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

learn more… | top users | synonyms

0
votes
2answers
33 views

Is it a good practice to write this integral in this form?

I'm trying to compute the following integral: $$\int e^{3x}\cos2x \;dx$$ Now I'm about to use the integration by parts. Suppose that I do not know what is the integral of $\displaystyle \int ...
0
votes
0answers
12 views

In the derivative of an integral, can I compare the size of the limit effect to the area effect?

I have minimal formal math training and sometimes encounter problems like this where I am not sure what relevant techniques are available to use. Thanks for any advice you can give. My integral looks ...
0
votes
1answer
50 views

Converging integral $\int_1^\infty {\frac{\sqrt{x}\cos{x}}{x+2013}}dx$

I want to show that $$\int_1^\infty {\frac{\sqrt{x}\cos{x}}{x+2013}}dx$$ is converging. I tried $${\frac{\sqrt{x}\cos{x}}{x+2013}}\leq {\frac{\sqrt{x}\cos{x}}{x}}\leq \frac{1}{\sqrt{x}}$$ but it ...
1
vote
0answers
23 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
0
votes
0answers
18 views

Approximate fraction of two integrals

could you propose a way to simplify or approximate (under some assumptions) $\bar{\eta}$ defined as below? $$ \bar{\eta} = \frac{\int f(t)dt}{\int\frac{f(t)}{\eta{(t)}}dt} $$ The $f(x)$ and ...
0
votes
2answers
31 views

Analysis of continuity and differentiability of a function

Find a,b,c $\in \mathbb{R}$ for which the function is a) continuous, b) differentiable. $$f(x)=\left\{\begin{array}{cc} ax^2+bx+c & x<0 \\ 2\sin x+cos x & x\:\ge 0 \end{array}\right.$$ ...
1
vote
0answers
35 views

Inner Product Properties And Applications

In every calculus or analysis class we are told that the concept of inner product is very important, and that its applications are vast, diverse, and extremely useful. I don't think there is a single ...
-3
votes
1answer
26 views

Proving volume of a general cone using Gauss theorem

Please assist me with answering this question: The parametrization of a general cone is $$(x(t,\mu),y(t,\mu),z(t,\mu))=\left((1-\mu)C_1(t),(1-\mu)C_2(t),\mu h\right)\quad\quad 0\leq t\leq b,\quad ...
2
votes
4answers
68 views

Is this correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$

I know the formula $ {d \over dx} x^x = x^x \cdot( 1+ \ln x ) $, but is below evaluation correct? $ {d \over dy} (1+xy)^y = (1+xy)^y \cdot (1+x \cdot \ln(1+xy))$
0
votes
2answers
46 views

Help with calculating the sum of series

I need help to calculate the following sum: $$\sum_{n = 1}^{\infty} \frac{n}{n+1} x^{n}$$ I managed to develop it to: $$\sum_{n=1}^{\infty}x^{n} -\sum_{n=1}^{\infty} \frac{1}{n+1} x^{n}$$ But now ...
0
votes
1answer
41 views

Evaluation of the integral $\int_0^1 e^{2t^2 -at} dt$

I would like to integrate a function in the range $[0,1]$. I tried a lot of ways including Mathlab. All solutions come in terms of some error function. I would like the answer in terms of $a$. ...
2
votes
4answers
104 views

Integrating $\sqrt{1-x^2}$ without using trigonometry

I am a beginning calculus student. Tonight I had a thought. Maybe I could calculate $\pi$ using integration, but no trig. The problem is that I don't really know where to start. I thought perhaps I ...
-1
votes
0answers
32 views

What are the main kinds of mathematics? [on hold]

I stumble upon as much on math I don't know (trascendal math, number theory) and math I know on the internet and elsewhere. I have a pretty good idea about differential and integral calculus, and I'd ...
0
votes
3answers
18 views

Radius of Convergence

Is the radius of convergence of $$\frac{n(x+3)^n}{4^n}$$ equals 4? I got $|x+3|\lt 4$ as the final result. How do you know, what is the radius from here?
1
vote
2answers
40 views

Proving a partial derivative identity

I'm currently studying for a resit and I've been faced with this partial differentiation question: If $z = f(y/x)$ show that $$x^2\frac{\partial^2 z}{\partial x^2}+2xy\frac{\partial^2 z}{\partial ...
4
votes
5answers
198 views

Showing a function is injective using that $f'(x)\ne0$

Given a differentiable function $f\colon \mathbb R\to\mathbb R$ which we must prove to be injective, does it suffice to show $f'(x)≠0$ for all $x$ (for which the function is defined)? It makes sense, ...
0
votes
3answers
34 views

Evaluating the closed integral of an elliptical path

I've been working on a problem that states: Evaluate $\int F*dr $ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k $ and C is the elliptical path given by $$ x^2+4y^2-8y+3=0 $$ in the xy-plane, traversed ...
2
votes
2answers
55 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
1
vote
4answers
201 views

Is the following Alternating Series Absolutely Convergent?

$$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}$$ I think it is Absolutely Convergent because it converges by direct comparison to Harmonic series? Am I right or wrong?
3
votes
1answer
75 views

Finding $\sum\limits_{k=0}^n k$ using summation by parts

This is another exercise from Smoryński's Logical Number Theory; not being a mathematician, I'm a bit new to this finite difference stuff, so, please, bear with me! In a previous exercise, Smoryński ...
2
votes
4answers
53 views

According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem?

In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does $$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$ simplify to ...
0
votes
1answer
13 views

Small angle approximation of the integrand of the arc length

I am confused by the following approximation made in a book: The justification made in the book is as follows: "In the case of a flat curve, the quantity (dy/dx)^2 is small in comparison with unity ...
0
votes
3answers
74 views

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$ [on hold]

I have great doubts solve this exercise by integral by parts $\int_{0}^1 \int_0^1 x\cdot e^{xy}\, dy\, dx$
1
vote
1answer
60 views

A mean value theorem involving two functions [duplicate]

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$ I ...
0
votes
1answer
15 views

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis

I am having a little trouble figuring out how to integrate this problem. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. ...
0
votes
1answer
20 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
142 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
91 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
69 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
57 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
47 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
43 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
58 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
129 views

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

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
36 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
22 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
62 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
25 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
70 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
118 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
55 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
27 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
55 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
35 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
27 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 ...