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

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2
votes
2answers
56 views

Evaluate the limit $\lim\limits_{n\rightarrow \infty}(\frac{1}{6}+\frac{1}{24}+\frac{1}{60}+\frac{1}{120}+\dots+\frac{1}{n^{3}-n})$

$$\displaystyle\lim_{n\rightarrow \infty} \left(\frac{1}{6}+\frac{1}{24}+\frac{1}{60}+\frac{1}{120}+\dots+\frac{1}{n^{3}-n}\right)$$ I am not able to find any technique to proceed. It might be simple ...
0
votes
1answer
8 views

Clarification on the idea of absolute maxima

$$f(x)=-|x|\:,\:\:x≠0$$ If f(x) did not have an point of discontinuity at x = 0, then it is obvious it would have an absolute maximum there. However, now that that point no longer exists, does it ...
1
vote
1answer
56 views

How to find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when both number in front and exponent have fractions?

I'm not sure how to solve this: $\frac{5}{9}x^\frac{2}{3}$. I applied the product rule and have $\frac{2}{3}\frac{5}{9}x^{-\frac{1}{3}}$. $\frac{30}{9}x^{-\frac{1}{3}}$, then ...
2
votes
1answer
91 views

How can we see that $ \sum_{n=0}^{\infty}\frac{2^n(1-n)^3}{(n+1)(2n+1){2n \choose n}}=(\pi-1)(\pi-3) $?

I wonder will it help me so prove it if I was to decompose it into partial fractions? Mathematica approves of the identity; it is converges. can anyone help me to prove it? $$ ...
-1
votes
0answers
17 views

Fourier series of a piecewise function

Over period $-3\lt t \lt 3$ $f(t) = \begin{cases} 2, & {-3\lt t \lt 0} \\ 0, & {0 \lt t \lt 3} \end{cases}$ Can the above be entered in WolframAlpha? I want to check my answer
1
vote
2answers
23 views

What happens if there's a number in front of x to the power of zero?

The problem is to find $\frac{dy}{dx}$ of $-\frac{4}{5}$x - $\frac{1}{3}$ I have (1)($-\frac{4}{5})x^0$ Is the answer 1 because of $x^0$ also affecting the number in front? Or is it $-\frac{4}{5}$ ...
1
vote
1answer
33 views

Cannot make sense of a derivative

Short version of the question: In this presentation http://www.slideshare.net/ShangxuanZhang/xgboost (page 74-75)I cannot understand how the gradient of the L function is calculated. $$ L = y_i log ...
0
votes
0answers
15 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
0
votes
1answer
19 views

Suppose $f:[a,b] \rightarrow [0, \infty)$ is bounded.Let $A(g)=\frac{1}{b-a}\int_{a}^{b}g(x)dx$

Suppose $f:[a,b] \rightarrow [0, \infty)$ is bounded.Let $A(g)=\frac{1}{b-a}\int_{a}^{b}g(x)dx$ for any bounded function $g:[a,b] \rightarrow [0,\infty)$.Show that $A(f)^2 \le A(f^2)$. I was thinking ...
4
votes
2answers
55 views

Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
0
votes
2answers
61 views

Antiderivative of y = $\dfrac {x+22} {x^{2}+2x-8}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
-3
votes
0answers
33 views

Differential calculus: function of two variables question (four parts) [on hold]

Let the function of two variables be given as $f(x, y) = \sin(x + y), x ∈ \mathbb{R}, y ∈ \mathbb{R} $ (a) Sketch the surface $z = \sin(x + y)$ and draw the level sets of this function for levels ...
3
votes
1answer
65 views

Brostein Integral 21.42

Good morning. I came across the following integral in some field theory calculation: $\int_0^\pi dx\,\log\left(a^2+b^2-2ab\cos x\right)=2\pi\log\left(\max\lbrace a,b\rbrace\right)$ for ...
5
votes
1answer
40 views

Chain Rule of Calculus as a Group Property?

I read that the chain rule and inverse function theorem are expressions of the group property of successive non-singular transformations. How do you say this more formally? My guess is that we are ...
1
vote
0answers
49 views

A very detailed book for calculus 1-3.

Is there a very good book covering the whole calculus in detail, explaining all topics in calculus 1-3 for self-learning? I'm in geometry I, so I will start calculus in two years, and finish in five ...
3
votes
3answers
27 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
0
votes
3answers
36 views

How to find minimum integer $k$ , $\frac{1}{e+1}+\ln(e+1)-1\leq \ln k$

I would appreciate if somebody could help me with the following problem: Q: How to find minimum integer $k$ ? $$\frac{1}{e+1}+\ln(e+1)-1\leq \ln k$$
1
vote
1answer
25 views

How do these partial derivative and derivative terms relate?

From the top line, this proof jumps to the integration and evaluation of the function. I'm not sure how the partial of $t$ and $dt$ play in the integration to give $(s,t)$ before evaluation. Any help ...
2
votes
1answer
53 views

Derive zeta values of even integers from the Euler-Maclaurin formula.

Euler showed: \begin{equation} B_{2 k} = (-1)^{k+1} \frac{2 \, (2 \, k)!}{ (2 \, \pi)^{2 k}} \zeta(2 k) \end{equation} for $k=1,2, \cdots$. We could from here find $\zeta(2k)$ in terms of the ...
3
votes
1answer
30 views

Drones and Integrals Project

Hello everyone and thanks for taking the time to read this post. So in my college calculus class we had the opportunity to fly a drone and get it's flight data. I have a spreadsheet featuring two ...
0
votes
0answers
29 views

Finding “equilibrium”

Say I have three amounts $A$, $B$ and $C$. And a set of conversions between them $K_{A->B}$, $K_{B->A}$ and $K_{B->C}$. The conversions denote what fraction and at what efficiency they ...
1
vote
1answer
38 views

How to prove derivative of logarithm with base $b$?

I learned how to derive a logarithm with any base. This is the formula: $$\frac{d}{dx}\log_bx=\frac{1}{x\ln b}$$ How can it be proved?
0
votes
0answers
9 views

Geometric interpretation of adding dependence on a otherwise constant in a vector field.

So if i have $$ \vec{u} = \Omega r \vec{e_{\theta}}$$ Now if i take the curl $$\omega = 2\Omega\vec{e_z}$$ This is what we expect, we have a "rotating flow". Who's curl would be pointing in the z ...
-1
votes
2answers
76 views

Easiest way to solve this integral [on hold]

I was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there is ...
-2
votes
0answers
32 views

Indefinite Integration 2 [on hold]

I have to calculate the following integral: $$\int \frac{4x^4-2x^3-3x^2-4}{(x^4 + x^3 + 1)^{3/2}} \ dx$$ I have no idea how to begin. Any help is appreciated.
0
votes
2answers
30 views

Differentiation of $x^TAx$

I have in my text that if I differentiate $x^TAx$ with respect to the vector $x$ I get $2xA$ - could I ask why? Here $x$ is a $3\times1$ vector, $A$ is a $3\times 3$ matrix - I am given the ...
1
vote
2answers
64 views

What is the sum of this series: $1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 +\cdots$?

Say I have a series like the following; $$1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 + \frac{1 \times 6 \times 11}{5 \times 10 \times 15}x^3 + \cdots.$$ How do I find the sum of this? ...
2
votes
1answer
61 views

Evaluate $\int \sin(3x)\cos(4x) \,dx$

Evaluate $$\int \sin(3x) \cos(4x) \; \mathrm{d}x$$ I do not know how to solve this as a whole. I tried all the substitutions known to me.
1
vote
0answers
24 views

Finding the inverse of a function involving logarithms

Let $A \asymp B$ mean that there exists universal constants $m,M >0$ such that $mA \leq B \leq MA$. Let $k,n \in \mathbb{N}$ be such that $\log n \leq k \leq n$. I want to prove that $$ k ...
0
votes
1answer
73 views

Evaluate $\int_{0}^1 f(x) dx$

Assume $f(x) = 0$ when $x$ is irrational and $f(m/n) = 1/n$ where $m$ and $n$ are relatively prime integers with $m>0$. Show $f$ is integrable over $[0,1]$ and evaluate $\int_{0}^1 f(x) dx$. ...
0
votes
1answer
25 views

Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
9
votes
5answers
293 views

Number of points of accumulation of a sequence

Can a sequence have infinitely many points of accumulation i.e. we can extract infinitely many subsequences from it s.t. they all converge to their respective point of accumulation? I have the ...
1
vote
1answer
19 views

Function with inflection points

Question: Let $g_a$ be the function given by $g_a(x)=(x^2-a)\cdot e^x$ with $a\in\mathbb{R}$. For which values of $a$ does the graph of $g_a$ have one or more inflection points? The second ...
2
votes
3answers
30 views

Minor flaw in understanding of the proof of the derivative of exponential functions

I understand the majority of the proof of the derivative formula for exponential functions of the form: (full proof at bottom of post) $\frac{d}{dx}a^x$ but I have a little trouble with the last ...
0
votes
0answers
18 views

Is it sensible to define the absolute value of the integral and the derivative with measures in this way?

Is it sensible to define the absolute value of the integral and the derivative with measures in this way? $\mu$ and $\nu$ are measures (functions that take a shape [a set of points] and give the ...
1
vote
3answers
51 views

What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$?

What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$? My initial idea was to divide by x in the numerator and denominator. However that would only solve the inner ...
4
votes
4answers
99 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
vote
0answers
18 views

showing a processes is martingale using ito's lemma

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt ...
3
votes
1answer
61 views

Infinite product include summation

I would like to find an infinite product of$$\prod _{n=2}^{\infty} \left(1+\frac{(-1)^{n-1}}{a_n}\right)$$ where $a_n = \sum_{k=1}^{n-1} \frac{n!(-1)^{k-1}}{k!} $ I tried to compute $a_2 , a_3 ...
-3
votes
1answer
49 views

Big O counterexample [on hold]

If a given function is said to have a Big O of N^(1/2+a), where a > 0, then what would be a counterexample? My understanding of Big O is that it is an upper bound to growth, but not necessarily the ...
2
votes
1answer
47 views

How to know if this points are maximum or minimum of function

$g: \mathbb{R} \rightarrow \mathbb{R}$ is 3 times differentiable and $g{'''}(x)>0$ $ \forall x \in \mathbb{R}$ and it has 2 points of extremum $\alpha$ and $\beta$ with $\alpha < \beta$. ...
0
votes
2answers
36 views

AP Calculus BC - Area integration question

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
2answers
41 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
1
vote
3answers
38 views

How to know differentiation of the function at zero

Suppose we have function $f(x)=\frac{x^2}{2+|x|}$. Can anyone tell me that this function is differential at zero or not? Thanks
2
votes
2answers
31 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
-1
votes
0answers
20 views

Mean value of periodic function

$f(t) = \begin{cases} A \sin\Omega t, & {-{T\over 2}\le t \le 0} \\ 0, & {0 \lt t \lt {T\over 2}} \end{cases}$ where $A, \Omega, T$ are constants If I want to calculate the mean value of ...
0
votes
3answers
44 views

Find area of shaded area in curve with range of values for $y$

The parabola in the diagram has equation $y = 32 - 2x^2$ The shaded area lies between the lines $y=14$ and $y=24$ Looking at the graph, I only need to find half the area and multiply by ...
-1
votes
3answers
88 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
1
vote
2answers
56 views

Prove that $n^2 < n \cdot (n - 1) \cdot (n -2) $ [on hold]

How to prove or disprove that: $$ n^2 < n \cdot (n - 1) \cdot (n -2) $$ for every $n > 0$
1
vote
1answer
51 views

Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and ...