure which rule to apply here, but I'm going to assume integration by parts:
\int u \frac{dv}{dx} = uv - \int v \frac{du}{dx}
The Attempt at a Solution
I'm a little thrown off because...
Homework Statement
I thought sometimes the integral of e^x is xe^x. Under what circumstances is the integral of e^x = xe^x? I think it has something to do with u substitution.
Homework Statement
I'm just trying to understand a few things about the Maclaurin series for e^x...
So, in one case, if you have a series from 1 to infinity of [(-1)^n * 3^n ]/n!, how is it that it is equal to e^-3 - 1? I understand the e^-3 part, as -3 is simply our x value from the...
I was searching for the proof of \frac{d}{dx} e^x = e^x.
and I found one in yahoo knowledge saying that
\frac{d}{dx} e^x = \lim_{Δx\to 0} \frac {e^x(e^{Δx}-1)} {Δx}
= \lim_{Δx\to 0} \frac {e^x [\lim_{n\to\infty} (1+ \frac{1}{n})^{n(Δx)}-1]} {Δx}
Let h= \frac {1}{n} , So that n = \frac...
I'm working on a take home exam in my Calculus 2 class. The exam is completely done except for one problem and I desperately need help. I've put so much time into this one problem that I'm ready to just miss it and take the hit.
Homework Statement
Indefinite Integral...
Hey,
I have to show:
Should I try to find the Fourier series from -2pi to 2pi? I have tried this already but I can't seem to get rid of the cos(nx/2) and sin(nx/2) to turn them into just sin(nx) and cos(nx) and the denominator stays as (n^2+4 instead of n^2+1.
Any suggestions would be...
Homework Statement
which increases faster e^x or x^e ?Homework Equations
The Attempt at a Solution
My attempt was taking the log of both, assuming it doesn't change anything (is this assumption correct?)
x*ln(e) ------------------------ e*ln(x)
now I took the derivative
1...
Homework Statement
In the last couple threads, it has become apparent that I need to organize my understanding of some of the derivative rules, specifically as they relate to exponential functions, such as e^x.
So I wrote out a couple possible ways of evaluating e^x. Could you tell...
Homework Statement
Find the indefinite integral (preferably using u-substitution):
∫(ex-e-x)2 dx
Homework Equations
N/A
The Attempt at a Solution
To be honest, I'm slightly confused as to which path I'm supposed to take with this, especially since I'm not sure what I should be...
Homework Statement
Evaluate limx->0 (e^x - 1- x - (x^2/2))/x^3
The Attempt at a Solution
I can't remember how to solve this limit. Do I need to evaluate each part seperately? I plugged in the 0 to find that the limit does exist. I just can't seem to figure out what to do next.
So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3?
I hope my question makes sense... thanks
Hello,
I have some background in complex analysis (a very minimal amount) but I did come up with a rather odd question.
Given a polynomial a + bx + cx^2 + dx^3... nx^n
There exists n or fewer solutions to the equation that each have a multiplicity of 1 to n.
Given that information...
Homework Statement
Integral of (ln(e^x + 1))^(1/3) / (e^x + 1)
See http://www2.wolframalpha.com/input/?i=integral+of+%28ln%28e**x+%2B+1%29%29**%281%2F3%29%2F%28e**x+%2B+1%29"Homework Equations
N/A
The Attempt at a Solution
First, substitute k = ln(e^x + 1) dk = dx(e^x)/(e^x + 1)
Then, used...
Homework Statement
I have to determine if [e][/x] is a convex function. If it is then show proof. I know its a convex function by looking at the graph, Iam stuck at prooving it mathematically though.
Homework Equations
The function is f(x)=e^x.
The Attempt at a Solution
I am...