Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [calculus]

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

0
votes
0answers
5 views

Why the 'gradient of the diffeomorphism at a point in the surface' perpendicular to the surface at that point?

This question is related to these two questions of mine: Intuition or motivation for the definition of an hypersurface. What are we actually trying to define? and Understanding this very generic ...
1
vote
1answer
13 views

Equation with $\text{Li}_2$ has a unique solution

Let $c \geq 0$ and $y \in [0,1]$, I would like to show that $$c^2\text{Li}_2(y)=\log^2(1-y),$$ has a unique solution $y=y(c).$ Here $\text{Li}_2(y)= \sum_{n \geq 1} \frac{y^n}{n^2}$ is the ...
0
votes
1answer
21 views

If H(b) $\leq$ b then $\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db $?

$H$ is a CDF over $[0,1]$ I want to prove that if $H(b) \leq b$ then : $$\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db $$ for all $k \in [0,1]$ I think this is true, but a rigorous ...
0
votes
1answer
15 views

Suppose $P(x,y,z)$ and $Q(x,y,z)$ have continuous partial derivatives on $\mathbb{R}^3$,

Suppose $P(x,y,z)$ and $Q(x,y,z)$ have continuous partial derivatives on $\mathbb{R}^3$, and for every smooth surface $\Sigma$, it holds $$\iint_{\Sigma}P \, \mathrm{d}y \, \mathrm{d}z + Q \, \mathrm{...
1
vote
1answer
29 views

Simple limit with asymptotic approach. Where's the error?

Simply calculus question about a limit. I don't understand why I'm wrong, I have to calculate $$ \lim_{x \rightarrow 0} \frac{2x - \sqrt[3]{8 - x^2}\sin x}{1 - \cos\sqrt{x^3}} $$ Using asymptotics, ...
0
votes
3answers
21 views

Partial fraction of $\frac{2s+12}{ (s^2 + 5s + 6)(s+1)} $ then inverse transform it

Find the inverse Laplace transform of $\mathcal{L}^{-1} \frac{2s+12}{ (s^2 + 5s + 6)(s+1)} $ I recognise I need to use partial fractions to solve it and that is where I got stuck. Here’s my working, ...
5
votes
0answers
86 views

Show that $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4…}}}$

I was solving a problem asking me to prove the identity $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4...}}}$ which was first posed by Ramanujan. The standard answer goes as $$3=\sqrt{1+2\cdot4}=\sqrt{1+2\sqrt{1+3\...
-5
votes
2answers
45 views

Help in limit computation [on hold]

Can you please help me computing $$ \lim_{n \to \infty} \frac{n^m}{x^n} $$ with $x \geq 1$ thanks
2
votes
3answers
49 views

Determine this limit using L'Hopitals rule

I couldn't find a way to get the answer for $$\lim\limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^{2}}}$$ From my knowledge of L'Hopital's Rule, I see that this is some kind of $1^{\infty}...
0
votes
2answers
27 views

Number of possible solution(s)?

What is the number of possible solution(s)of the equation $\int_{0}^{x} t^2-8t+13 dt=xsin(a/x)$? I tried applying Leibniz's rule and differentiated both sides. I got the equation $x^2-8x+13=sin(a/x)-a/...
1
vote
0answers
29 views

Calculus: application of chain rule seemingly contradicts what I’ve learned so far on deriving functions.

Please note: I am a person studying high school mathematics on my own (well after high school). Furthermore, English is not my first language, which may affect my use of math terminology. The ...
2
votes
4answers
97 views

Help evaluating $ \int \frac {\sqrt{x^2-1}}x \ dx$

Before anything, I'd like to clarify that I have no background in calculus (I'm still in school). I only try to learn calculus as a hobby. Please be gentle. I'm trying to evaluate the aforementioned ...
2
votes
1answer
43 views

Assume $f'$ is continuous on$[0,2]$, Prove $\max_{x\in[0,2]} |f(x)|\le|\frac{1}{2}\int_0^2f(x)\text dx|+\int_0^2|f'(x) |\text dx$

Assume $f'$ is continuous on$[0,2]$. Prove $$\max_{x\in[0,2]} |f(x)|\le \left|\frac{1}{2}\int_0^2f(x)\text dx \right|+\int_0^2|f'(x) |\text dx$$ I think it's about Taylor expansion. But really don't ...
1
vote
1answer
30 views

Use CLT to find the probability

Not a homework. The entire problem could be found at Find the Expected Value of the Sum of Random Variables. But it is not that relevant. My work: In order to use CLT we assume that the variable is ...
1
vote
1answer
19 views

How to find the expected value of the random sum of variables?

$C$ is a constant Not a homework. The detail is formatted as a picture. The whole problem (but not very relevant) could be found at Find the Expected Value of the Sum of Random Variables $\mathbb E[...
0
votes
0answers
17 views

Set of zeros for the difference of two CDF

Let $F$ and $G$ be two CDF on $[0,1]$ and let $F\leq G$. (note: $F$ and $G$ are two non-decreasing, non-negative and right-continuous real-valued functions with $F(x)=G(x)=1$ for $x\geq 1$ and $F(x)...
0
votes
1answer
29 views

Convergence of 3 dimension integral with a singularity refer to $\int_{B_{r_0}(0)}d^3x'\frac{\cos^2(\frac{|x'|\pi}{2r_0})}{|x-x'|}e^{i\omega |x-x'|}$.

I want to check if the following integral converges: Given $x\in\mathbb{R}^3$ $$\int_{0}^{L_x}\int_{0}^{L_y}\int_{0}^{L_z}d^3x^{'}\frac{\sin^2(\frac{x_1^{'}\pi}{L_x})\sin^2(\frac{x^{'}_2\pi}{L_y})\...
1
vote
0answers
20 views

Continuous joint distribution function and the cdf

$f$ is a jointly continuous distribution function $f(x,y)=Cx^2 y^2$ where $0<y<1$,$y^3<x<y^{1/3}$ C is a constant. Find $\mathbb P\{0<x<1/4\}$ and $\mathbb P\{0<x<...
0
votes
3answers
44 views

Why in a continuous function derivative is defined? [on hold]

I was taught that in a continuous function (for example R-->R) the derivative is defined for every point in the domain. Even if this seems intuitively correct, mathematically the fact that the ...
0
votes
1answer
14 views

How to find the probability density function by the joint distribution?

$f$ is a jointly continuous distribution function $f(x,y)=Cx^2 y^2$ where $0<y<1$,$y^3<x<y^{1/3}$ C is a constant. Find $f_X (x)$ and $f_{(X|Y)} (x│y)$ This is not a homework ...
3
votes
0answers
24 views

Finding an elementary evaluation of $B_{1/2}(a,1-a)$

I'm trying to prove $$B_{1/2}(a,1-a):=\int_0^{1/2}x^{a-1}(1-x)^{-a}dx=\int_0^1\frac{x^{a-1}-x^a}{1-x^2}dx$$ $(a>0)$ (where $B$ denotes the incomplete beta function) with elementary method. I have ...
0
votes
0answers
33 views

Find the Expected Value of the Sum of Random Variables

R is discrete random variable defined by the density function: $f(n)=a\frac{2^n}{n!}e^{-2}$ when $n\geq 0$; $f(n)=a3^n$ when $n<0$ $S=-R$ if $R<0$; $S=0$ otherwise. $T=X$ if $R\geq 0$; $T=0$ ...
-1
votes
2answers
52 views

What is limit point?

I'm new in calculus and can't understand what the limit point is.... Here, the definition from textbook that I've a question If E=(1,2) 1.It' say that 1 and 2 is also limit point,I can't understand ...
1
vote
2answers
23 views

Removing absolute value from argument of logarithm in $\frac{(x)^{1/3}\log|x+1|}{e^x-1}$ to find asymptotic

I want to find the asymptotics of the following function when $x\rightarrow0$. This is what my textbook does: $$\frac{(x)^{1/3}\log|x+1|}{e^x-1}=\frac{(x)^{1/3}\log(x+1)}{e^x-1}=\dots$$ I wonder ...
0
votes
2answers
34 views

Justify $e^{2x}\log{x}\sim (x-1)$

How do I justify that for $x\rightarrow1$ $$e^{2x}\log{x}\sim (x-1)$$ I know that if $x\rightarrow 0$, then $\log{x}\sim(x-1)$ because $\log(1+x)\sim x$ for $x\rightarrow0$. This limit does not ...
1
vote
2answers
59 views

Evaluate $ \lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$ where $[\cdot]$ denotes the greatest integer function. [duplicate]

Evaluate $$\lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$$ where $[\cdot]$ denotes the greatest integer function. Can anyone give me a hint to proceed? I know that $$\frac {\sin x}{x} &...
0
votes
1answer
13 views

Inverse Laplace transform (first shifting theorem)

This is my textbook from what I am studying and the “green” highlighted part is where I am questioning myself. Let’s pull out the expression- $$\mathcal{L} ( \frac{ \frac{19}{25} (s) + \frac{14}{25}...
0
votes
0answers
12 views

Ways to express sum of $n$th-power partial derivatives?

In the context of multivariable calculus, we can express the sum of the first and second power partial derivatives of a function as follows: $$\vec 1 \cdot \nabla f = f_x + f_y + f_z$$ $$| \nabla f |...
0
votes
1answer
32 views

Understanding Taylor Approximations

I am curious about what quantity a Taylor approximation actually optimizes, when it produces, as they say, the "best" possible nth-degree approximation of a function around the given x-value. ...
2
votes
1answer
15 views

Minimize the cost of a box of fixed volume if the sides are twice as expensive as the base and top

Suppose, to build a box (a rectangular solid) of fixed volume and square base, the cost per square inch of the base and top is twice that of the four sides. Choose dimensions for the box that ...
-5
votes
1answer
40 views

How to solve the integral $\int \frac{x}{\sqrt{x^2+1+{(x^2+1)}^{3/2}}}dx$?

$$\int \frac{x}{\sqrt{x^2+1+{(x^2+1)}^{3/2}}}dx = ?$$ Rewriting to $$\int \frac{x}{\sqrt{x^2+1}} \frac{1}{\sqrt{1+\sqrt{x^2+1}}} dx$$ Seems to be a good start, but it doesn't help too much.
0
votes
0answers
32 views

Deriving $t(x)$ for a cubic Bezier curve

I'm having some trouble deriving the function $t(x)$, given a Bezier curve with $x(t) = a(1-t)^3 + 3b(1-t)^2t + 3c(1-t)t^2+dt^3$ for its $x$ component, with further restriction that all coefficients ...
0
votes
0answers
28 views

$f^{(j)}(a) = 0$ except for the last derivative. Under which conditions $a$ can be a minimizer of $f$?

Let $f:\mathbb{R}\to\mathbb{R}$ and suppose that $f^{(j)}(a) = 0, j=0,\cdots,n-1$ and $f^{(n)}(a) \neq 0$. Under which conditions the point $x=a$ can be a minimizer of $f$? Based on your answer: $f(...
2
votes
1answer
130 views

Convergence/Divergence of an Infinite Series with Natural Logarithms

I've spent a good week and half manipulating and trying different tests to find the convergence or divergence of this series: $$\sum_{n=0}^\infty \frac{1}{(\ln n)^{\ln n}}$$ I've tried all the ...
1
vote
0answers
50 views

Can I switch the order of taking minimums?

Suppose I have some function $F(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}$. Is it always the case that $$ \min_{1 \leq x \leq N} (\min_{1 \leq y \leq M} |F(x,y)|) = \min_{1 \leq y \leq M} (\min_{1 \...
2
votes
1answer
33 views

Minimizing the criterion function $f(a) = \int_0^1 [g(x) - p(x)]^2\ dx$ by a polynomial

We can approximate a function $g$ in the interval $[0,1]$ by a polynomial of degree $\le n$ by minimizing the criterion function: $$f(a) = \int_0^1 [g(x) - p(x)]^2\ dx$$ where $p(x) = a_0 +...
0
votes
1answer
19 views

Given k food items, with constraints being the total weights of various nutrients, how do I find the minimum price p? [on hold]

Let's say I have six food items; each of them have four attributes: the amount carbohydrates (in grams), the amount of protein, the amount of fat, and the price (in USD). I'd like to buy a combination ...
0
votes
0answers
11 views

Minimize $\sum_{i=1}^n|x-a_i|$, $\max \{|x-a_i|, i=1,\cdots,n\}$, $\sum_{i=1}^n|x-a_i|^2$ and maximize $\Pi_{i=1}^n|x-a_i|$

$a_1\le a_2\le \cdots \le a_n$ real numbers. a) Minimize $\sum_{i=1}^n|x-a_i|$ b) Minimize $\max \{|x-a_i|, i=1,\cdots,n\}$ c) Minimize $\sum_{i=1}^n|x-a_i|^2$ d) Maximize $\Pi_{i=...
1
vote
2answers
34 views

Show that for each number $n > 0$, $nx^{1/n} < \ln(x)$ for $x > 1$

Show that for each number $n > 0$ $$ nx^{1/n} < \ln(x) $$ for $x > 1$. Update as pointed below, original problem set has a misprint, and the inequity should be $$ nx^{1/n} > \ln(x)...
0
votes
1answer
63 views

Example 20.10 Fitzpatrick's Advanced Calculus

According to the book Advanced Calculus by Fitzpatrick: The following example shows it is not true that a parametrized path $\gamma :[a, b] \to \mathbb{R^n}$ that has only the property that $\gamma ...
2
votes
1answer
40 views

How do I find the set of functions that would make this non-linear operator diverge?

I have this non linear operator $$H(p) = -\sum_{n=0}^ {\infty} p_n ln(p_n)$$ where $p_n$ are given by a function $p(n)$ when $n$ is a whole number. I want to find what set of $p(n)$ makes $H(p)$ ...
1
vote
1answer
45 views

Is this proof of Cauchy about limits valid?

I've read in some places that the following proof from Cours d'Analyse of Cauchy is not correct, but I can't find the mistake. Theorem: if the difference $f(x+1)-f(x)$ converges towards a certain ...
2
votes
2answers
65 views

Lagrange Theorem with $\mathbb{tan}(x)$

I take $\mathbb{tan}(x)$ in $[30°, 45°]$ and I want to find $f'(c)$. The hypothesis are satisfied. I compute: $$f'(c)=\frac{\mathbb{tan}(45°)-\mathbb{tan}(30°)}{45°-30°}=\frac{1-0.577..}{15°}\simeq0....
1
vote
0answers
26 views

Defining limit of a function in terms of neighborhoods

I am reading from the book Multidimensional real analysis vol I by Duistermaat and Kolk and am trying to understand the following theorem from it: Theorem 1.3.2: In the notation of Definition 1.3.1 ...
3
votes
1answer
69 views

Prove that if $\dfrac{ax^2+2bx+c}{\alpha x^2+2\beta x+\gamma} (\alpha\neq 0)$ has $3$ inflection points, then all of them lie on one line?

Prove that if the rational function $f(x)=\dfrac{ax^2+2bx+c}{\alpha x^2+2\beta x+\gamma} (\alpha\neq 0)$ has three inflection points, then all of them lie on one line? (All the parameters are real ...
0
votes
0answers
16 views

proof of Poincare Lemma in Spivak's calculus on manifolds

I wonder how the equation about $I(d\omega)$ holds in Spivak's book: $I(d\omega) = \sum_{i_1<...<i_l}\sum_{j=1}^n(\int_0^1t^lD_j(\omega_{i_1,...,i_l})(tx)dt)x^j dx^{i_1}\wedge...\wedge dx^{i_l} $...
2
votes
1answer
67 views

Why do we evaluate $-x^{z-1}e^{-x}$ as zero when explaining the gamma function through integration by parts?

The gamma function is the integral of $x^{z-1}e^{-x}$ If you integrate by parts you get two terms. The first one is $-x^{z-1}e^{-x}$ and this is bound by infinity and zero. If you plug in infinity, ...
6
votes
3answers
716 views

Problem involving Fundamental Theorem of Calculus

I'm working on the following question: Simplify the following: $$\frac{d}{dx}\int_x^{x^2}\frac{t}{\log t}\,dt$$ The solution key says this simplifies to: $$2x\frac{x^2}{\log(x^2)}-\frac{x}{\log(...
0
votes
1answer
24 views

Taylor in two variables: Can we know that two functions are different.

Consider the following setup: Two functions $f,g:\mathbb{R}^2\to\mathbb{R}$ that are twice continuously differentiable such that: $f(0,0)=g(0,0)$ $f_x(0,0)=g_x(0,0)$ $f_y(0,0)\neq g_y(0,0)$ Can we ...
0
votes
2answers
34 views

Pointer needed to integrate using u-substitution

I have been at this problem for a bit too long now, using a calculator online I got an answer that is ~6 but my answers are always in the ~2 range, I would really appreciate if someone can point me to ...