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Recent questions in Differential Calculus

logiski9s 2022-07-13

How do you prove that the function $f (x) = {(x + 2 x^{3})}^{4}$ is continuous at a =-1?

Sonia Ayers 2022-07-13

How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?

antennense 2022-07-13

Evaluate $lim_{x \to \infty} e^{- x^{2}} \int_{x}^{x + \ln (x) / x} e^{- t^{2}} d t$

Palmosigx 2022-07-12

Evaluate $lim_{x \to 0} \frac{\tan (\tan x) - \sin (\sin x)}{\tan x - \sin x}$

Lucia Grimes 2022-07-12

Evaluating $lim_{x \to 0} \frac{2 \cos (a + x) - \cos (a + 2 x) - \cos (a)}{x^{2}}$

Janet Forbes 2022-07-12

What is $lim_{n \to \infty} n (\sum_{k = 1}^{n} \frac{1}{\sqrt{n^{2} + k}} - 1)$ ?

ntaraxq 2022-07-12

Is there any branch of constructing inverse function exist that is inverse of $n$ functions?

Nickolas Taylor 2022-07-12

While doing a sum I was stuck in a particular step:
$r_{1} = \frac{4 a \cos θ}{\sin^{2} θ}$
and
$r_{2} = \frac{4 a \sin θ}{\cos^{2} θ}$
How to eliminate $θ$ ?

rjawbreakerca 2022-07-12

Let be $X$ a topological space and we suppose that $E$ and $F$ are homeomorphic though a map $f$ from $F$ to $E$ . So I ask to me if $E$ is open/closed when $F$ is open/closed but unfortunatley I was not able to prove or to disprove this so that I thought to put a specific question where I ask some clarification: in particular if the result is generally false I'd like to know if it can be true we additional hypotesis, e.g. Hausdorff separability, First Countability, Metric Topology, etc....

Riya Hansen 2022-07-11

Limit of $\frac{\sqrt{1 - \cos x}}{x}$ using l'Hôpital

hornejada1c 2022-07-10

Consider the following:
$\int_{x}^{3 x} \frac{1}{\arctan t} d t$

2d3vljtq 2022-07-10

Evalutating $lim_{x \to \infty} (x + 1)^{k} - (x)^{k}$ , given $0 < k < 1$

grenivkah3z 2022-07-10

I have one question about limits: it is required to find the limit $lim_{n \to \infty} \sqrt[n]{3^{n} + 2^{n}}$

babyagelesszj 2022-07-09

We have the following limits:
$lim_{x \to \frac{π}{4}} \frac{1 - \tan^{2} x}{\cos 2 x}$ and $lim_{θ \to 1} = \frac{θ - 1 + \sin (θ^{2} - 1)}{θ^{2} - 1}$
Using L'Hôpital I obtained that the second limit is $θ = \frac{3}{2}$ . For the first limit I was able to rewrite it to $lim_{x \to \frac{π}{4}} (1 - \tan^{2} (x)) \sec (2 x)$ , but I don't really know what to do with that.

Frank Day 2022-07-09

Find the inverse function of
$f (x) = x - (2 \sqrt{x}) + 1$
the domain of definition is $x \geq 0$

aggierabz2006zw 2022-07-09

Compute $lim_{x \to 0} \frac{\sin^{3} x}{(2 x)^{3}}$

cdsommegolfzp 2022-07-09

I want to solve:
$lim_{n \to \infty} \frac{f l o o r (x \cdot 10^{n})}{10^{n}}$

uplakanimkk 2022-07-09

$f, g : R \to R$ continuous functions such that $f (g (x)) = x$ . Prove that $g (f (x)) = x$ .
Since $f$ is surjective, we basically have to prove that $f$ is injective. For $g$ is clearly, but how can I prove that $f$ is injective ? Please, help me!

Janet Forbes 2022-07-08

Limit as $x \to 2$ of $\frac{\cos (\frac{π}{x})}{x - 2}$

Ryan Robertson 2022-07-08

Definition: Here a neighborhood of the point $\vec{c} \in R^{n}$ is the set $N_{r} (\vec{c}) = {\vec{x} \in R^{n} : ‖ \vec{x} - \vec{c} ‖_{2} < r}$ for some $r > 0$ , and a subset $U \subset R^{n}$ is said to be convex if $\vec{x}, \vec{y} \in U, t \in [0, 1]$ implies $t \vec{y} + (1 - t) \vec{x} \in U$ .

My Attempt: Suppose $\vec{x}, \vec{y} \in N_{r} (\vec{c})$ and $t \in [0, 1]$ , we then have $‖ \vec{x} - \vec{c} ‖_{2} < r$ and $‖ \vec{y} - \vec{c} ‖_{2} < r$ . Now,
$‖ t \vec{y} + (1 - t) \vec{x} - \vec{c} ‖_{2} = ‖ t \vec{y} - t \vec{x} + \vec{x} - \vec{c} ‖_{2} \leq ‖ t \vec{y} - t \vec{x} ‖_{2} + ‖ \vec{x} - \vec{c} ‖_{2} < | t | ‖ \vec{y} - \vec{x} ‖_{2} + r \leq | t | (‖ \vec{y} - \vec{c} ‖_{2} + ‖ \vec{c} - \vec{x} ‖_{2}) + r < | t | (r + r) + r = (1 + 2 | t |) r$
Screeching halt.

My Question: How should I make $‖ t \vec{y} + (1 - t) \vec{x} - \vec{c} ‖_{2}$ less than $r$ ? My intuition is that I need the Cauchy-Schwarz inequality; but I am unsure where to apply it. Any hint would be greatly appreciated.

One of the most famous and challenging aspects of learning calculus and analysis is dealing with differential calculus equations that are not too complex per se if you have an example of differential calculus sample problems that contain both answers and questions. It is recommended to use your logic and see how differential calculus problems are formed with the help of several equations where you must be solving equations with decimals, functions and their derivatives. These are used to calculate the flow of electricity, model some motion, or set the energy flow of an object in a graphical form.

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