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

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2
votes
2answers
34 views

Integration by parts and $dx$ notation

Please overview this integral evaluation: $$ \int x^3 \arctan(x^2)dx = \frac{x^4}{4}\arctan(x^2) - \int \frac{1}{1+x^4}2x dx $$ Let's evaluate the right term: $$\int \frac{1}{1+x^4}\color{Blue}{2x ...
13
votes
3answers
320 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
1
vote
1answer
25 views

How to divide trigonometric ratios using identities?

$$\frac{1-\tan^2x}{1+\tan^2x}$$ We know: $$\frac{1-\frac{\sin^2x}{\cos^2x}}{1+\frac{\sin^2x}{\cos^2x}}$$ Now what? Flip denominator and times numerator? Which equals ??? Please help - Thanks
3
votes
2answers
114 views

Find $\int_0^a{f(x)}\, dx$

SMT 2013 Calculus #8: The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, ...
0
votes
0answers
19 views

monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
0
votes
1answer
30 views

If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, then prove that $\sum_{n=1}^\infty a_n $ converges

Let $\{a_n\}, \{b_n\}$ be positive sequences. Let $c_n= b_n-\dfrac {b_{n+1}a_{n+1}} {a_n}$. Prove that : If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, ...
1
vote
2answers
40 views

Convergence of $\sum_{n=1}^\infty \frac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $

Test convergence of $\sum_{n=1}^\infty \dfrac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $ Attempt: By using the the $n^{th}$ root test : $\lim_{n \rightarrow \infty} a_n^{1/n} = \lim_{n \rightarrow \infty} ...
0
votes
0answers
34 views

Differentiable only at $x=0$ and $f'(0)>0$

Is there a function $f$ satisfy (1) Only differentiable at $x=0$ (2) $f'(0)>0$ but $f$ is not increasing state on $x=0$?
-1
votes
0answers
16 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
-1
votes
0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
0
votes
0answers
43 views

What is umbral calculus, really? [duplicate]

I've seen this page on umbral calculus as well as wikipedia and and another question asked on this website (What's umbral calculus about?), but I still cannot realize what really umbral calculus ...
0
votes
0answers
36 views

Tricky multiple choice question about the properties of $\int_0^x\cos x\ e^{kx^4}\ \mathrm dt$

The question Let $$F(x) = \int_0^x\cos x\ e^{kx^4}\ \mathrm dt,$$ with $k \in \mathbb{Z}$. Which of the following is FALSE? a) $F$ does not have a horizontal asymptote for $k = 3$. b) ...
0
votes
1answer
49 views

$\int_0^{\pi/2}\ln(\sin(x))$?

From this paper: http://math.ucsd.edu/~ebender/20B/7_DefInt.pdf Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
1
vote
0answers
25 views

A limit of the hyperfactorial and Barnes G-function

I'm doing some work on the various means (arithmetic, geometric, etc.) of some sequences of binomial coefficients, and I'm having some trouble proving a result regarding a ratio of the Hyperfactorial ...
0
votes
0answers
56 views

Help with calculus. what does a subscript k represent in (1)

I would like some help solving these problems. In (1) the maximum can be found by derivation right? where the derivative = 0, but what does " find the maximum a(subscript k)" mean? i keep seing it ...
2
votes
2answers
65 views

Evaluating $\int_0^\infty \frac{1}{(k-1)!} (\frac{x}{y})^{k+1} (1-y)^{-x/y} \, dx$

EDIT: I CHANGED THE QUESTION (I HAD THE WRONG BOUNDS!) THE ACTUAL QUESTION WAS FROM 0 TO INFINITY, NOT 0 TO 1! I'm stuck with evaluating this integral and I need some help! $$\large\int_0^\infty ...
1
vote
1answer
13 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
2
votes
1answer
35 views

Change of variables in multi-variable calculus?

About the last equality, I know it is change of variables. Let $\xi=x+t,\eta=-x+t$, but I don't know how to get the integration domain? I have been thinking for an hour and I can't get the ...
1
vote
2answers
51 views

Prove or disprove that this function is continuous

If $f(x,y)$ is a real valued continuous function defined in $A \times B$ where $A$, $B$ are compact sets in $\mathbb R^n$ and $\mathbb R^m$ respectively. Let $g(x)=\min_{y \in B}f(x,y)$. Prove or ...
0
votes
1answer
25 views

cross sections using disks or washers method to rotate about the line $y=4$

I am dealing with a cross-sections problem that I cannot seem to solve. I have graphed it and gotten this equation for the volume, but it does not seem to lead me to the proper answer. $$ ...
-3
votes
0answers
24 views

What is the maximum area of a trapezium with 3 known sides and unknown angles. [on hold]

The Question: A major company in your city has both new equipment capable of making guttering in the shape of an open top trapezium. The sheet metal used is 22 cm wide and bent such that the base s ...
-4
votes
0answers
31 views

Trigonometric math problem [on hold]

A camera is mounted at a point 3000 ft from the base of a rocket launching pad. The Rocket rises vertically when launched, and the camera's elevation angle is continually adjusted to follow the bottom ...
1
vote
2answers
58 views

Integration by parts- using a u and v that are not inside of the original integral?

For instance, if I want to integrate some function $\frac{df(x)}{dx}=f'(x)$ $$\int_a^b f'(x)\,dx$$ And I use integration by parts, is it acceptable to set $u=x$ and $dv = f'(x)$ even though $x$ ...
2
votes
1answer
60 views

Differentiating position with respect to 'modified time'

I've been reading a book on orbit determination, and I've hit a block with the calculus. Why do this and is the resulting value even acceleration? This differentiation is very odd to me. He defines ...
0
votes
1answer
15 views

Cross sections for disk or washers when rotating about the $y$-axis

I am having trouble with this cross-sections question, simply because I am not sure what to do with the function. Am I supposed to change it so it's in terms of $x$ and then graph it, or leave it how ...
1
vote
2answers
38 views

Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative

The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$ thanks for helpers
0
votes
1answer
23 views

Effective Acceleration for Non-Constant Acceleration Motion

This question uses the same symbols as "Effective Acceleration" is Distance-Averaged Acceleration?. One of the kinematics formulas for constant acceleration is: $\Delta x=v_0*\Delta ...
0
votes
1answer
30 views

three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
0
votes
1answer
40 views

Radians or degrees?

In problem 2 from this page: http://www.analyzemath.com/calculus/Problems/rate_change.html The last couple steps including the equation: $$\frac{da}{dt} = \left[-\frac{\sin ...
1
vote
1answer
43 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
0
votes
0answers
49 views

Gradient function for restricted likelihood with respect to terms that influence Sigma

Is there a straightforward/generalized way to calculate partial derivatives of the restricted multivariate log-likelihood function $\ln\mathscr{L}=C+\ln\lvert ...
2
votes
7answers
228 views

Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$ But I don't know how to explain it. I understand that the derivative of the equation is a ...
2
votes
2answers
221 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
1
vote
0answers
41 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
0
votes
4answers
82 views

Integrating $\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt$.

I am trying to compute $$ \int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt. $$ This is what I got so far: $t=\sec(x)$ and $dt=\sec(x)\tan(x)x\,dx$ So plugging this in gives me $$ \int ...
-1
votes
2answers
126 views

Is $x/x$ continuous at $0$? [on hold]

Just wondering, while studying limit, if $x\over x$ is continuous at $0$. $f(0)={0 \over 0}$ ,, but $x/x=1$. In this case, is it continuous at $0$?
1
vote
0answers
51 views

Problem proving monotonicity while showing uniform convergence of $\sum_{k} \frac{1}{k} \sin \left ( \frac{\pi k^2}{x + k} \right )$

Remark This is a homework type of question and since there is no homework tag anymore I ask you to tell me how would you solve this (hints) and let me solve it for myself so I can practice. I can ...
3
votes
3answers
151 views

Determining the best possible substitution for an integrand

What substitution is best used to calculate $$\int \frac{1}{1 + \sqrt{x^2 -1}}dx$$
4
votes
1answer
45 views

Derivative of $x-\sqrt { x } $

Compute $f'(x)$ using the limit definition $$f(x)=x-\sqrt { x } $$ Steps I took: $$f'(x)=\lim _{ h\rightarrow 0 }{ \frac { x+h-\sqrt { x+h } -(x-\sqrt { x } ) }{ h } } \quad $$ $$f'(x)=\lim _{ ...
6
votes
3answers
136 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1) dx$$ to five significant digits. I've used Mathematica, ...
0
votes
1answer
33 views

help understanding calculus derivation of sum of squares

The photo is from Strang's Calculus. He is trying to find the discrete function $f$ for the sum of $n$ squares. He knows that the integral is $x^3/3$ and uses this as a starting point and then ...
1
vote
1answer
42 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
2
votes
1answer
22 views

Find a the value of a point on the tangent line

Suppose that the line tangent to the graph of $y = h(x)$ at $x = 3$ passes through the points $(-2, 3)$ and $(4, -1)$ with a slope of $-2/3$. Find $h(3)$. Hey guys, here's a question from my ...
4
votes
2answers
67 views

An easier way to prove this

Suppose $f\in C^1[a,b]$, and $f''$ exists on $(a,b)$. Show that, for any $c\in (a,b)$, there is $\xi\in(a,b)$ s.t. ...
1
vote
1answer
58 views

Is this equality true? Why? Why not?

Let $$ \lim_{a\to 0} \frac{1}{2} \left( \left( \sum_{n=-\infty}^\infty \frac{1}{(n+a)^2} - \frac{1}{a^2} \right) \right) = \sum_{n=1}^\infty \frac{1}{n^2}$$ I already know that ...
0
votes
3answers
20 views

How to solve this system of equations (Lagrange Multipliers)

I was doing a question on Lagrange multipliers and stucked when trying to evaluate the point. The system of equations that I can't solve is this: $$y^2-x^2+3x-3y=0$$ $$-y^2-yx+3y-xy=0$$ I just ...
0
votes
1answer
20 views

Finding hypervolume lying between Gaussian function and x-y-z plane over $\mathbb{R}^3$

Define the 3-variable Gaussian function by $G(x,y,z) = e^{-(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})}$. Find the hypervolume lying between this surface and the x-y-z hyperplane, over the ...
1
vote
1answer
25 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
1
vote
5answers
64 views

Finding $\lim_{n \rightarrow \infty} \frac{\log n}{\sqrt{n}}$

Compute $$\lim_{n \rightarrow \infty} \frac{\log n}{\sqrt{n}}$$ It seems pretty obvious, but I have tried Stolz-Cesaro and other tricks and I still can't get a solution.
-1
votes
1answer
33 views

if $\lim_{x \rightarrow \infty } |f(x)| = \infty$ then $\lim_{x \rightarrow \infty } f(x) = \infty$? [on hold]

Let $f: \Re \mapsto \Re$ be continuous function. How to prove that if $\lim_{x \rightarrow \infty } |f(x)| = \infty$ then $\lim_{x \rightarrow \infty } f(x) = \infty$ or $\lim_{x ...