# Questions tagged [calculus]

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

106,166 questions
Filter by
Sorted by
Tagged with
40 views

### Where this solution of $f ' (x) = g(x)$ comes from?

I would have liked to solve the following first-order linear ODE for $f(x)$: $$f'(x) = g(x)$$ I attempted to solve it like this: $$\int f'(x)\,dx = \int g(x)\,dx$$ $$f(x) = \int g(x)\,dx+C$$ ...
44 views

### Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5 + x^9 …$ where $\lvert x\rvert$ < 1

Show that $\int_{0}^x \frac{1}{1+t^4} dt = x - x^5 + x^9 .....$ where $\lvert x\rvert$ < 1 I tried expanding using binomial theorem, but I am unable to prove the series and the integral will be ...
51 views

### Understanding differential equations — why can't all ODEs be integrated?

I'm in the process of learning about differential equations and something keeps bothering me. I know the idea that a differential equation relates a function and its derivatives and I can do simple ...
32 views

### can radical and rational equations be linear?

I am a student and I am just started the title of linear equations. I want to learn about linear equations deeply and linear equations make me confused. I have some question to ask you that explain to ...
15 views

### The nearest point of a curve to the certain point problem

Question: Find the nearest point to point $(3,0)$ and lie on the curve of $y = x^3- 2x^2 + 3$ Honestly, it's not the question what i'm dealing with. I have another question, but the answer is ...
17 views

### Is this function, containing a cross product continuous?

Is the following function continuous ? f : $\mathbb{R^3}\times \mathbb{R^3} \rightarrow \mathbb{R^3}$ $x,y \mapsto x \times \frac{y-x}{|y-x|}$ for $x \neq y$, else $0$. Now I would have argued that ...
15 views

### Cardinality of some special sets

Consider a set $\mathcal{R} = \{\{(i,j),(k,l)\}:i,j,k,l\in\{1,2,\ldots,K\},\;\mbox{and}\; i<j, k<l\}$. The cardinality of $\mathcal{R}$ is ${K\choose 2}^2$. The set $\mathcal{R}$ can be written ...
31 views

### Optimizing a multi-parameter quadratics solution set

So, recently I was doing a physics problem and I ended up in getting this quadratic in middle of the steps: $0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$ So, this is ...
20 views

47 views

33 views

### Finding $\frac{d^n}{dx^n} f_1(x)$ from $\frac{d}{dx} f_{k-1}(x)=f_k(x)-f_{k-1}(x) f_1(x)$

Suppose the following recursive equation holds: \begin{align} \frac{d}{dx} f_{k-1}(x)=f_k(x)-f_{k-1}(x) f_1(x) \end{align} where $f_0=1$. Question: Can we use this recursion to find \begin{align} f^{(...
15 views

### P(x)=(x-a)^2*Q(x), find Q(x) and show there exists at most one line $\ell$ that is tangent to the graph of $P(x)$ at two places. [closed]

Show that if $P(x)$ is a polynomial such that $P(a)=P'(a)=0$ then there exists a polynomial $Q(x)$ such that P(x)=(x-a)^2 * Q(x) and show that if $P(x)$ is a quartic polynomial then there exists at ...
### Is $\lim_{s \to \infty} \int f(x) g(s)dx$ equal to $\int f(x) (\lim_{s \to \infty}g(s) ) dx$?
$$\lim_{s \to \infty} \int f(x) g(s)dx = \int f(x) (\lim_{s \to \infty}g(s) ) dx$$ Is this equality true? Can you move the limit operator inside of the integral, since we're not integrating with ...