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.
View More On Wikipedia.org
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.
View More On Wikipedia.org
-
I Is the exponential function, the only function where y'=y?
Hello,I was wondering. Is the exponential function, the only function where ##y'=y##. I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point. Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...- Phylosopher
- Thread
- Replies: 11
- Forum: Differential Equations
-
I What is the exponential decay equation for a bouncing ball?
This is not a physics question. Each time a ball bounces it will bounce to, let's say 75% of its previous height. (I am not interested in the time, energy or velocity, of the ball.) So if we drop it from 100 cm it will bounce back up to 75 cm, and on the next bounce it goes up to 56.25 cm and...- wolf1728
- Thread
- Replies: 6
- Forum: General Math
-
S
Sum of (n+1) terms in exponential series
Homework Statement S = 1+ x/1! +x2/2! +x3/3! +...+xn/n! To find S in simple terms. Homework Equations None The Attempt at a Solution I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.- ssd
- Thread
-
- Tags
- Exponential Series Sum Terms
- Replies: 7
- Forum: Precalculus Mathematics Homework Help
-
A How to Convert a Complex Logarithm to a Complex Exponential
Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...- CalcExplorer
- Thread
- Replies: 5
- Forum: Calculus
-
I Decomposing a Certain Exponential Integral
There is nothing wrong with the well known $$e^{i\theta}=\cos\theta+i\sin\theta$$ for real ## \theta## but what about $$\int_{-\infty}^\infty~e^{i\theta(p)}\mathrm{d}p=\int_{-\infty}^\infty~\cos\theta(p)\mathrm{d}p+i\int_{-\infty}^\infty~\sin\theta(p)\mathrm{d}p$$ I have been trying to use... -
Y
MHB Exponential Functions Problem, find k and a in f(x)=ka^(−x)
Hello all, I am trying to solve the following problem: In the given graph, we see the function: \[f(x)=ka^{-x} , x\geq 0\] 1) Find k and a 2) Find x1 3) Show that an increase of 2 units in x brings a 50% reduction in the value of the function f. I have tried solving it, but taking two...- Yankel
- Thread
-
- Tags
- Exponential Functions
- Replies: 5
- Forum: General Math
-
Bi-exponential function fitting in Origin Lab SW
I can't understand the exponential function fit for this set of data works well: ExpDec2 exponential function fit 0 3,04 10 2,77 20 2,52 30 2,27 40 2,09 50 1,92 60 1,75 70 1,62 80 1,51 90 1,43 100 1,36 110 1,29 120 1,24 130 1,19 140 1,14 150 1,09 160 1,05 170 1,02 180 0,99 190 0,97 200 0,95 210...- Jakub
- Thread
- Replies: 1
- Forum: MATLAB, Maple, Mathematica, LaTeX
-
E
I Probability density of an exponential probability function
I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this? P(r) = ∫p(r) dr = exp(-μr) p(r) = dP(r)/(4πr²dr) One way I tried to... -
B Exponential Growth: Modeling growth in months vs years
Say I have a function that represents the population growth of a certain country that can be written as ##f\left(x\right)=1.25\left(1.012\right)^t##, where t is in years. I can graph this function and it will look a certain way exponentially. I've looked at a ton of examples, and they're all...- opus
- Thread
-
- Tags
- Exponential Growth Modeling Years
- Replies: 3
- Forum: General Math
-
Fourier transform of exponential function
1. The problem statement, all variables, and given/known data Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...- Peter Alexander
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
-
F
MHB Exponential Equations solve 27^x=1/√3
I'm taking an online class and I was doing some very simple exponential equations when this was thrown at me, and I have no clue how to solve it. 27^x=1/√3- Fright535
- Thread
-
- Tags
- Exponential
- Replies: 2
- Forum: General Math
-
A
Exponential decay of a pendulum oscillation amplitude
Homework Statement I found an answer on the internet for this problem, but I'm not sure on one of the steps. The solution says, "Take ln of both sides to get rid of Ae. If we do that, then the right side will be ln(Ae^t/T). I don't see how using ln will get rid of Ae? Homework Equations Refer...- asilvester635
- Thread
- Replies: 1
- Forum: Introductory Physics Homework Help
-
D
MHB Complex Exponential Function
Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f). I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?- Deanmark
- Thread
- Replies: 2
- Forum: Topology and Analysis
-
M
MHB Exponential distribution - inequality
Hey! :o We consider the exponential distribution. I want to show that $$\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right )\geq \frac{\lambda^4-1}{\lambda^4}$$ I have shown so far that \begin{align*}\mathbb{P}\left (\left |X-\frac{1}{\lambda}\right |\leq \lambda \right...- mathmari
- Thread
- Replies: 4
- Forum: Set Theory, Logic, Probability, Statistics
-
R
B Exponential decay convolved with Gaussian
Hello all, I have a data which look like reversed exponentially modified Gaussian (EMG) function and interested to fit the data with with reversed EMG function. After searching on internet I found the EMG function, which is given below...- Rajini
- Thread
- Replies: 5
- Forum: Other Physics Topics
-
M
MHB Proving Limits of Exponential Series at Infinity
Hey! :o I want to show that $\displaystyle{\lim_{x\rightarrow \infty}\frac{e^x}{x^{\alpha}}=\infty}$ and $\displaystyle{\lim_{x\rightarrow \infty}x^{\alpha}e^{-x}=0}$ using the exponential series (for a fixed $\alpha\in \mathbb{R}$). I have done the following: $$\lim_{x\rightarrow...- mathmari
- Thread
-
- Tags
- Exponential Limits Series
- Replies: 7
- Forum: Topology and Analysis
-
Determinant of exponential matrix
Homework Statement Homework EquationsThe Attempt at a Solution [/B] Det( ## e^A ## ) = ## e^{(trace A)} ## ## trace(A) = trace( SAS^{-1}) = 0 ## as trace is similiarity invariant. Det( ## e^A ## ) = 1 The answer is option (a). Is this correct? But in the question, it is not...- Pushoam
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
-
J
MHB Integral of Rational Exponential
Hi, I'm new to this forum. This semester I took Calculus I and just took the final yesterday. There were a few questions that were unexpected that I didn't know how to handle. This integral has got me stumped.\int_{0}^{1} e^{x}/(1 + e^{2x}) \,dx The techniques I know at this point include u...- Jarhead1
- Thread
-
- Tags
- Exponential Integral Rational
- Replies: 5
- Forum: Calculus
-
M
Getting the Horizontal Asymptotes
Homework Statement Homework EquationsThe Attempt at a Solution I understand there is no vertical asymptotes and can usually get the horizontal ,but can't understand with the exponential.- MatthewR
- Thread
- Replies: 4
- Forum: Calculus and Beyond Homework Help
-
F
Simplification of a complex exponential
Homework Statement Is there a way to simplify the following expression? ##[cos(\frac {n \pi} 2) - j sin(\frac {n \pi} 2)] + [cos(\frac {3n \pi} 2) - j sin(\frac {3n \pi} 2)]## Homework Equations ##e^{jx} = cos(x) + j sin(x)## The Attempt at a Solution ##cos(\frac {n \pi} 2)## and...- fonz
- Thread
- Replies: 9
- Forum: Precalculus Mathematics Homework Help
-
M
MHB Can we somehow modify the Lagrange form to get a tighter bound? (Curious)
Hey! :o I am looking at the following: Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$. Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...- mathmari
- Thread
- Replies: 8
- Forum: Topology and Analysis
-
M
MHB Normal and exponential distribution
Hey! :o I am looking at the following: 1) A machine produces $100$ gram chocolate. Due to random influences, not all bars are equally heavy. From a long series of observations it is known that the mass X of a chocolate is distributed normally with parameters $\mu = 100$g and $\sigma = 2.0$g...- mathmari
- Thread
- Replies: 5
- Forum: Set Theory, Logic, Probability, Statistics
-
A
Solution of an exponential equation
Homework Statement I am trying to solve an equation. 128^b - 127^b = 147.058. Homework EquationsThe Attempt at a Solution I have tried numerical methods like Bisection method and Newton-Raphson, but I need analytical solution. Thank you.- Ali Baig
- Thread
-
- Tags
- Exponential
- Replies: 2
- Forum: Precalculus Mathematics Homework Help
-
Trace of the Exponential of a Square Matrix
Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...- VSayantan
- Thread
-
- Tags
- Exponential Inverse Matrix Square Trace
- Replies: 5
- Forum: Introductory Physics Homework Help
-
Coefficients for an exponential Fourier Series
I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...- ElPimiento
- Thread
- Replies: 3
- Forum: Calculus and Beyond Homework Help
-
B
Matrix Exponential Homework: Eigenvalues & Eigenvectors
Homework Statement Show that if ##λ##and ##V ## are a pair of eigenvalue and eigenvector for matrix A, $$e^Av=e^λv$$ Homework Equations ##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n## The Attempt at a Solution I don't know where to start.- BearY
- Thread
-
- Tags
- Exponential Matrix
- Replies: 3
- Forum: Calculus and Beyond Homework Help
-
J
Exponential forcing function to an RL circuit -- poles & zeros
For reference: Engineering Circuit Analysis, Hayt & Kemmerly, 4th ed, 1986, page 345. Given a series RL circuit, the phasor current is I(s) = Vm/(R+sigma L), where S = sigma, w (omega) = 0. Thus we are dealing with only a exponential forcing function. Obviously I(s) goes to zero as sigma...- Joseph M. Zias
- Thread
- Replies: 12
- Forum: Electrical Engineering
-
B
Solutions to Equations Involving Exponential and Trig Functions
Homework Statement Show that ##e^x = x## does not have any solutions, and show that ##\sec x = e^{-x^2}## has only one solution. Homework EquationsThe Attempt at a Solution Here is my proof of the first proposition: Since ##e^x## is concave up on ##\Bbb{R}##, it must lie above all of its...- Bashyboy
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
-
T
B What are the equations for the quadratic and power laws in unstable systems?
http://www.nat.vu.nl/~tvisser/nonexponential.pdf I always thought that an unstable system will decay exponentially but I recently learned that it obeys the quadratic law in the short time and power law in the long time. Can somebody tell me the equation that governs the quadratic law and power law?- Trollfaz
- Thread
-
- Tags
- deviation Exponential Law
- Replies: 7
- Forum: High Energy, Nuclear, Particle Physics
-
J
I Proving Matrix Exponential Theorem: Unipotent & Nilpotent
Hi, I'm kind of stuck with this theorem stating that: if A is an unipotent matrix, then exp(log A) = A and also if X is nilpotent then log(exp X) = X Does anyone know any good approaches to prove this? I know that for unipotent A, logA will be nilpotent and that for nilpotent X, exp(X)...- Josh1079
- Thread
-
- Tags
- Exponential Matrix
- Replies: 1
- Forum: Linear and Abstract Algebra
-
O
Exponential Distribution, Mean, and Lamda confusion
Homework Statement Accidents at a busy intersection follow a Poisson distribution with three accidents expected in a week. What is the probability that at least 10 days pass between accidents? Homework Equations F(X) = 1- e-λx μ = 1/λ The Attempt at a Solution Let x = amount of time between...- Of Mike and Men
- Thread
- Replies: 6
- Forum: Calculus and Beyond Homework Help
-
J
Analytic Integration of Function Containing the Exponential of an Exponential
Homework Statement Can this function be integrated analytically? ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),## where ##a##, ##b## and ##L## are some real positive...- junt
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
-
E
B Logarithmic growth vs exponential growth
From the book Calculus made easy: "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call a logarithmic rate of growing." From Wikipedia: "Exponential growth is feasible when the growth rate of the value of a mathematical function is...- Elias Waranoi
- Thread
- Replies: 4
- Forum: General Math
-
S
MHB Exponential Equation: If X is one more than twice Y, what is the value of X?
The square of X is equal to 4 times the square of Y. If X is one more than twice Y, what is the value of X?- ScrewedIre
- Thread
-
- Tags
- Exponential Value
- Replies: 2
- Forum: General Math
-
M
MHB How can I solve more complex exponential equations?
I can solve equations like 4^(x) = 16 or 5^(x + 1) = 25. However, there are exponential equations that a bit more involved. The following equation has two exponentials on the left side. Solve for x. 5^(x - 2) + 8^(x) = 200- mathdad
- Thread
-
- Tags
- Exponential
- Replies: 2
- Forum: General Math
-
Exponential and logarithmic Equation Problems
Homework Statement Evaluate each of the following expressions without using a calculator. 1) log216√8Solve for the unknown value in each of the following equations without using a calculator. 2) 3(x+4)−5(3x)=684 3) 7(42x)=28(4x) Homework Equations Exponent law for multiplication The...- onemic
- Thread
- Replies: 12
- Forum: Precalculus Mathematics Homework Help
-
C
I Exponential Distribution Question
Hi all, Can anyone teach me this problem ? Thanks The life of a tiger is exponentially distributed with a mean of 15 years.If a tiger is 10 years old, what is the expected remaining life of the tiger? A 5 years B 10 years C 15 years D Longer than 15 years- Caution
- Thread
- Replies: 3
- Forum: Set Theory, Logic, Probability, Statistics
-
I Exponential decay: I need an expression for N at time t
I have a large quantity N, which starts off equal to a determinable value N0. Over a short time ∆t, the value of N changes by -∆t*(B*N - C) where B and C are determinable constants. Am I correct in thinking I can turn this into: dN/dt = -(B*N - C) How do I get this into a formula for N at...- Jehannum
- Thread
- Replies: 6
- Forum: General Math
-
Exponential Forcing Differential Equation
Homework Statement Solve ## \frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}## and find the amplitude of the resulting oscillation when ##t \rightarrow \infty ## given ##y=dy/dt=0## at ##t=0##. Homework EquationsThe Attempt at a Solution I have found the homogenious solution to be: ##y_h = A\cos\omega...- Alettix
- Thread
- Replies: 1
- Forum: Calculus and Beyond Homework Help
-
T
I Why Does Unstable Particle Decay Follow an Exponential Curve?
Given that an unstable particle has a constant probability of decaying per unit time, why is it said that its chance of surviving falls exponentially?- Trollfaz
- Thread
- Replies: 2
- Forum: High Energy, Nuclear, Particle Physics
-
How to prove that two exponential terms are congruent to 7?
Homework Statement "Prove: ##∀n∈ℕ,7|[3^{4n+1}-5^{2n-1}]##" Homework EquationsThe Attempt at a Solution (1) "We take the trivial case: ##n=1##, and notice that ##3^5-5=238## and ##7|238## because ##7⋅(34)=238##." (2) "Now let ##n=k## for some ##1<k∈ℕ##. Then we assume that...- Eclair_de_XII
- Thread
-
- Tags
- Exponential Terms
- Replies: 4
- Forum: Calculus and Beyond Homework Help
-
M
MHB How to change the subject when exponential is involved
It's been a long time since I've worried about this - but could someone help me make Pr(x) the subject (I can't remember if it's possible, if it's not, I'd love a brief explanation): T = S [(1-Pr(x))^N] + Pr(x) Thanks in advance!- MWD02
- Thread
-
- Tags
- Change Exponential
- Replies: 4
- Forum: General Math
-
P
A Closed form for series over Exponential Integral
Is there a closed form for the constant given by: $$\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(2))}{n}$$ (Where Ei is the exponential integral)? Could we generalize it for: $$I(k)=\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(k))}{n}$$ ? My try: As it is given that k will be a positive integer, I have... -
Z
Exponential of hermitian matrix
Homework Statement Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential. a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are...- ZCOR
- Thread
-
- Tags
- Exponential Hermitian Matrix
- Replies: 4
- Forum: Introductory Physics Homework Help
-
F
MHB Find x in Exponential Equation: 2^x=8x
Find x, if 2^x =8x.- fasakintitus
- Thread
-
- Tags
- Exponential
- Replies: 2
- Forum: General Math
-
How can we prove ##e^{ln x}= x## and ##e^-{ln(x+1)}= \frac 1 {x+1}##?
Homework Statement they say 1. ##e^{ln x}= x ## and 2.##e^-{ln(x+1)}= \frac 1 {x+1}## how can we prove this ##e^{ln x}= x ## and also ##e^-{ln(x+1)}= \frac 1 {x+1}##? Homework EquationsThe Attempt at a Solution let ## ln x = a## then ##e^a= x, ## a ln e= x,## →a= x, where ## ln x= x- chwala
- Thread
-
- Tags
- Exponential Proof
- Replies: 5
- Forum: Calculus and Beyond Homework Help
-
Y
MHB Derivative and Limit of an Exponential Function
Hello all, I have a complicated function: \[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\] I need to find it's derivative and it's limit when x goes to infinity. As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct? How should I approach... -
D
Exponential curve fit using Apache Commons Math
Homework Statement I have the following data which I would like to model using an exponential function of the form y = A + Becx. Using wolfram mathematica, solving for these coefficients was computed easily using the findfit function. I was tasked however to implement this using java and have...- Duo Who Ow
- Thread
- Replies: 9
- Forum: Calculus and Beyond Homework Help
-
B Simplifying an exponential with a square root
I have the expression ##e^{\frac{1}{2} \log|2x-1|}##. I am tempted to just say that this is equal to ##\sqrt{2x-1}## and be done with it. However, I am not sure how to justify this, since it seems that then the domains of the two functions would be different, since the latter would be all real...- Mr Davis 97
- Thread
- Replies: 4
- Forum: General Math
-
Derivative of an Nasty Exponential Function
Homework Statement The question is given just like this: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds)## = ? I assume they want me to find the derivative of the whole thing. Homework EquationsThe Attempt at a Solution I'm thinking the first step is: ##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x...- Drakkith
- Thread
- Replies: 7
- Forum: Calculus and Beyond Homework Help
