0
votes
0answers
15 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
1
vote
2answers
69 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
4
votes
1answer
63 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n(z)=(1+\frac{z}{n})^n$ for $\:n\in \mathbb{R^+}$. Then $\frac{d}{dz}g_n(z)=n(1+\frac{z}{n})^{n-1}\cdot\frac{1}{n}=(1+\frac{z}{n})^{n-1}$ Define $g_{\infty}(z)=\lim ...
3
votes
3answers
281 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
0
votes
4answers
80 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
-4
votes
1answer
33 views

Find the percent increase or decrease

For each function, find the percent increase or decrease that the function models $y=1298 \cdot 1.63^x$ $f(x)=2 \cdot 0.65^x$
-1
votes
1answer
32 views

Don't understand answer from exponential growth question

"A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute, and with the amount she currently has, she ...
4
votes
5answers
234 views

Why $e^x$ never equal $x$?

Je veux savoir pourquoi $x=e^x$ n'a aucune solution dans $\Bbb R$. Lorsque j'ai essayé de tracer le graphe de la fonction $e^x$, j'ai trouvé en fait qu'elle est une fonction strictement croissante ...
2
votes
1answer
71 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
19
votes
3answers
332 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
1
vote
2answers
152 views

Which is greater, $e^{\pi}$ or $\pi^e$? [duplicate]

I'm familiar with a simple method of demonstrating that $e^\pi$ is greater: $f(x) = \ln|x|/x$ $f'(x) = (1 - (\ln|x|))/(x^2)$ so f's max is at $(e, 1/e)$ so $1/e > \ln(\pi)/\pi$ and $e^{\pi} > ...
3
votes
2answers
50 views

Proof of the derivative of $a^x$ [duplicate]

I've tried for a while myself from first principles and applying various rules, but always end up going in circles. I've gotten as far as $$ y = a^x $$ $$ \frac{dy}{dx} = a^x \left( \lim_{x ...
0
votes
1answer
27 views

Calculus exponential function/ slope and equtation

Consider the function $f(x) = 3(1 − e^x)$. Use exact values when answering the following questions: Find the slope of the graph of $f(x)$ at the point where it crosses the $x$-axis. Find the ...
1
vote
1answer
16 views

Derivative of exponential function

1) $f(t) = (\ln 5)^t$ what is the $f'(t)$? I tried $t\ln(5)$ but it was wrong. 2) $f(x) = x^{\Large π^6} + (π^4)^x$ This one I did not attempt in it because I find it confusing little bit.
2
votes
0answers
37 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
1
vote
4answers
60 views

Need an example of piece wise function continuous but not differentiable

I Need an example of piece wise function continuous but not differentiable. One of the functions has to be trigonometric and the other has to be exponential. Please
14
votes
3answers
2k views

Is this question too easy or am I getting it wrong?

In my homework, I am asked to find the limit $$\lim\limits_{x\to0}{\frac{x}{e^x}}$$ But obviously, you could just substitute $x = 0$: $$\lim\limits_{x\to0}{\frac{x}{e^x}} = ...
7
votes
3answers
460 views

Evaluate a limit (probably involving L'Hôpital rule)

Evaluate the limit: $$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$ My attempts didn't yield a result. I'd be glad for a guidance. Thanks!
2
votes
1answer
66 views

The only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x)$ is $f(x)=ce^{x}$

Prove that the only differentiable function $f \colon \mathbb R \to \mathbb R$ such that $f^\prime(x)=f(x) \mspace{1ex} \forall x\in \mathbb R$ is $f(x)=ce^{x}, \forall x\in \mathbb R$, and for some ...
3
votes
1answer
113 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
2
votes
4answers
59 views

Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
0
votes
2answers
38 views

How to prove the following? $\frac{d}{dx}a^x=(\ln a)a^x$

How to prove that the following holds? $$\frac{d}{dx}a^x=(\ln a)a^x.$$ Just a hint will do it.
26
votes
3answers
856 views

Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't ...
1
vote
1answer
101 views

Double integrals of exponential functions

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to ...
0
votes
1answer
82 views

Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
2
votes
8answers
273 views

How come $1^{\infty}$ = undefined, while $2^{\infty} = \infty$ and $0^{\infty} = 0$? [duplicate]

$1^\infty$ = undefined $2^\infty = \infty$ $0^\infty = 0$ Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and ...
-14
votes
4answers
180 views

How is the limit definition of e, actually equal to e? [duplicate]

$$\lim_{n\to\infty} (1+\dfrac{1}{n})^n = e.$$ To me this seems like $\dfrac{1}{n}$ goes to $0$, then you get $1^\infty$, which is equal to $1$. So why is it $e$?
0
votes
1answer
26 views

Definition of Derivative And Exponential Functions

Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative. The definition of the derivative of $f(x)$ is $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ The derivative of $f(x) = ...
0
votes
1answer
34 views

Simulation - Find the maximum of a function with exponential decay

I need to run a program to calculate the integral of the following function with exponential decay $$t(x) = \exp(-Lx)(a\sin(bx) + d\cos(ex))$$ and for a simulation purpose, I need to find maximum of ...
0
votes
1answer
59 views

Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
1
vote
1answer
37 views

How to evaluate the integral $\int_0^{\ln3} e^{x-e^x}\,\mathrm dx$?

How to evaluate the following definite integral? $$\int_0^{\ln3} e^{x-e^x}\,\mathrm dx.$$ Should I use some sort of U Substitution?
0
votes
1answer
72 views

Rate of exponential decay

Good day all I have this curve (it is a solution of a partial differential equation that am working on) and I want to calculate numerically the rate of exponential decay but I don't know how to go ...
0
votes
0answers
17 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
1
vote
1answer
44 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
2
votes
4answers
73 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
1
vote
3answers
135 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
1
vote
1answer
39 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
0
votes
1answer
35 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
0
votes
1answer
45 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
0
votes
1answer
35 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
0
votes
1answer
39 views

Length of a very basic exponential curve

I have the beginning points (0,1) and end points (180, 141.732) of a curve. The function I am currently using is f(x) = Ae^kx. However, when deriving the original function, I end up with 0 (from ...
2
votes
1answer
35 views

Evaluating a limit with two steps - Right/Legal?

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {\left( {{{4{n^2}} \over {(2n + 1)(2n - 1)}}} \right)^{1 - {n^2}}} = \mathop {\lim }\limits_{n \to \infty } {\left( {{1 \over {{{(2n + 1)(2n ...
1
vote
1answer
36 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
2
votes
4answers
196 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
11
votes
3answers
454 views

Proving that $e$ is irrational

Show that $e$ is irrational. Recall $\mathrm{e} = \exp(1)$ so assume $\mathrm{e}$ is rational , then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some positive ...
1
vote
3answers
36 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
0
votes
2answers
28 views

Calculus Exponential Functions Again

This one wants us to evaluate the following limits of this exponential function. $$\lim_{x \to \infty} \frac6{e^x-6}$$ I'm not sure how to approach this problem. I did easily figure out this version ...
1
vote
2answers
40 views

Derivatives of Logarithmic functions

I am stuck in these problems. $\displaystyle \frac{d}{dx} (\log_2 x^8)$ $\displaystyle \frac{d}{dx} (e^x \ln x)$ I think for the first problem the answer is $\dfrac{2}{x^7}$, whereas for the ...