# calculus's questions - English 1answer

82.437 calculus questions.

### In defining the derivative, what definition of a limit is being used?

The following definition of the derivative of a function is taken from Analysis on Manifolds by James Munkres: Now what I'm confused about is what definition of a limit is being used above. There are ...

### What is the state of convergence or divergence $\sum_{n=1}^{\infty} {\frac {5^{n}}{(3^{n+1})n}}$?

4 answers, 47 views calculus sequences-and-series
Please tell us about the convergence or divergence in the series? \begin{matrix} \sum_{n=1}^{\infty} {\frac {5^{n}}{(3^{n+1})n}}\end{matrix} thanks

### 3 Are integrable functions always bounded?

3 answers, 49 views calculus integration definition
So I was going through some questions, and these two made me curious: 1) An integrable function is always bounded 2) If a function is defined on an unbounded interval, then it cannot be integrable. ...

### -2 Generalising trivial solutions of $\int_a^bf(x) dx=c , f(x)=?$

Trying to Find trivial solutions of $\int_a^bf(x) dx=c , f(x)=?$ 2.Generalise 1, in number of possible ways and then generalise those ways as well. Motivation : there are unaccountably many ...

### Solve $x(dy/dx) = x^4y^3 - y$ (Bernoulli's equations)

2 answers, 20 views calculus differential-equations
Use method of Bernoulli's equations to solve the equation: $$x(dy/dx) = x^4y^3 - y$$ I don't really can understand how to use Bernoulli's equations as I had it only once and I wasn't able to ...

### 1 Prove, that the sum $\sum_{k=1}^{2n} (-1)^k (2k+1)$ is proportional to $n$

Prove, by induction, that the sum \begin{matrix} \sum_{k=1}^{2n} (-1)^k (2k+1)\end{matrix} is proportional to n, and find the constant of proportionality thanks

### 1 Let $f(x)=\frac{1}{x^3+3x^2+3x+5}$, then what is $f^{(99)}(-1)$?

2 answers, 53 views calculus real-analysis
Let $f(x)=\frac{1}{x^3+3x^2+3x+5}$, then what is $f^{(99)}(-1)$? By letting $g(x)=x^3+3x^2+3x+5$, it is easy to see that $g'(-1) = g''(-1) = 0$, and so $f'(-1) = f''(-1) = f'''(-1) = 0$. However, I ...

### Differential Equations - Arbitrary and fixed constants

2 answers, 1.681 views calculus differential-equations constants
I am having a problem understanding, Whether to keep a constant (arbitrary or fixed) in the solution of a differential equation Figuring which is an arbitrary constant and which a fixed constant ...

### The limit behaviour of $f(r,\theta ,\phi)=\frac {1}{r\sin \theta}$

I have been examining a function $f(r,\theta,\phi)$ where $f(r,\theta ,\phi)=\frac {1}{r\sin \theta}$ for $\theta \gt \pi /4$ $\;$ and $\;$ $f(r,\theta ,\phi) = 0$ for $\theta \lt \pi /4$. I am ...

### 2 Induction principle for $n!<n^n$

7 answers, 70 views calculus induction

### 1 Determining coefficients of a trigonometric function with given tangent lines

2 answers, 50 views calculus algebra-precalculus
I am given the following problem: Find a function of the form $f(x)=a+b\cos{(cx)}$ that is tangent to the line $y=1$ at the point $(0,1)$ and tangent to the line $y=x+\dfrac{3}{2}-\dfrac{\pi}{4}$ ...

### 2 Test $\int_{0}^{\infty}\frac{e^{ix}}{\log(x)}, \int_{0}^{\infty}\frac{\cos(x)}{x^p}, \int_{0}^{\infty}\cos(x^2)$ for convergence

I need to test these integrals for convergence $\int_{0}^{\infty}\frac{e^{ix}}{\log(x)}, \int_{0}^{\infty}\frac{\cos(x)}{x^p}, \int_{0}^{\infty}\cos(x^2)$ with $p\in\mathbb{R}$. However I suck ...

### What is the relation between $\frac{d^2y}{dx^2}$ and $\frac{d^2x}{dy^2}$?

1 answers, 120 views calculus derivatives inverse-function
What is the relation between $\frac{d^2y}{dx^2}$ and $\frac{d^2x}{dy^2}$ ? For example $\frac{dy}{dx}$=$\frac{1}{\frac{dx}{dy}}$

### 1 Does $4(x^3+2x^2+x)^3(3x^2+4x+1) = 4x^3(3x+1)(x+1)^7$?

Please provide proof The answer in the back of the book is different to both my calculations and also the online calculator I crosschecked my answer with... The Question: "Differentiate with respect ...

### Integral Computation check

2 answers, 32 views calculus integration approximation

### 1 Optimize: $\arg\min_{\theta>0} \quad \big\| x - \sum_{n} c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$

Can somebody help me get started with the following problem? I want to solve: $$\theta^* := \arg\min_{\theta>0} \quad \big\| x - \sum_{n=1}^N c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$$ where $x$ ...

### 1 How to find the minimum of $f(x)=\frac{4x^2}{\sqrt{x^2-16}}$ without using the derivative?

Find the minimum of function $$f(x)=\frac{4x^2}{\sqrt{x^2-16}}$$ without using the derivative. In math class we haven't learnt how to solve this kind of problems (optimization) yet. I already know ...

### 3 If $p^2=a^2 \cos^2 \theta+b^2 \sin^2 \theta$, show that $p + p'' = \frac{a^2b^2}{p^3}$

if $p^2=a^2 \cos^2 \theta+b^2 \sin^2 \theta$, show that $p + p'' = \frac{a^2b^2}{p^3}$ My try : $2pp' = (b^2-a^2)\sin 2\theta$ $p'^2 + pp'' = (b^2-a^2)\cos 2\theta$ Thats it ! it doesn't ...

### 4 How to find the finite limit of this function?

1 answers, 48 views calculus limits