# Tagged Questions

For question involving exponential functions and questions on exponential growth or decay.

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### Algebra question about simplying a constant from exponential [closed]

i've a question, i'm doing an exercise of differential equations, but my result is wrong due to a step that compared with wolfram alpha i don't understood. You can check the screenshot, how the $C$ ...
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Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&... 2answers 149 views ### Proving$\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ... 2answers 123 views ### Is it valid to write$1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$? I just want to clarify one thing I was never really sure on. First the question: $$1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$$ is ... 3answers 53 views ### how so simplify this exponential equations ((a^3/2)/(b^3))/((a^-1)/(b^2)) I tried to solve this problem many times, however I tend to get the wrong answer. Here is the method I tried (((a^3)^1/2)/(b^3))*... sorry I get confused i got (... 2answers 72 views ### How can I get the expression of x? If there is $$x^2e^{A\sqrt x}=B$$ then what is the expression of$x$? If this cannot be solved, is there any approximation? 2answers 71 views ### Proofing that the exponential function is continuous in every$x_{0}$Given: $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also$e = \exp(1)$. For all$x \in \mathbb{R}$with$\left | x \right | \leq 1$: $$\left | \exp(x) -... 1answer 61 views ### Proving that the exponential function is continuous We aren't allowed to use many tricks such as difference quotient / integral calculus... Prove that \exp is continuous at x_{0}=0 ..................................................................... 0answers 28 views ### Integrate this expression containing exponential I am trying to integrate$$ \int x^a f'(x)\lambda \exp(f(x)\lambda) dx$$I'm not that great at these things, but I noticed that f'(x)\lambda \exp(f(x)\lambda = \frac{d}{dx} \exp(f(x)\lambda). I ... 1answer 29 views ### How can I prove this inequality involving the exponential function? Given$$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$also e = \exp(1). For all x \in \mathbb{R} with \left | x \right | \leq 1:$$\left | \exp(x) - ... 1answer 57 views ### Calculate:$f(a)=\int\limits_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x$Given the following function: $$f(a)=\int_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x$$ For which values of$a$is it possible to give an exact value for this function? I only know$f(2)=\sqrt{\pi}$... 1answer 29 views ### Please explain how to do this, base e [closed] Express$ 3^X$,$x^\pi$,$x^{\sin x} $using base$e$. 0answers 7 views ### How for continuous compounding of interest, the difference in balance over difference in time is equal to interest rate times balance at the instant? I was reading the chapter on exponential growth and decay in Morris Klein's Calculus book. He says$\delta A=A(0.04)\delta t$, 0.04 the interest is in unit percent per year. A is balance. t is time. ... 0answers 20 views ### Determining bounds for a sum with nested infinite series I am computing the inner product of the characters of the trivial and the$k$-th irreducible two dimensional representations of the dihedral group$D_n$of order$2 n$when$n$is even. The ... 1answer 92 views ### Prove$e^x - e^y \leq e |x-y|$for$x$belonging to$[0,1]$[closed] I'm not sure how to go about this. Does it involve using MVT? I got as far as saying$e = \frac{e^x - e^y}{x-y}$. 2answers 89 views ### I can't complete the integration of$e^{\sqrt{x}}$Compute$\displaystyle\int_0^1e^{\sqrt{x}}\,dx$That's a picture of how far I could get while trying to integrate$e^{\sqrt{x}}$. I tried the substitution method first, (boxed part) and then went ... 1answer 25 views ### Exponential decay + a recurrence relation I'm not sure if I get this right, some pointers could be helpful. Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3.... 1answer 95 views ### What is the value of$e^{-10000}$? What is the value of$e^{-10000}$? We know that the function$e$does not attain value$0$anymore. But in R and Matlab the value of$e^{-10000}$is given as$0$which is not correct anymore. I ... 1answer 19 views ### Bounding an exponential integral I'm having trouble seeing this bound I've seen on a proof. Let$f$be a polynomial, and$F$the polynomial obtained from$f$by replacing each coefficient by its absolute value. Then: $$\bigg{|}\... 0answers 40 views ### Prove Exponential series from Binomial Expansion I try to prove the Exponential series :$$\exp(x) = \sum_{k=0}^{\infty} \dfrac{x^k}{k!}$$From the definition of the exponential function$$\exp(x) \stackrel{\mathrm{def}}{=} \lim_{n\to\infty} \left(... 2answers 66 views ### Exponential of a number [closed] What is exponential of a number?$E^{10}=22026.4657948$What is the mathematical calculation behind the above calculation? Regards, Philip 1answer 40 views ### Proof that$a^x$goes towards infinity as x goes towards infinity I'm tasked to prove that$a^x \rightarrow \infty $when$x \rightarrow \infty$provided that (a > 1). I've found a very rigorous proof for this. But my question is, why can't it be logically realized ... 0answers 16 views ### How do I determine and sketch the images$g(\mathbb{R})^2$as sets and as geometric objects? It's given function$g(x, y) = \begin{pmatrix}e^x \cos y\\ e^x \sin y\end{pmatrix}$. How do I determine and sketch the images$g(\mathbb{R})^2$as sets and as geometric objects? 4answers 82 views ### Prove that for all$a > 0$:$\int_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \frac{\pi}{2} - \int_0^a\frac{\sin x}{x}dx$Prove that for all$a > 0\$: $$\int\limits_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \cfrac{\pi}{2} - \int\limits_0^a\cfrac{\sin x}{x}dx$$ I have no idea how to solve it. But the task looks very ...
So I have: $$\log_2(5x) + \log_2 3 + \frac{\log_2 10}{2}$$ I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first ...