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

learn more… | top users | synonyms

0
votes
1answer
11 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
1
vote
4answers
83 views

How to find this function's limit?

Let $$ \lim_{x\rightarrow 0} \frac{(x + 1)^{\frac{1}{x}} - e}{x} = ? $$ How would you calculate it's limit? I thought using l'hopital rule, but it then becomes something nasty, as the differentiate ...
2
votes
2answers
75 views

How to calculate $\int \frac{\sin x}{\tan x+\cos x} \, dx$

How to calculate $$\int \frac{\sin x}{\tan x+\cos x} \, dx\text{ ?}$$ I got to $$\int \frac{-u}{u^2-u-1} \, du$$ while $u=\sin x$ but can I continue from here?
0
votes
1answer
67 views

Using the ratio test when the limit of ratio is infinity

If the limit in ratio test is infinity. Does the sequence converge? I suspect not as it is infinity and not some finite value but I'm not sure. Any help?
1
vote
3answers
51 views

Solving a double integral using change of variables.

$$\int ^{1}_{0} \int^{1}_{y}e^{-x^{2}}\,dx\,dy$$ To solve this I know one must use change of variables, but the problem is that I do not know how to approach the actual change. Just thinking out ...
0
votes
0answers
43 views

Can there be a unique function for $\int_{\frac{1}{n}}^n \sinh(x) \, dx$ when including parameters of $ 0 < n < 1 $ [on hold]

$$ \int_{\frac{1}{n}}^n \sinh(x) \, dx$$ For this function, depending on what the n value is, you end up getting different areas. For all values of $n$, the function creates: $ ...
1
vote
1answer
41 views

Fourier series converges

Suppose $S_N(x)$ is the Fourier series of $f(x)$, a continuous function. Now, I've understood that if $S_N(x)$ converges uniformly to some $g(x)$ then is must be that $f\equiv g$. What about the ...
3
votes
4answers
53 views

Is it allowed to write $\min\{\delta_k\mid\forall k\in\Bbb N\}$ when nothing is known about the $\delta s$

I encounter this case many times in calculus problems, but I'm never really sure it is legal. The question is, can I write $\min\{\delta_1, \delta_2, \dots\}$ when nothing is known about the ...
0
votes
0answers
15 views

multivariable Taylor polinomials

I'm trying to find the Taylor series of \begin{equation*}e^{-(x^2+y^2)}\cos(xy) \textrm{ : up to 4'th order around } (0,0) \end{equation*} \begin{equation*}e^y\tan(x) \textrm{ : up to 3'rd order ...
-4
votes
4answers
121 views

How to prove that series $\frac{1}{n+1}$, as $n\to \infty$ is zero.

Can somebody explain how to prove that series $\frac{1}{n+1}$, as $n \to \infty$? I mean infinite series, not sequence, and I want to understand how to define the partial sum when n goes to infinity. ...
0
votes
1answer
35 views

Why does this inequality hold, formally looking at it? Can someone prove this?

$$d_2, d_1-\text{metrics in } R^k$$ $$d_2(x,y)=(\sum_{i=1}^{k}|x^i-y^i|^2)^{1 \over 2} \\ d_1(x,y)=\sum_{i=1}^{k}|x^i-y^i| \\ d_2(x,y) \leq d_1(x,y) \leq \sqrt{k}\ d_2(x,y)$$
1
vote
2answers
21 views

Question on metric spaces.. 2 properties which I don't know whether they apply

Do these two properties hold in all metric spaces. In my textbook, it says they hold in spaces, that have defined scalar products, but I am interested if they hold in generally metric spaces: $$1.) ...
0
votes
1answer
39 views

how to solve this formula limits's formula [duplicate]

I have already known one way to solve this formula but I just want to know the easier way to do so: $$\lim_{ u \to 0} \frac{\sin(u)}{u}=1$$ Please kindly help me!! Thank You.
1
vote
3answers
39 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
2
votes
1answer
59 views

What did I do wrong?

So, I have found the following problem. This problem is a multiple-choice one, and I have to pick the correct answer. The problem, gives a function $f:D \to R$, $$f(x)=\frac{xe^x}{e^x-a}$$ with $a$ ...
0
votes
0answers
18 views

A problem on Constrained Motion

Q. A particle is moving in a smooth curve under gravity and its velocity varies as the actual distance from the highest point. Prove that the curve is a cycloid. Attempt: The eq. of motion is ...
0
votes
0answers
24 views

Change of variable in double and triple integrals?

I learn double and triples integral as same as change of variable and then surface integral in my class so there is some conflict between how to do double integrals Here is how the text book say ...
2
votes
3answers
67 views

Very difficult sequence

How can I show this? I tried with the definiton of the exponential function but it didn't work. $$\displaystyle \lim\limits_{n\to\infty} \frac{(4n^7+3^n)^n}{3^{n^2}+(-9)^n\log(n)}=1$$ I hope ...
4
votes
5answers
115 views

Differentiate expression involving reciprocal of square roots.

I need to differentiate $$5\over 2+\sqrt{1+3x}$$ I can get the answer from Wolfram Alpha but I'm trying to understand the working. Do I use the chain rule? My calculus is at the basic level.
2
votes
1answer
51 views

Why are the two limits equal?

I want to show that if $g$ is continuous at $a$ and $f$ at $g(a)$, then $$\lim_{x \to a}{\frac{f(g(x))-f(g(a))}{g(x)-g(a)}} = \lim_{x \to g(a)}{\frac{f(x)-f(g(a))}{x-g(a)}}$$ Now I know that ...
2
votes
1answer
25 views

Find all $n \in \mathbb N$ such that $g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$ is differentiable $\forall x$

Find all $n \in \mathbb N$ such that $$g(x) = 100|x+1| - \sum_{k=1}^{n}|x^k+1|$$ is differentiable $\forall x$. It's my high school calculus problem. Is it possible to solve this problem in the high ...
1
vote
1answer
14 views

Implicit differentiation and linear approximations

Consider the implicit function $$(w(x)+1)e^{w(x)}=x.$$ I need to approximate $w(1.1)$ using the fact that $w(1)=0$. Could you give me any hints?
3
votes
1answer
51 views

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, ...
3
votes
1answer
32 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
0
votes
1answer
26 views

Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
0
votes
1answer
25 views

All surfaces through a common “concur-line” [on hold]

Find all second degree surfaces passing through a common given parameterized space curve of intersection: $$ (x,y,z) = (\pm \sqrt {2 t ( 1-t)} , t , (1-t) ) $$ using a single variable parameter ...
0
votes
2answers
20 views

Bounding $\delta$ without the loss of generality while proving non uniform continuity

To show $f(x)$ isn't uniformly continuous on an interval $I$, I show that there is an $\epsilon$ such that for every $\delta$ there exist $x, y \in I$ such that $|x-y|<\delta$ and ...
1
vote
2answers
33 views

Proving $\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$ diverges

Consider $f(t)$, continuous on $[0,1]$, and $\alpha > 1$, and: $$\int_0^1 \frac{f(t)}{t^{\alpha + 1}} \ dt$$ How can we tell this integral diverges? Basically since $f$ is continuous it reaches ...
4
votes
2answers
51 views

Prove $f_n\to f'$ uniformly on $[0,1]$

Let $f:[0,2]\to \Bbb{R}$ be a continuously differentiable function. Let us define $f_n:[0,1]\to \Bbb{R}$ by $f_n(x)=n(f(x+{1 \over n})-f(x))$. Prove $f_n\to f'$ uniformly on $[0,1]$. I know that ...
0
votes
1answer
32 views

A strange growth speed equation

This question has had me stumped for months, now... It is as quotes: The population of fish in a bay (measured in thousands of fish) at time $t$ is described by the function $p(t) = t^4 + t^2 + ...
4
votes
4answers
403 views

Why can we treat infinitesimals as real numbers in integration by substitution?

During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, ...
0
votes
2answers
40 views

Verify a property of limits

Is it generally true that for two functions $f$ and $g$, $$\lim_{x \to g(a)}{f(x)} = \lim_{x \to a}{f(g(x))}$$ as long as both exist? This seems to be true when $f$ is continuous at $g(a)$ and $f ...
1
vote
1answer
32 views

Calculate the derivative

I'm asked to find the derivative of the following: $$ \sqrt[4]{x} + \sqrt[3]{3x} $$ I attempted to solve the problem and got the following result, but my book says I am wrong. $$ \frac 14x^{-\frac ...
2
votes
0answers
35 views

Differentiable and concave functions with the following properties? [on hold]

What are all differentiable and concave function $f: [0, \infty) \to [0, \infty)$ with the following properties: $f'(0) - 1 = 0$. $f(f(x)) = f(x)f'(x)$, whenver $x \in [0, \infty)$.
5
votes
5answers
126 views

A limit problem: $\lim\limits_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$

I need help in solving the limit below: $$\lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$ What I've done is to simplify ...
-3
votes
2answers
68 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$ [on hold]

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$\cos^2 x + \sin^2 x=1$$ Express the number $2$, ...
0
votes
1answer
52 views

Zeros of an analytic function

How to prove zeros of a real analytic function (non-zero function) is always countable?
0
votes
1answer
25 views

Infinite sub-sequences that make up a sequence

A sequence $\{a_n\}$ can be broken into sub-sequences, $\{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}$, if every element in $\{a_n\}$ belongs to at least one of the sub-sequences. I had to ...
0
votes
4answers
40 views

Find $y'$ and $y''$ : $ y=x^2\ln(2x)$

for $x> 0$ : $ y=x^2\ln(2x)$ Product rule: $$(x^2)\cdot[\ln (2x)]'+ (\ln (2x))\cdot[x^2]' $$ $$y'= x^2\frac{1}{2x}\cdot (2)+\ln(2x)\cdot(2x) =2x\ln(2x)$$ ...
0
votes
1answer
68 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{\sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2\sin(t)\cos(t)} \implies \lim_{t \to 0} \frac{2t}{\sin(2t)}$$ ...
0
votes
2answers
15 views

How to prove this claim about this function?

I thnk first I need to show the that the function is monotonic. Or maybe use it for the proof of this claim : for all $$c > 0 % MathType!MTEF!2!1!+- % ...
0
votes
0answers
26 views

Derivative of indicator function and summation

I would like to take the derivative of $G = \sum_{i=1}^{n}\Big(\mathbb{1}\{i \geq x + k\} v(x) + \mathbb{1}\{i < x + k\} v(i)\Big)$ with respect to $x$, where $\mathbb{1}\{\cdot\}$ is the indicator ...
1
vote
5answers
55 views

Differentiate the Function: y=$\ ln\ tan^2x$

y=$(ln\ tan^2x)$ = $2(ln\ tan\ x)\cdot (ln\ tan\ x)'$ =$2(ln\ tan\ x)\cdot \frac{1}{(tan\ x)'}$ =$\frac{2(ln\ tan\ x)}{sec^2x}$ Is this right if not what am I doing wrong?
0
votes
2answers
41 views

Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
3
votes
1answer
48 views

Finding $\sum\limits_{k=0}^n (f(k)g(k))$ (calculus of finite difference)

So, I'm working though Smoryński's Logical Number Theory and I'm stuck on the following exercise. Define $\Delta f(x) = f(x+1) - f(x)$. Given this, it's not difficult to show that $\Delta (f(x) g(x)) ...
2
votes
3answers
64 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
1
vote
2answers
45 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
-3
votes
0answers
45 views

What comes after Spivak/Apostol/Courant?

Calculus texts by these authors are the essential archetypal references for undergraduate students. Subsequent courses: linear algebra, multivariable calculus, differential equations. Any in the ...
0
votes
2answers
41 views

Differentiate the Function: $g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$

$$g(u)=\ln\left(\frac{\ln\ u}{1+\ln\ (2u)}\right)$$ $$=\ln\ (\ln\ u)-\ln(1+\ln\ (2u))$$ This is the part where I get a little confused. Keep in mind I am using this formula $$\frac{d}{dx}[\ln ...
0
votes
1answer
53 views

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge?

For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge? I have seen a duplicate of this question but the answer there, though very good and creative, isn't clear about negative values. When ...