Exponential Functions Mastery: Pro Tips & Tricks

Recent questions in Exponential Functions

kituoti126 2022-11-08

IProve that exponential functions $a^{n}$ have different orders of growth for different values of base a>0.
It looks obvious that when a=3 it grows faster when compared to a=2. But how do i make a formal proof for this? Thanks for your help.

assupecoitteem81 2022-11-07

I can visualize the exponential function with the graph $y = e^{x}$ , but I can do that for almost any function.
In addition to its graph, the function $f (x) = x^{n}$ can be visualized as the volume of a box with sides of length x in n-dimensional space, and the trigonometric functions can be interpreted as side lengths of certain right triangles.
Is there a similar geometric interpretation of the exponential function?

Demarion Ortega 2022-11-06

What are biological or biochemical processes that are described (stochastically or deterministically) by exponential function $f (x) = e^{a x}$ ? What about examples that are not related to the population growth? I'm especially interested in biochemical examples.

Jadon Johnson 2022-11-06

Exponential functions always have a linear variable as a power of constant base and an increasing monotonic graph. Are these $e^{\sqrt x}$ and $e^{x}$ functions also exponential functions. What is the unique characteristic for which a function is said exponential and do these functions has the same traits. Why they do not have a monotonic graph? Which $e^{x}$ has. Do they need to satisfy the pre defining properties an exponential function should satisfy i.e. ( base>0, base must not be 1, and power always belongs to real set (R)). Please clarify over each point.

perlejatyh8 2022-11-05

How to change the form of an exponential function. Can someone explain the process in which the function below
$T (t) = e^{- 0.0407409 t + 3.89467} + 26$
is changed into this form.
$T (t) = 49.1398 e^{- 0.0407409 t} + 26$

apopiw83 2022-11-04

Given the vector space, $C (- \infty, \infty)$ as the set of all continuous functions that are always continuous, is the set of all exponential functions, $U = {a^{x} ∣ a \geq 1}$ , a subspace of the given vector space?As far as I'm aware, proving a subspace of a given vector space only requires you to prove closure under addition and scalar multiplication, but I'm kind of at a loss as to how to do this with exponential functions (I'm sure it's way simpler than I'm making it).My argument so far is that the set U is a subset of the set of all differentiable functions, which itself is a subset of $C (- \infty, \infty)$ , but I doubt that argument would hold up on my test, given how we've tested for subspaces in class (with closure).

pin1ta4r3k7b 2022-11-02

Solve this one using exponential function
$\int \frac{2^{\frac{1}{x}}}{x^{2}} d x$

gasavasiv 2022-11-01

I have a topological field K that admits a non-trivial continuous exponential function E, must every non-trivial continuous exponential function E′ on K be of the form $E^{'} (x) = E (r σ (x))$ for some $r \in K$ * and $σ \in A u t (K / Q)$ ?
If not, can you explain for which fields other than $R$ is this condition met?

raapjeqp 2022-10-31

Does $\begin{aligned} lim_{n \to \infty} \int_{a}^{b} f (x)^{p_{n} (x)} d x = \int_{a}^{b} f (x)^{p (x)} d x \end{aligned}$ hold in general?

racmanovcf 2022-10-29

Suppose we know the definition of rational exponents of positive real numbers. If we define the exponential function $y = a^{x} (a > 0)$ by extending rational exponents to real exponents by the following limiting process:
$a^{x} = lim_{r_{i} (\in Q) \to x} a^{r_{i}},$
then how can we prove the exponential function is continuous in R?

Tessa Peters 2022-10-27

We have been taught that linear functions, usually expressed in the form y=mx+b, when given a input of 0,1,2,3, etc..., you can get from one output to the next by adding some constant (in this case, 1).
$\begin{array}{cl} Input & Output \\ 0 & 1 \\ 1 & 2 \\ 2 & 3 \end{array}$
But with exponential functions (which are usually expressed in the form $y = a \cdot b^{x}$ ), instead of adding a constant, you multiply by a constant. (In this case, 2)
$\begin{array}{cl} Input & Output \\ 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 8 \end{array}$
We can keep going, can't we?
$\begin{array}{cl} Input & Output \\ 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 16 \\ 4 & 256 \end{array}$

Evelyn Freeman 2022-10-22

Is the exponential function integrable?
is there, in general, a difference between integrable and integrable in the Riemann sense?

Cristofer Watson 2022-10-21

solve the below equation with respect to x
$0.6 \cdot \exp (\frac{- 40}{x}) + 0.4 \cdot \exp (\frac{10}{x}) = 1$

faois3nh 2022-10-21

How do you prove that the exponential function is continuous at x=0 and how do you prove it is continuous for all real x?

Brianna Schmidt 2022-10-20

Given the series definition of the exponential function, i.e. $\exp (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{k!}$ . Given that I have already proven that polynomials are continuous, does from this fact follow the continuity of the exponential function ?

Maribel Mcintyre 2022-10-20

$5 e^{2 x} = 6 + 29 e^{x}$
I would ususally multiply each side by a log value to get those exponents by themselfs, but that 6 is throwing me off. How do I get rid of it? Or do i just have to multiply it by log too?

Krish Logan 2022-10-20

What is the difference between this notation of the exponential function
$(1 + \frac{1}{n})^{n} \to e a s n \to \infty$
and this notation:
$lim_{n \to \infty} (1 + \frac{x}{n})^{n}$
Why is there a variable x in the second equation, and a 1 in the first equation?

Taniya Melton 2022-10-18

What is example of functions that grow faster than the exponential functions and/or factorial functions?

Payton George 2022-10-18

$y = e^{- (x - 3)}$ . This graph has a reflection over the y-axis and is shifted right 3 units.
Why right instead of left, though. Considering that this is equivalent to $y = e^{(- x + 3)}$ I thought that the shift would be opposite of the sign being that it is in parentheses.
Take $y = e^{x + 3}$ for example. The graph of this function is shifted left 3 because of the parentheses.
I want to supposed to figure out the shifts in the graphs. Am I missing a detail here?

ebendasqc 2022-10-17

The exponential function is the unique function satisfying
f′=f and f(0)=1
however, unless I've made a mistake, we have
$\frac{\partial}{\partial x} (a x)^{x} = x (a x)^{x - 1} a = a x (a x)^{x - 1} = (a x)^{x}$
and
$(a 0)^{0} = 0^{0} = 1$
so I feel like I must be missing something special about $e^{x}$ . Any pointers would be greatly appreciated.

Coming up with calculus and analysis studies, the chances are high that you will deal with the exponential functions for your calculations. While exponential functions problems will not be overly challenging as you study the answers to the questions asked below, graphic exponential functions always take time as you need to connect the verbal explanation with the graphical representation of your problem. Sometimes equations may be used to solve the challenges with the help of a single methodology where an analytical part has been omitted. These challenges are also reflected below in most engineering tasks where exponential functions are required.

Calculus 1

Exponential Functions Explained: In-Depth Examples, Equations, and Expert Insights