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

learn more… | top users | synonyms

1
vote
2answers
45 views

Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
2
votes
4answers
46 views

$f$ is integrable & continuous over $[a,b]$ , $\int_{a}^{b}f(x)dx \geq 0$ for any subinterval $(\alpha,\beta)$ of $(a,b)$, then $f \geq 0$ in $[a,b]$

Some known things about this problem are: if $f(c) < 0$, $a < c < b$, then $f(x) < f(c)/2$ in some neighborhood of $c$, but I am not exactly sure how to use this to get to my goal of ...
1
vote
2answers
41 views

If $f$ is integrable on $[a,\ b]$ and $\int_a^b f(x) \mathrm dx >1$, then there exists a point $c$ in $(a,\ b)$ such that $f(c) > \frac{1}{b-a}$

So far for this problem, to my understanding, for something to be integrable means that $U(p,\ f) - L(p,\ f) < \epsilon$ but not sure how exactly to move beyond there to show that there exists a ...
-3
votes
1answer
40 views

evaluate the following integration [closed]

I'd like help with this question : $\int_{0}^{3}\sqrt{x}, $ dx
-1
votes
1answer
44 views

Please solve this… [duplicate]

Please prove that $$e^x - x$$ is always bigger than or equal to 1
2
votes
1answer
55 views

Evaluate $\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}(1-\frac{t}{\sqrt{\alpha}})^{-\alpha}$

How does one show $$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$ Not homework, this is from this proof that the gamma distribution ...
2
votes
0answers
32 views

When is $\int { \frac { \alpha { x }^{ 2 }+2\beta x+\gamma }{ \left( \alpha { x }^{ 2 }+2bx+c \right) ^{ 2 } } } dx$ a rational function?

Question is: In which case(s) is this integral a rational function? $$\int { \frac { \alpha { x }^{ 2 }+2\beta x+\gamma }{ \left( \alpha { x }^{ 2 }+2bx+c \right) ^{ 2 } } } dx$$ Thanks in ...
1
vote
3answers
59 views

prove continuity

Let $ f:\Bbb R \to \Bbb R $ satisfy the property $ f(x+y)=f(x)+f(y)$ for all $x,y$ in $ \Bbb R $ I have to show that 1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in ...
2
votes
1answer
40 views

Second Mean Value Theorem for Integrals Meaning

The Second Mean Value Theorem for Integrals says that for $f (x)$ and $g(x)$ continuous on $[a, b]$ and $g(x)\ge 0$ $$\int_a^bf(x)g(x)\,dx=f(a)\int_a^cg(x)\,dx+f(b)\int_c^bg(x)\,dx$$ I have a ...
3
votes
1answer
23 views

Generalized linear combination of probability density functions

I am working with linear non-unity combinations of independent variables in the equation form of: $$Y_i=\sum_{j=1}^N a_{ij} X_j ~~~~\forall~ a_{ij} \in \mathbb{R}, a_{ij}\neq 1$$ I am aware of the ...
4
votes
1answer
47 views

double integral problem $\iint e^{\frac{x}{x+y}}dxdy$

I'm trying to integrate $$\iint e^{\frac{x}{x+y}}dxdy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$. I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have ...
1
vote
1answer
64 views

Multiple Integrals

$$\int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \ln(w+x+y+z) }\ dw\; dx\; dy\; dz } } }$$ Unfortunately I cannot think of how to approach this problem. The only ...
-2
votes
0answers
26 views

psql and Eisenstein series [closed]

I found using PSQL the following relation; $$\sum _{k=1}^{\infty } \left(\frac{(-1)^{k+1} 2^{-3 k} k^3 \pi ^{-6 k}}{1-2^{-3 k} \pi ^{-6 k}}-\frac{8 \pi ^4 k^3 \text{csch}\left(\frac{2 \pi ^2 k}{3 \log ...
-1
votes
2answers
70 views

Infinite Series for Arctan [closed]

$$ \sum_{k=1}^{\infty}\arctan\left(\frac{1}{k^2}\right) $$ Does anyone know how to determine if this infinite series diverges or converges and if it converges, what its value is?
3
votes
1answer
38 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
-1
votes
2answers
42 views

How to solve $\lim_{x\to-\infty} \frac{\left|x + 1\right|e^{-x}}{x} $?

I'm trying to solve this limit $$\lim_{x\to-\infty} \frac{\left|x + 1\right|e^{-x}}{x} $$ but I get stuck with $$\lim_{x\to-\infty} -\frac{xe^{-x} + e^{-x}}{x} $$ I've tried to transform it in ...
2
votes
1answer
65 views

Is this simple calculus proof formal enough and correct?

There is function $f$ differentiable at $x=0$ and $f'(0) = m > 0, f(0) = 0.$ I need to prove that there is $K > 0$ and $\delta > 0$ that for every $0<x<\delta$ : $f(x) > Kx$. So I ...
-5
votes
1answer
26 views

How to find the values of $c$ that satisfy the conclusion of the Mean Value Theorem for $f(x)=x^{1/3}$ on the interval $[0,8]$? [closed]

Verify that the function $f(x)=x^{1/3}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0,8]$ . Then find all numbers $c$ that satisfy the conclusion of the Mean Value ...
-3
votes
0answers
40 views

Find $\lim_{x\to 0^+}\frac1x\int_0^{2x}\sin^t(t)\ \mathrm{d}t$ [closed]

Find the $$\lim_{x\to 0^+}\frac1x\int_0^{2x}\sin^t(t)\ \mathrm{d}t$$
0
votes
1answer
47 views

compute $\nabla f$ for a function over a cone

Let $D$ be the cone $D=\{rt:r>0, t\in\Omega\}$ with $\Omega\subset S^{n-1}$. I want to show that $$ \frac1{r^2}\int_{B_r}\frac{|\nabla f(x)|^2}{|x|^{n-2}} dx= C(n,g)r^{2(a-1)} $$ where $C(n,g)$ is ...
1
vote
1answer
41 views

Differential of a tricky function

I have a function that I'm strugling to take the differential of. $$F(t) = F(t-a)G(t).$$ My attempt is the following: $$ dF(t) = F(t-a)dG(t) + G(t) dF(t-a)) $$ but I have a feeling something is not ...
0
votes
1answer
47 views

Sum on integration and binomial theorem.

If $(1+x)^n = \sum_{r=0}^n \binom{n}{r}x^r$ and $$\sum_{r=0}^n \frac{(-1)^r}{(r+1)^2} \binom{n}{r} = k\sum_{r=0}^n \frac{1}{r+1}$$ Then prove that $$k=\frac{1}{n+1}.$$
-2
votes
2answers
105 views

Calculate definite integral $\int_0^{\pi/2} 3\sin x\cos x/(x^2-3x+2)\; dx$ [closed]

Please help to calculate definite integral $$ \int_0^{\pi/2} \frac{3\sin x\cos(x)}{x^2-3x+2} \, dx. $$ I feel that there is a trick somewhere, but I cannot understand where?
1
vote
4answers
105 views

Limit $(e^x+x)^{1/x}$, when $x\to 0$

Can I expand e^x in the limit $$\large{\lim _{x\to 0}(e^x+x)^{1/x}},$$ just as $1+x$ according to the Taylor expansion? I mean is it normal to think about limits of a form $(1+x+o(x))^{1/x}$ just as ...
2
votes
4answers
38 views

Translating basic limit intuition to epsilon delta definition

Early on in precalculus/calculus I learned that a $$\lim_{x\to c} ~ f(x)$$ does not exist if $$\lim_{x\to c^{+}} ~ f(x) \neq \lim_{x\to c^{-}} ~ f(x)$$ I'm having trouble understanding how to ...
-3
votes
1answer
44 views

A problem about prism with triangular bases

Consider a prism with triangular base . The total area of the three faces containing a particular given vertex is $k$ . Then is the maximum possible volume of the prism $\sqrt {\dfrac {k^3} {54} } $ ? ...
-1
votes
1answer
42 views

ode and area of triangle

Question: find a curve $x$ so that the area bounded between it's tangent at some point $t$ and the time axis on the interval between the point of contact of $x$ and it's tangent ( $t$ ), and the ...
0
votes
0answers
27 views

Minimal boundary conditions for divergence theorem

I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ...
1
vote
3answers
69 views

If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?

I am asked to prove something that assumes this. But is it true that derivative is necessarily defined at the "edges" of the domain of the definition of its function? Does it matter if the original ...
0
votes
3answers
71 views

Finding the maximum value of the function $( x^2-x+1)^{ 1/3}$ on the interval $[0,1]$

While finding the maximum value of the function $( x^2-x+1)^{ 1/3}$ on the closed interval $[0,1]$ the point where the derivative is $0$ is $1/2$ and $f(1/2)=(3/4)^{1/3}$. Then, for $f(0)=1$ and ...
0
votes
1answer
19 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
0
votes
0answers
33 views

Proof that all derivatives at zero equal zero [duplicate]

Trying to prove that given $$ f(x)=\begin{cases} e^{-{\frac {1}{x^2}}} & \text{if $x\ne0$}\\[6px] 0 & \text{if $x=0$} \end{cases} $$ that $\ f^{(n)}_{(0)}=0$ for every n$\ \in\mathbb N$ ...
2
votes
1answer
27 views

Why not an Absolute maximum in an open interval?

The function $x^3+x^2\: \text{has a maximun value at}\: x=-\frac{2}{3} \text{in (-1, 0) }.$ My question is why call it a Local Maximun and not an Absolute Maximum when it is the highest value in that ...
0
votes
0answers
46 views

Limit tending to infinity

Given that $f(x)$ is continuous on $[0,\infty]$. If $\lim\limits_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists then evaluate $\lim\limits_{x\to\infty}f(x)$
1
vote
1answer
32 views

how x y and z become equal in this solution?

I'm trying to understand an example given in my book but not able to understand it as I am quite weak in mathematics. In the below images I don't get how x, y and z become equal to each other. Please ...
1
vote
1answer
23 views

When are monic polynomials of fourth degree divisible?

Note that this might be an X/Y problem, therefore I'm posting the original question too. I am asked to prove that given a monic polynomial of fourth degree which has a non-zero root, must have at ...
6
votes
1answer
110 views

How to compute the integral $ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $

I am currently working on this question and the following integral came up: $$ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $$ for a suitable c. I would like to compute it in terms of ...
0
votes
3answers
48 views

How derivatives of integrals like $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$ are computed?

I need to find the derivative of $G(x)=\int_{x-\sin x}^{\sin x}\arcsin(t)dt$. I know I don't have to find the integral, but I just have troubles computing the derivative. I suppose I get: ...
2
votes
2answers
77 views

If $|u+v| = |u| + |v|$ then $u = \lambda v$. How do I prove $\lambda \ge 0$?

I'm trying to prove that if $|u+v| = |u| + |v|$ implies $u = \lambda v$ for $\lambda \ge 0$. To this end I have $|u + v|^2 = (|u| + |v|)^2 \Rightarrow |u|^2 + |v|^2 + 2|u||v| = |u|^2 + |v|^2 + ...
0
votes
1answer
39 views

Prove that for every $x$ in the area of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$

I need to prove 2 things: Prove that for every $x$ in a neighbourhood of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}.$ What I did is that I calculated the derivatives of ...
4
votes
1answer
80 views

Simplify P(n), where n is a positive integer : $ P(x)=\sum \limits_{k=1}^\infty \arctan\left(\frac{x-1}{(k+x+1)\sqrt{k+1}+(k+2)\sqrt{k+x}}\right). $

This is what I have tried, but I don't know what to do next, so I need help : $ P(x)=\sum \limits_{k=1}^\infty \arctan\left(\frac{x-1}{(n+x+1)\sqrt{n+1}+(n+2)\sqrt{n+x}}\right). $ $ ...
5
votes
3answers
99 views

Solution of limit $\lim\limits_{x\to 0} \frac {e^{-1/x^2}}{x} $

Small question, I'm trying to solve this limit but I just can't wrap my head around this problem. $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$ L'Hopital just seems to make it messier. It's probably ...
0
votes
1answer
47 views

wrong result for $\int_{0}^{+\infty} e^{-\sqrt{x}}dx$

I have some problem with the result of this integral: $$\int_{0}^{+\infty} e^{-\sqrt{x}}dx$$ The result should be 2 but I get ...
0
votes
0answers
25 views

measurement of acceleration of an object over time

If I have a battery that weighs 26kg and this battery is placed in a car and the car hits a 15 cm bump (Angle 20 degrees) (to slow down the car), is it possible to calculate the speed of the car ...
1
vote
2answers
71 views

Why is $\int_0^{2\pi}{\sin x\over x}$ bigger than 0?

Why is $\int_0^{2\pi}{\sin x\over x}$ bigger than 0? I really try to understand the concept. If I were asked if $\int_0^{\pi}{\sin x\over x}$ bigger than 0, I would know it is $\int_0^{\pi}{\sin ...
3
votes
1answer
82 views

Evaluate $\int_0^a \frac{1}{x}\mathrm dx$

This would be $\ln(a)-\ln(0)$. Because $\ln(0)$ is undefined, is this integral undefined too?
3
votes
1answer
50 views

Evaluate $\int \dfrac{2\pi y}{2y^3-1}dy$

Evaluate: $$\int \dfrac{2\pi y}{2y^3-1}dy$$ I've been struggling with this for a while. If it had just been $y^3$ instead of $2y^3$ in the Denominator, Partial Fraction Decomposition, although ...
0
votes
1answer
33 views

how to calculate integral of product of exponential function and trigonometry function?

Let $x_0$ and $\sigma$ are constants. How to calculate this? $$ \int^{L}_{-L}e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos x dx $$ I think i can solve that with integration by parts. But I'm confused how to ...
2
votes
2answers
23 views

Property of polynomials proof

Let$$P(z)=\sum_{k=0}^n a_kz^k=a_0+a_1z+...+a_nz^n$$ be an N-th degree polynomial of a complex variable z, where the $a_k$ are complex constants. Now,$$\vert a_0\vert-\vert a_1\vert x-...-\vert ...
0
votes
0answers
17 views

Find $U \subset D $ with $|U|$ minimal s.t $v = (1/2,1/2,1/2)$ belongs to the convex hull of $U$.

Consider the convex hull $\text {conv} \{e_1,e_2,e_3,(1,1/2,1/2),(1/2,1,1/2), (1/2,1/2,1)\}$ in $\mathbb R^3$ and the vector $v = (1/2,1/2,1/2)$. I want to compute $U \subset ...