I have a problem with an exponential function. I am wondering if an exact solution is possible, or if I have to write the solution as a logarithm of an unknown.
A formula says that E(z)=E(0)^{-kz}, where E is light intensity and z is depth in water. My objective is to find the constant k. I...
I am having trouble solving this problem. I'm not sure how to solve this problem... Assume X and Y are independent exponential random variables with means 1/x and 1/y, respectively. If Z=min(X,Y): Is Z exponentially distributed as well (if so, how do you know)? What is the expectation of Z...
Homework Statement
for x and y satisfying x-y=2 and 2^{x}-2^{y}=6
find 2^{x}+2^{y}.
The Attempt at a Solution
x-y=2 ; x=2+y
log_{2} 2^{x}-log_{2}2^{y}=log_{2}6
log 6/log2=2.585
x-y=2.585 but this violates the intitial condition x-y=2
where am i going wrong?
I feel like I'm doing everything correctly but my answer for (b) doesn't make any sense.
Assume that bacteria in aculture grows according to an exponential growth model. If the number of bacteria grows from 50 to 1000 in 12 hours:
a)How many bacteria will be present after 18 hours?
b)How...
hey I am doing some questions outta a txt book, i sort of understand complex numbers, like multiplying and dividing, ..
The question asks to rearrange for z,
e^(iz) = i - 1
im not sure what to do with the exponential function.
thanks for the help
Homework Statement
10(1 + e-x)-1=3
Homework Equations
The Attempt at a Solution
I'm supposed to solve for x, but I don't know how to go about this. I tried dividing the 3 by the 10, but after that I don't know what to do. I believe I should use ln on both side, but that's after...
Homework Statement
Show that
\overline{e^{i\theta}} = e^{-i\theta}
Homework Equations
The Attempt at a Solution
So I what's going through my mind is that the problem above is pretty much the same as saying \bar{z} = z^{-1}
Then to prove it is all I need to say is that...
Homework Statement
e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}}Homework Equations
The Attempt at a Solution
For some reason every I multiply (cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2) I am getting
(cos\theta_1 cos\theta_2 + sin\theta_1 sin\theta_2) + i(sin\theta_1...
Homework Statement
I was studying L'Hopital's rule and how to deal with indeterminate forms of the type 0^0.
It's not clear to me how lim e^f(x) = e^lim f(x).
In wikipedia http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" (under other indeterminate forms)
it says "It is valid...
Hello,
I am trying to find an analytic solution to the following:
\int_{-1}^{1}\exp(-p\sqrt{1-x^{2}}-qx)dx
where p,q > 0.
Does anyone have any ideas? Thanks.
Homework Statement
This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function
Homework Equations
e^x=1+x+(x^2)/2!+(x^3)/3!+...
The Attempt at a Solution
then how about the e^(-x)=...
Homework Statement
This topic is under linear system differential equation.Solve the system by using exponential method. Just want to ask the expansion of exponential function
Homework Equations
e^x=1+x+(x^2)/2!+(x^3)/3!+...
The Attempt at a Solution
then how about the e^(-1)=...
I haven't been taking math for 3 years so I have a question about the following:
is 14m6n2 the same as 2mn x 7m5n
basically asking this because I am not sure whether or not I can factor the terms out like this
if this information is insufficient I can post the whole problem. Thanks in advance.
Will anyone help me to find out the analytic expression
of the following 2^N\times2^N exponential?
exp[t(X\otimes X\otimes I\ldots\otimes I+I\otimes X\otimes X\otimes I\ldots\otimes I+\ldots+I\otimes I\otimes\ldots I \otimes X \otimes X+X\otimes I\ldots I\otimes X)],
where...
Homework Statement
I'm unable to integrate the following function with respect to x [-inf, +inf]:
Homework Equations
exp(y*x)*(1+exp(x))^(-n)dx
The Attempt at a Solution
I tried to expand the function by distributing the exponent (-n) across the rightmost product, but I don't think its...
it is suspected that cells in a sample are dividing so that the number of cells present at anyone time t (measured in seconds) is growing exponentially according to the relationship y = 64 x 2^2t. it would be hard to check this relationship accurately by plotting measurements of y against t, so...
Hey everyone, I'm new to using the Maxima software and I'm having some trouble. When I enter the following formula to be evaluated:
diff(1-exp^(-t/R*C),t);
I get the following output:
\frac{log\left( exp\right) \,C}{{exp}^{\frac{t\,C}{R}}\,R}
This doesn't look right, even if I...
Hello,
I've been solving a problem which forces me to answer the question: "Is there a boolean function with exponential number (in variable count) of prime implicants of the length n - 1?"
Anyway, during solving this problem I came to this point:
Is the following sum exponential in 2n...
I am trying to read through this paper on the standard model. The ideas seem straightforward enough, but as always, I'm tripping over the "physicist's math" it uses. I was wondering if I can get some clarification or general guidance...
J.J. Sakurai Modern Quantum Mechanics p. 74
It says,
[A,H] = 0;
H|a'> = Ea' |a'>
where H is the Hamiltonian A is any observable |a'> is eigenket of A
then,
exp ( -iHt/h)|a'> = exp (-iEa't/h)|a'>
where h is the reduced Planck's constant.
I want to know WHY ?
and besides, I would...
Hello,
Suppose that:
Z=X_1+X_2+X_1X_2
where X_i for i=1 and 2 are independent and identically distribuited exponential RVs.
can we find the PDF of Z?
Regards
As a kid, I remember my father saying "there's a small chance that an electron in your body is on the moon" Well, today I decided to calculate the odds. Among the assumptions I made to make math easier.
*Ground state wave function of Hydrogen
*the moon is a cube of sides 2r. where r is...
Homework Statement
An investment pays 8% interest, compounded annually.
a) Write an equation that expresses the amount, A, of the investment as a function of time, t, in years.
b) Determine how long it will take for this investment to 1. Double in value, and 2. Triple in value.
c) Determine...
Homework Statement
The measure of effectiveness of a sleep deprived student studying for a test on a scale of 0 to 1 is given by the formula M(t) = ln(t^2) / 3t, where 't' is the time in hours that a student spends studying, 0 < t ≤ 4.
Determine the maximum measure of effectiveness of...
Hi everyone.
My final is coming up for Precalc, and I'm studying my butt off.
I was really needing help with this Exponential Growth Equation to find variable t.
8.0e^(.033t) = 59.8e^(.001t)
(8 times e to the .033t equals 59.8 times e to the .001t)
I would greatly appreciate this...
Q. A certain type of memory chip is known to have an exponential life distribution with a failure rate of 0.15*10^-5.
a) What is the probability that a memory chip will survine 20,000 hours of use?
b) What is the probability it will fail in the next 35,000 hours if it has survived 20,000...
Firstly, I don't get why the at term on the exponential turned positive (red arrow).. can someone explain that please?
And how do I start on this? How do I split it up such that I can do it for t>0 and t=<0?
Do I just integrate e^2t between -inf and 0 and integrate e^-t...
Homework Statement
Find the limit as x tends to zero of: (e^-x - cos x)/2x
Homework Equations
lim_x->0 e^-x = 1
lim_x->0 cos x = 1
lim_x->0 sin x / x= 1
The Attempt at a Solution
Hi everyone,
Here's what I've done so far:
(e^-x - cosx)/2x = [(e^-x)^2 - (cosx)^2] / 2x(e^-x...
Homework Statement
Prove that:
sinh(3x)=3sinh(x)+4sinh^{3}(x)
2. The attempt at a solution
I know that:
sinh(3x)=0.5(e^{3x}-e^{-3x})
and:
3sinh(x)=1.5(e^{x}-e^{-x})
But I have no idea how to rewrite 4sinh^{3}(x) in exponential form...
Homework Statement
f(x)=f'(x) for all x in R
S.T there exists a c in R such that f(x) = c exp(x) for all x
Homework Equations
The Attempt at a Solution
By defining g = f/c, I was able to show that c= f(0)
But i am also supposed to show that c Not equal to any other value
I...
Homework Statement
Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2)
O.Vallee gives this formula in his book, "Airy Functions and Applications to Physics"
but there are no proof of this formula. I tried to prove this formula, but I failed.
Would you...
Homework Statement
Given x' = Ax where A =
( 0 1 )
( -1 0 )
Compute the matrix exponential and then find the solution such that x(0) =
( 1 )
( 2 )
Homework Equations
The Attempt at a Solution
I computed the matrix exponential and obtained the matrix,
e^(A) =
( cos(t)...
Hi,
Does anyone knows how to solve this 2nd order non linear differential equation with exponential components?
d"V/dx" = A*exp(-B*V)-C*exp(B*V)
where A, B, C are constants.
Thanks
Homework Statement
L(t)=15(0.5^(t/26))
Find the rate of L, when t=60
Homework Equations
The Attempt at a Solution
L'(t) = (15/26)(1/2)^(t/26)ln(1/2)
L'(60)=(15/26)(1/2)^(60/26)ln(1/2)
= -0.08
Did I do this right? If I did it wrong, please say where, I am having great...
Homework Statement
Given the matrix: A=[-2 0 0;4 -2 0;1 0 -2], find e^AT.
I found the eigenvalues to be -2,-2,-2.
How do I solve this problem with 3 identical eigenvalues? Do I cube to matrix after plugging in my eigenvalue -2 and then solve it that way?
After plugging in the...
Homework Statement
Solve the equation e^x+10e^-x=7
Homework Equations
Logarithm rules and the natural logarithm
The Attempt at a Solution
Not really a calculus question but one I'm lost on nevertheless. I don't know how to go about solving this question at all. My first attempt...
Homework Statement
Species A doubles every 2 hours and initially there are 6 grams. Species B doubles every 5 hours and initially there are 14 grams.
Homework Equations
The Attempt at a Solution
I've tried graphing this, but I don't think I have the right equations down. I don't...
(x^2).e^(-2a(x^2))
how would i integrate this? By parts?
If so using |udv = uv - |vdu (hope that's right)
would i let u = x^2 and dv = e^(-2a(x^2)) ?
but how do I integrate dv = e^(-2a(x^2)) to find v ?
any help appreciated :)
Homework Statement
The actual problem shows a graph however I can state all the information. The graph is of a sinusiodal waveform where the amplitude is decaying exponentially. The formula for the graph is given by the equation:
T = Ae-Ktsin(wt + ø)
The question is to find A,K,w and ø...
Homework Statement
I'm trying to verify the Fourier transform but am getting stuck on the integration. Here is the pair:
f(x) = e^{-ax^2}
\hat{f}(k) = \frac{1}{\sqrt{2a}}e^{-k^2/4a}
a>0
Homework Equations
I know that
\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}dx
The Attempt at a...
Hey all, I'm really having a hard time figuring out a couple of problems in which I have to differentiate:
1: \frac{e^3^x}{\ln x}
I just don't know how to put it together... I know that e^3^x is 3e^3^x, and I know that you can't different \ln x, so I don't know what to do from there...
Homework Statement 3^x + x = 4
solve for x.
Homework Equations
I'm thinking of using the logarithm laws. log(a.b) = log a + logb log(a/b) = log a - log b
log(a^b) = bloga
The Attempt at a Solution
Well if I isolate the exponential
3^x = 4-x
take logs on both sides
xlog3 = log(4-x)
I...
Homework Statement
If spraying of grasshoppers with 800kg/km^2 of a certain insecticide will kill 40% of these insects, how much insecticide is needed to kill 98% of the grasshoppers?
(assume exponential relationship)
Homework Equations
\intdP/dt=\intkp
lnP=kt+c
e^(kt+c)=PThe Attempt at a...
Homework Statement
The population of a country is growing exponentially. The population in millions was 120 in 1970 and 150 in 1980.
(a) What is the population t years after 1970?
(b) How long does it take the population to double?
(c) When will the population be 400 million...
Homework Statement
a. 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...