What is Exponential: Definition and 1000 Discussions
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.
Homework Statement
Find the exponential function that passes through the points (2, 1) and (5, 7).
Homework Equations
y= Ce^ (kt)
The Attempt at a Solution
I can only get as far as substituting in 2 for t and 1 for y, then I'm completely stuck.
Homework Statement
A curve passes through the point (0,6) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
Homework Equations
Y=Yoe^kx is a solution of (dy/dx)=ky, where k is constant
The Attempt...
In general what do you do when some kind of data is missing from some data that follows exponential distribution. For example, say 3 observations are made by an instrument where x1=5, x2=3, but for x3 the instrument can not give a specific answer because it can't measure past 10. So the only...
Homework Statement
X is exponentially distributed. 3 observations are made by an instrument that reports x1=5, x2=3, but x3 is too large for the instrument to measure and it reports only that x3 > 20 . (The largest value the instrument can measure is 10)
a)What is the likelihood function...
I am in a trouble to work out the sum, i.e.
"sum of exponential of[-a^2.n^2]" , where, n is an integer... and sum over 1 to infinity, and "a" is "not small...", I've calculate two asymptotic behaviours , i,e when "a<<1" , in this case I've integrated it and have got, sqrt(pi)/2a-1, and in the...
Homework Statement
Solve the equation:
(x+xy^{2})dx + e^{x^{2}}y dy =0
Homework Equations
n/a
The Attempt at a Solution
i) xdx(1+y^{2}) = -e^{x^{2}}y dy
(multiply both sides by 2 to prepare for integration:
ii) \frac{-2xdx}{e^{x^{2}}}= \frac{2y dy}{1+y^{2}}
integrate and get:
iii)...
¡¡get matrix exponential Please!
Homework Statement
I have a exam and i don't know how get matrix exponential:
| 2*t t|
| 3*t -t|
it is a 2x2 matrix.
where 't' is not a constant ,it is a variable
somehere could help me,please.
Homework Equations
The Attempt at a Solution...
Homework Statement
This isn't a problem, I'm just obsessed with analyzing a trig/exp equation algebraically instead of with a calculator.
8*sin(t) - 16*cos(t) = 9*exp(-t/2)
Homework Equations
See part (1.) above...
The Attempt at a Solution
I tried converting the exp part...
Can a convenient value for a be found without resorting to substituting numerical values for h in this expression?
EDIT: I am trying to indicate, "as h approaches zero".
EDIT: neither of the formattings worked; hopefully someone understands what I am asking...
Homework Statement
I'm new to integration, and was attempting the integral of ex with respect to x.
Homework Equations
\intex dx
The Attempt at a Solution
Should I keep it in the format y=ex so that I use this in the calculation of the area? Or should I convert to ln y? I got stuck...
Hi!
I want to know how to intergrat e^(-(x^2)/2) . Someone told me to use the intergration of e^x= 1+ x + (x^2) /2!+ ...(X^n)/n! ... is that sounds right?? and if it is i got pi as an answer for e^(-(x^2)/2) , is that the right answer?
Theorem
Let A be a square matrix nXn then exp(At) can be written as
exp(At)=\alpha_{n-1}A^{n-1}t^{n-1} + \alpha_{n-2}A^{n-2}t^{n-2} + ... + \alpha_1At + \alpha_0 I
where \alpha_0 , \alpha_1 , ... , \alpha_{n-1} are functions of t.
Let define
r(\lambda)=\alpha_{n-1}\lambda^{n-1} +...
Hi,
I'm trying to solve the following:
f(x) = \int^\infty_{-\infty}ce^{yx-y^2/2} dy
where c is a constant
My only idea thus far was that since it is an even function, the expression can be simplified to:
= 2c\int^\infty_0 e^{y(x-{1/2}y)} dy
but I'm stuck here.
Anyone know how to do...
Homework Statement
Show that (ez)*=ez*
note: * is the conjugate
Homework Equations
The Attempt at a Solution
So I wasn't sure what form to put this in...either in exponential reiѲ or cos Ѳ + isin Ѳ...Either way, I think I'm just making it too easy...
Please help!
Homework Statement
What is the derivative of ex3? also what is the derivative of (ln1/x)2Homework Equations
The Attempt at a Solution
is it 3x2e3x2?
2(ln1/x)(x)?
Hello,
In a paper, the authors defined an exponential Random Variable (RV) as X_1 \mbox{~EXP}(\lambda) where \lambda is the hazard rate. What will be the distribution of this RV:
f_{X_1}(x)=\lambda e^{-\lambda x} or
f_{X_1}(x)=\frac{1}{\lambda} e^{-\frac{x}{\lambda}}
Thanks in advance.
Homework Statement
The problem asks to graph tangent lines to the given function y = (ln(x)/x, and gives the points (1,0) and (e, 1/e)
Homework Equations
The Attempt at a Solution
I got the answer by taking the derivative and finding the slope, and at the point (e,1/e) the...
Homework Statement
Hello. I'm afraid it's quite a simple and unexciting problem I have. Basically, I can't remember how to rearrange the equation for an exponential curve:
The capacitor has a capacitance of 0.63 mF and the resistance in the discharge circuit is
2.4 kΩ.
(i) Calculate...
1. It is estimated that x years from now the value of an acre of farmland will be increasing at the rate of \frac{0.4x^3}{sqrt(0.2x^4+8000)} dollars per year. If the land is worth 500 per acre, how much will it be worth in 10 years?
2. Use integral
Since the the function of money...
Homework Statement
The distance between major cracks in the highway follows an exponential distribution with a mean of 5 miles. What is the probability that there are two major cracks in a 10 mile stretch of the highway?
Homework Equations
exponential dist: f(x) = Le(-Lx) where L...
I have seen the following identity used.
Exp[iw/2]-Exp[iw/2]=Exp[iw]-1
I can't find this in any book and I can't prove it myself.
The left side equals 2isin(w/2)
The right side equals cos(w)+isin(w)-1
On the face of it, that seems to make the identity absurd
How can one go about proving...
Homework Statement
Consider a horizontal slab of air whose thickness(height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for Dp/dz, the variation of pressure...
Hello Physics Forums,
I sometimes come across statements like this (from http://beige.ucs.indiana.edu/B673/node62.html" ):
Now assume a solution of the following form: x(t) = Aeat
What is the motivation/background to make such an assumption? In this case the equation to be solved is x''...
Homework Statement
Derive (from the equation of motion of a neutral gas and an assumption of constant gravitational field) an expression showing why the concentrations of neutral molecules decrease approximately exponentially with increasing altitude, and why the concentration of atomic oxygen...
Solve for real-valued x, e-aX + e-bX = 1, where a and b are arbitrary known constants > 0.
For example, e-48.12/50 + e-48.12/100 ~ 1.00
In this case X = 48.12 (to two decimals), a = 1/50 and b = 1/100.
For any specific values of a and b, a computational solution can easily be determined...
Homework Statement
Viruses are reproducing exponentially, while the body eliminates the viruses.
The elimination rate is constant, 50000 per hour. I decided to take down on the minute level, so it would be 50000/60.
Pinitial is 10^6
k is ln(1,6)/240, since the growth rate is 160% in 4...
Homework Statement
Viruses are reproducing with rate of k ,in t minutes, the function is:
f(x) = Po x e^(kt)
However there is an elimination rate of a viruses per minute.
Homework Equations
The Attempt at a Solution
We can't say that the new function will be:
f(x) = Po x e^(kt)...
It is known that
\[\sum\limits_{k = 0}^\infty {\frac{{N^k }}
{{k!}}} = e^N
\]
My question is
\[\sum\limits_{k = 0}^M {\frac{{N^k }}
{{k!}}} = ?
\]
where $M\leq N$ an integer.
This is not an homework
Homework Statement
e^(2x+9) - 4e^x -5 = 0
Homework Equations
The Attempt at a Solution
I changed this into: e^(2x+9) =4e^x + 5
I took logarithm of both sides:
2x+9=ln(4e^x +5)
but i don't know what to do with the right hand side.
what will be the easiest way to solve this...
Homework Statement
A=\begin{bmatrix}-1 & 2 & 0\\-2 & -1 & 0\\ 0 & 0 & -3\end{bmatrix}
Use matrix A and compute exp(tA) explicitly.
Homework Equations
I am having trouble figuring out how to start this. I know how to look at each component of matrix A and to use exp(tA)=I+tA...
Homework Statement
Integrate the following equation for average energy from -infinity to infinity
\int(c*x^4)*(e^(-c*x^4)/KT)dx
Homework Equations
c, K, T are constants
\int(e^(-c*x^4/KT)) = (KT/c)^(1/4)*(2\Gamma(5/4))
The Attempt at a Solution
I tried using integration by parts...
Homework Statement
let be A_{i,j} a Hermitian Matrix with only real values then
\int_{V} dV e^{iA_{j,k}x^{j}x^{k}}= \delta (DetA) (2\pi)^{n} (1)
Homework Equations
\int_{V} dV e^{iA_{j,k}x^{j}x^{k}} = \delta (DetA) (2\pi)^{n}
The Attempt at a Solution
the idea is that...
Need help solving this.
\int^{\infty}_{-\infty} e^{\frac{-x^{2}}{\sigma^{2}}} e^{-ikx}dx
That's the integral of the product of the exponentials , couldn't get latex to make it look right.
Supposedly usefull information(I can't see how);
\int^{\infty}_{-\infty}...
Its true that if you integrate an exponential function from some time t0 to infinity it will converge to a finite value.
However, is the same true if it is multiplied by say t, t^2, t^3,t^n.
i.e. t*exp(-t) for example.
the exp is decaying to zero faaster than t is, so it goes to zero...
Homework Statement
Suppose two exponential functions, f(x) and g(x) are symmetric with respect to x = 2.
f(x)=a^{bx-1}
g(x)=a^{1-bx}
Prove f(2) = g(2)
Homework Equations
The Attempt at a Solution
This isn't actually a problem. This is a property used to solve a different...
Is the series
\sum_{n=0}^{\infty}\frac{z^n}{n!}
uniformly convergent for all z in the complex plane? It is uniformly convergent for all z in any bounded set, but the complex plane is unbounded. My instinct is that it is NOT uniformly convergent for all z in C.
This is not homework.
Hi folks, hope somebody can help me understand this one please?
Gven an expression i = ia-ic = icorr(exp(n/Ba)-exp(-n/Bc)), we are told that if -n<<Bc then exp(-n/Bc) tends to 0 & the equation becomes i (is approx) = ia = icorr exp(n/Ba).
I find that exp(-n/Bc) tends towards 0 if I...
let be the exponential sum
S= \sum_{n=1}^{N}e( \frac{f(x)}{p})
e(x)= exp( 2i \pi x)
my conjecture is that since the complex exponential takes its maximum value '1' when x is equal to an integer then
Re(S)= \Pi (f,N) with \Pi (f,N) is the number of solutions on the interval...
Homework Statement
I am in the process of doing a physics problem with a differential equation that has the form:
y = Acos(kx) + Bsin(kx)
According to my notes, this can also be written as y =Aejkx + Be-jkx, unfortunately I just don't see how to write the original equation like that...
Homework Statement
Series k=1 to k=Infinity of:
k^2/k!
a) e
b) 2e
c) (1+e)(e-1)
d) e^2
e) Infinity
Homework Equations
e^x = Series from n=1 to n=Infinity of:
x^n / n!
The Attempt at a Solution
I was guessing this would end up being C but the answer is infact B. As...
Exponential Attenuation and Beta Particle Range (solved)
Homework Statement
I am given a beta emitter and its atomic mass, as well as the atomic mass of its daughter, which have a difference of 0.0034 amu. I am to determine the range of the particles in air.
Homework Equations
I am given...
Homework Statement
1) 5x+4-5x-1<24
2) 2x+2-1>=22x+3
3) 21-x-2x+1/2x-1<0
Homework Equations
At the 2nd I do like this:
2x+2-22x>=4 divide by 2
2x+1-22x-1>=2
x+1-2x+1>=1
-x>=-1 divide by (-1)
x=<1
but when i put 0 in x the inequation is incorrect, what I have mistake, please...
Homework Statement
Please , can anybody help me to do these and to explan to me?
I tried tried, but i can't do anything.
1. 4x-1 + 4x + 4x-1=84
2. 15 * 2x+1 + 15 * 2 x-1 =135
3. 8x + 18= 2* 27x
4. 3x-8*3x/2 +15=0
5. 5x - 53-x=20
Homework Equations
At the 5th i do...
Homework Statement
Howdy,
Given a matrix \left[\begin{array}{ccc}x_{11} & x_{12}\\x_{21} & x_{12}\end{array}\right]
Which has the exponential matrix e^{t\cdot a}
When given the eqn x'= Ax + b where b = \left[\begin{array}{c}b_1 \\ b_2\end{array}\right]
I know that had it only...
Suppose I have a sample X_1, ..., X_n of independently, identically distributed exponential random variables.
One result I deducted was that the ratio of any two of them (eg. X_1 / X_2) is independent of the sample average 1/n * \sum_{i=1}^{n} X_i.
(Aside: that ratio, as a random variable...
Hi,
can I do the following?
If I am asked to find as lim x -> inf of (3^n/2^n), can I do:
3^n = e^(n*ln3)
2^n = e^(n*ln2)
assume C = lim (3^n/2^n).
C = lim e^(n*ln3)/e^(n*ln2)
ln C = lim ln [(exp(n*ln3)/(exp(n*ln2)]
In C = lim ln(exp(n*ln3)) - ln(exp(n*ln2))
ln C = lim n*ln3 -...
Homework Statement
For each n \in \mathbb{N}, let f_n(x) = e^{nx} for x \in \mathbb{R}. Prove that f_1, ... , f_n are linearly independent vectors in {\cal F}(\mathbb{R}, \mathbb{R})
Homework Equations
The Attempt at a Solution
I know that the simple way to prove this for n=2...