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What are the differences between e^x and e^ix both graphically and theoretically?And my second question is what is the precise intuition behind rotation of 2D number and exponential function?

Jairo Decker 2022-10-12

Why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves?
i.e.
$y = e^{x^{2}} \frac{d y}{d x} = 2 x e^{x^{2}}$
I am interested in a verbal explanation.

shiya43 2022-10-12

"Factorials grow faster than exponential functions, but much slower than double-exponential functions."
The author doesn't provide a link let alone a proof of that fact and I don't find it obvious. After all, factorial functions grow really fast but the author says they grow much slower than double exponentials. Does anyone know a proof that
$lim_{x \to \infty} \frac{e^{e^{x}}}{x!} = \infty$

Eliza Gregory 2022-10-11

Why is the expectation of an exponential function:
$E [\exp (A x)] = \exp ((1 / 2) A^{2}) ?$

abelynybco 2022-10-08

Solve the exponential equation $32^{x + 3} = 4^{3 x + 5}$ for x
{-4}
{5}
{}
{-1}

Chasity Kane 2022-10-08

Finding the derivative of an integral $g (x) = \int_{2 x}^{6 x} \frac{u + 2}{u - 4} d u$

tonan6e 2022-10-07

Prove for any real number k, prove that the exponential function $e^{z}$ is a bijection (z is a complex number) from the strip $k < I m (z) \leq k + 2 π$ to the complex plane minus the point 0, $C - {0}$

dannyboi2006tk 2022-10-07

$\int_{0}^{\infty} x^{n} e^{- a x^{b}} d x = \frac{1}{b} a^{- \frac{n + 1}{b}} Γ [(n + 1) / b]$
Is there a way to evaluate this integral with a lower limit K>0?

Diya Nayak2022-10-06

Muhammad Shaiq2022-10-05

Lisantiom 2022-10-05

Is the exponential function continuous in $R^{n}$ ? How can one show this? Let's say for a function like $f : R^{2} ⟶ R, f (x, y) = x^{y}$

Kassandra Mccall 2022-10-02

Using the definition of the integral, find $\int_{0}^{a} x^{2} d x$

Aubrie Mccall 2022-10-01

Evaluate the following limit: $lim_{x \to - \infty} \frac{8 x^{4} + 3 x^{2} - x}{\sqrt{9 x^{8} - 2 x^{5} + 4}}$

Aidyn Crosby 2022-09-30

Evaluate $\int x^{2} e^{- x^{3}} d x .$

howardfamilyranch44 2022-09-29

If f and g are the functions whose graphs are shown, let
u(x) = f(g(x)),
v(x) = g(f(x)), and w(x) = g(g(x)).
Find each derivative, if it exists. If it does not exist, explain why. (If an answer does not exist, enter DNE.)
(a)
u'(1) =
It does exist.u'(1) does not exist because f '(1) does not exist.     u'(1) does not exist because g'(1) does not exist.u'(1) does not exist because f '(3) does not exist.u'(1) does not exist because g'(2) does not exist.

(b)
v'(1) =
It does exist.v'(1) does not exist because f '(1) does not exist.     v'(1) does not exist because g'(1) does not exist.v'(1) does not exist because f '(3) does not exist.v'(1) does not exist because g'(2) does not exist.

(c)
w'(1) =
It does exist.w'(1) does not exist because f '(1) does not exist.     w'(1) does not exist because g'(1) does not exist.w'(1) does not exist because f '(3) does not exist.w'(1) does not exist because g'(2) does not exist.

charlygyloavao9 2022-09-29

Definite integration of logarithmic function
$\int_{0}^{π} \log (5 - 4 \sin x) d x$
I tried to proceed with this question using integration by parts but got stuck over
$\int_{0}^{π} (\frac{- 4 x \cos x}{5} - 4 \sin x) d x$

koraby2bc 2022-09-26

The devariate of a functions of f at x is given by
$f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
provide the limit exists
Use the definition of the derivative to find the derivative of $f (x) = 8 x^{2} + 9 x + 1$ .Enter the fully simplified expression for f(x+h)-f(x).

Celinamg8 2022-09-26

Find limit as h approaches 0 of
$\frac{e^{x (h + 2)^{2}} - e^{4 x}}{h}$

Lustyku8 2022-09-23

Solve for x
$\log_{12} x = \frac{1}{2} \log_{12} 9 + \frac{1}{3} \log_{12} 27$

Aidyn Meza 2022-09-23

Given a constant sum of $x_{n}$ values:
$\sum_{n = 1}^{N} x_{n} = C$
where $x_{n} \geq 1$
Find the maximum of the following expression:
$\sum_{n = 1}^{N} a_{n} \log (x_{n})$
where the $a_{n} > 0$ values are constants, and the $x_{n}$ values are free to change, given that their sum will remain constant.

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