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

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0
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1answer
29 views

Little o(h) limit about h=0

I understand that generally if a function $f(h)$ is described as $o(h)$ that $f(h)$ has a smaller rate of growth than $h$ (like it would have to be $\sqrt{h}$). i.e. $\sqrt{h} = o(h)$, just like (for ...
2
votes
1answer
35 views

Proof that the equation $x^2=\sin x $ has only one real solution different than $0$

I started doing it as following: Let $f(x) = \sin x - x^2$ Using the fact that $\sin x> x-\frac{x^3}{3!}$, I got that $f(\frac{1}{2})>0$ and, as $0<\sin 1< 1 , f(1)<0$. So, as ...
0
votes
1answer
17 views

If $∇f(a)\cdot y ≤ 0$ for every vector $y$, why does $\nabla f(a)$ have to be zero?

If $f$ is differentiable at every point in $B(a)$ and $f(x)≤f(a)$ for all $x$ in $B(a)$, prove that $∇f(a)=0$. I actually did some work and found out that $∇f(a)\cdot y ≤ 0$ for every vector $y$. ...
0
votes
2answers
39 views

What are the standard defintions of “counterclockwise” and “clockwise” in 3d space?

I'm in Calc III right now, and I'm a little confused as to what constitutes "clockwise", and "counterclockwise" rotations when dealing with the various planes in 3d-space. Of course, it's obvious in ...
2
votes
2answers
68 views

How to solve such an integration analytically?

$\displaystyle\int^{2\pi}_{0} e^{ia \cos{\theta}}d\theta$ where $a$ is some constant. Can it be solved with some substitution? I tried it by expanding the exponential series but that was not proper ...
1
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0answers
18 views

Let $f$ be a scalar field such that $f ' (a ;-y)$ exsits [on hold]

Let $f$ be a scalar field where derivative of $f$ at point a with respect to vector $-y$ exists, $f '(a;-y)$ exists. Is it always true for any nonzero vector $y , f '(a;-y) = - f '(a;y)$?
1
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2answers
47 views

What is $\int \frac{1}{\sqrt{25y^2-10y-3}}dy$

$= \int \dfrac{1}{\sqrt{(5y-1)^2-4}}dy$ $=\int \dfrac{1}{\sqrt{u^2-4}}\dfrac{du}{5}, \quad U$ substitution $=\int \dfrac{1}{10\cos(\theta)} 2\cos(\theta) d\theta, \quad$ Trig substitution $= ...
0
votes
2answers
24 views

What does it really mean by a derivative in a sense of something per unit.

Suppose we are given the differential equation $\frac{dP(t)}{dt}=kP$ where $P(t)$ is a function of population with variable time measured in years. And say $k>0$ is the relative growth rate of the ...
1
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2answers
24 views

Finding a good comparison/bound for determining the convergence of a series

The series is defined as follows: $b_0=1,b_1=-7,b_k=2b_{k-1}+b_{k-2}$. I need to find a good comparison sequence to determine whether $\sum_{k\geq0}1/b_k$ converges. I considered using $1/k^2$, which ...
0
votes
1answer
80 views

Do every math operation derive from sum?

I've been told sometimes, that every math operation (sum, subtraction, exponentiation, square rooting, so on) can be transformed to a sum of operands. For example, subtraction can be made as ...
1
vote
1answer
79 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
1
vote
1answer
20 views

nullity and rank of the linear transformation $T: T [ p (x)]= p(x+1)$

Let $V$ be the linear space of all polynomials $p(x)$ of degree $\le n$. if $p$ belongs to $V$ and $q = T(p)$, means that $q(x) = p(x+1)$ for all real $x$. find nullity and rank of the linear ...
1
vote
1answer
29 views

Nullity and rank of the linear transformation $T[f(t)] = \int_a^b f(t) \sin (x-t) dt ~\forall~x \in [a,b]$

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ Then, the nullity and rank ...
2
votes
2answers
61 views

Show that total energy is conserved

Question is as followed $\textbf{F} = f(r) \textbf{r}$ where $r = |\textbf{r}|$ $$U(r) = -\int rf(r)dr$$ $$K=\frac{1}{2} m|\textbf{v}|^2$$ Show that $E=K+U$ is constant by deriving ...
0
votes
1answer
34 views

Finding $a$ and $b$ so that the function is continuous

$$f(x) = \begin{cases} \displaystyle\frac{x^2-4}{x-2}&\quad x<2\\[0.4em] ax^2-bx+3&\quad 2 \leq x <3\\[0.3em] 2x-a+b&\quad x \geq 3 \end{cases}$$ I can't make the right limit of ...
1
vote
0answers
38 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ .

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
1
vote
1answer
19 views

Linear Motion and derivatives

A particle moving along a line has position $s(t)=t^4-18t^2m$ at time $t$ seconds. At which times does the particle pass through the origin? At which times is the particle instantaneously motionless ...
3
votes
3answers
57 views

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. I tried few things but it wouldn't work out. I would appreciate your help.
1
vote
2answers
57 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
1
vote
1answer
20 views

Showing $\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$

Let $$y(h)=1-2\sin^{2}(2\pi h) , f(y)=\frac{2}{1+\sqrt(1-y^2)} $$ Justify the statement $$f(y(h))=2-4\sqrt{2}\pi+O(h^2)$$ where $$\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$$
0
votes
1answer
24 views

Find the equation of the tangent line to the curve $3x^3 + 3y^2-11=4xy-x$ at the point $(1,-1)$.

The answer choices are given below: a) $5x + 7y = -2$ b) $-7x+5y = -12$ c) $-5x + 7y = -12$ d) $7x+5y = - 12$ e) $-7x + 5y = 2$
3
votes
0answers
38 views

Non-analytic smooth function

The Wikipedia page (http://en.wikipedia.org/wiki/Non-analytic_smooth_function) proves that $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is a ...
2
votes
1answer
36 views

Confusion in understanding a proof in Apostol's Calculus I

I'm using Apostol's Calculus I as my introductory book to the subject and I'm stuck trying to understand the proof of Theorem 1.13 (page 79) which, as I understand, provides a way to compute the value ...
1
vote
2answers
34 views

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$ I've seen a proof which goes like this. $$ \lim_{x\to\infty} \frac{\frac{\arctan x}{x^2}}{\frac{1}{x^2}} = \frac{\pi}{2} > ...
0
votes
0answers
20 views

Are 'similar' differentials equal? Specifics enclosed.

I'm trying to show that two infinitesimally small changes in an angle are actually equal, i.e. I want to say that $d\phi = d\phi '$, where the change in $\phi '$ is caused by a change in $\phi$. Here ...
1
vote
0answers
41 views

is the sequence $[ne]$ convergence?

Is the sequence $a_n=[ne]$ convergence or partially convergence? ($e$ is the Euler's number and the bracket mean the integer part function.)
2
votes
4answers
117 views

Convergence of $\int_0^{\infty} x \cos (x^6)\,dx$

I feel that $\int_0^{\infty} x \cos (x^6) dx$ is convergent using the regular first year definition of an integral. I have been trying to convince a university professor of this, but according to him ...
2
votes
2answers
65 views

Approximating solutions for the ODE $y'=\exp(y/x)$

I am currently trying to solve excercise 1-38 from Mathews and Walker. In this excercise I am asked to consider the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\exp(y/x)$$ for two ...
1
vote
1answer
33 views

Finding extrema of a continuous, univariate function.

Problem: Let $f:[0,1]\to\mathbb{R}$ be given by $f(x)=a(x-b)^2+c$, where $a,b,c$ are parameters. Find the minimum and maximum of $f$ depending on the values of $a,b,c$. I understand how to do this, ...
2
votes
1answer
62 views

finding $\int\frac{1}{(t^2+25)^2} dt$ without trig substitution

Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate $$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$ without using ...
0
votes
1answer
38 views

Existence of solution for an equation including polynomial and trinogometric sum

Prove that the following equation has at least a solution in $[-\pi, \pi]$ : $$ x^5+\sum^{n}_{k=1}(a_k\cos kx+b_k\sin kx)=0 $$ I think the existence of the solution on $[-\pi, \pi]$ strongly depends ...
1
vote
2answers
48 views

Integration of the following

What is the definite integral of $$ \int_0^1 \left(\frac{g(x)}{f(x)}\right)'\cdot\frac{1}{g(x)}\,dx, $$ where the conditions are as follows: $f(0) = 2 $ $f(1) = 3 $ $f'(x) $ is continuous For all ...
0
votes
1answer
53 views

Evaluate the integral $\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$

Evaluate the integral $$\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$$ My Try: substituting $t = 2^x$ we get: $$\ln 2 \int_2^\infty \left(\frac{1}{2}\right)^t dt = \frac{\ln 2}{\ln 0.5} \left( ...
1
vote
1answer
39 views

Why are Fourier series important?

Are there any real life applications of Fourier series? Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this ...
1
vote
1answer
25 views

Maximum and minimum of a fractional function

Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the ...
3
votes
1answer
28 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
1
vote
1answer
31 views

Bounding the increments of $\sin(e^{-x})$

Show that $|\sin(e^{-b}) -\sin(e^{-a})| \leq {b-a \over e^{-a}}$ for all $a \leq b$ This is part of a basic calculus class so i would appreciate answers suitable for my knowledge.
0
votes
1answer
32 views

Prove that $\Delta^{n} f = 0$ i f and only i f $f$ has the form $f ( x ) = a_0(x) + a_1 ( x ) x +. . . + a_{n-1}(x)x^{n-1}$

For$f$ a real valued function on the real line, define the function $\Delta f$ by $\Delta f ( x )= f (x + 1) - f ( x ) $. For $n>1$ , define recursively by $\Delta^{n} f = \Delta(\Delta^{n-1} f)$. ...
3
votes
3answers
46 views

Prove this inequality.

Let $S=a_1+...+a_n<1$ where $a_i>0$. Prove that $1+S<(1+a_1)\cdot ... \cdot (1+a_n)<{1\over 1-S}$. I started with the right inequality but I am not sure it iss plausible (I did something ...
0
votes
0answers
45 views

Finding the limit of the series .

I am dealing with the following: $$\lim_{x\to 0}\left(\lim_{n\to\infty}\sum_{i=1}^n i^{\frac{1}{\sin^2(x)}}\right)^{\sin^2(x)}$$ What is the limit, it is of the type infinity raised to the power ...
1
vote
1answer
49 views

For which $\alpha > 0$ we can interchange the integral with the summation?

For which $\alpha > 0$ the equality holds: $$ \int_0^1 \sum_{n=0}^\infty x^\alpha e^{-nx} dx= \sum_{n=0}^\infty \int_0^1 x^\alpha e^{-nx} dx $$ We've learned that an interchange can be ...
0
votes
0answers
33 views

Please guide me what are the topics i need to study in maths from basic. [on hold]

I am not having good knowledge in maths.Please guide me what are the topics i.e (algebra,calculus,diff.eqn...)i need to study by step by step. please guide me.
0
votes
1answer
26 views

If $a$ is $1$-periodic, then $\Delta(af) = a \Delta f$

Why if $a$ is 1-periodic, then $\Delta(af) = a \Delta f$ for any function $f$ ? $$\Delta f(x)= f(x+1)-f(x)$$
0
votes
0answers
19 views

What is the meaning of sub-constant error?

The error is defined as $E \geq \frac{1}{2ab(1+ab)}$, where $a$ and $b$ are both positive . The claim is: if we fix the value $a$, then to get a sub-constant error $E$, we must ensure that ...
2
votes
5answers
86 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
1
vote
3answers
24 views

lim sup iff conditions - please help explain

Deciphering the definition of the upper limit, we see that $\limsup x_n=L$ is and only if the following two conditions are fulfilled: (a) $\forall\epsilon>0\;\;\exists N\in\mathbb{N}$ such that ...
2
votes
2answers
256 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
0
votes
3answers
41 views

how to show that this function is continuous for all real numbers?

I'm having hard time playing with trigonometric functions. I want to show that this piecewise function is continuous for all real numbers (from $-\infty$ to $\infty$) a) $g(x)=$ \begin{cases} ...
0
votes
1answer
34 views

Suppose $f(g(t))$ is differentiable. Does this neccesarily imply that $g(t)$ is differentiable?

Suppose $f: \mathbb R \rightarrow \mathbb R$ given by $t \rightarrow f(g(t))$ is differentiable. I've been thinking whether this imply $g(t)$ must be differentiable ? I know the chain-rule, but ...
0
votes
1answer
19 views

Proof of floor function identity.

Let $f(x) = \lfloor x \rfloor$ and let $l$ be the greatest integer $\le x$ How do I prove $l + 1 > x$ I see that: $x \ge \lfloor x \rfloor = l$ No complete answers, just hints