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

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0
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2answers
32 views

Change integration limits, multivariable calculus.

Good night, i have a serious problem changing the integration limits, i read two books but i don't understand, i put an example... $\int_{0}^{1}\int_{0}^{1-x}\sqrt{x+y}\left(y-2x\right)^{2}dydx$ I ...
2
votes
1answer
374 views

Confused about notation and derivatives inside integrals

EDIT: To make what I am asking more clear. I've simplified it and have a more direct question. Let's say I am writing out an expression, and I want to write: $$\int_0^xF'(y)\,dy$$ However, for ...
4
votes
2answers
58 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. ...
2
votes
1answer
28 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
2
votes
4answers
62 views

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$ I used the definition of derivative: $f'(x)=|\lim_{h ...
0
votes
1answer
31 views

Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
1
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0answers
29 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
946 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
0
votes
1answer
20 views

How to find the upper bound of an error by Taylor polynomial approximation

I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... $f: ]-3;+\infty[ \rightarrow ...
2
votes
1answer
82 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 ...
0
votes
1answer
28 views

Finding the domain of the following integral in polar coordinates

Question: Convert the following integral into polar coordinates and solve $$\int_0^\frac{\sqrt{2}}{2}\int_x^\sqrt{1-x^2}xy \ dy\,dx$$ My attempt: I managed to get this: ...
-2
votes
1answer
24 views

Question about applying the Chain Rule with multiple variables

Let $z = u(x,y)$ and $y = y(x)$ and $u(x,y(x))$ = 0. What is the second derivative of the function $y(x)$? I tried to use chain rule but I keep making mistakes
1
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0answers
21 views

Converse of this theorem about existence of Green's function

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
2
votes
1answer
44 views

Are $\lim_{h\to0}f(a+h)=f(a)$ and $\lim_{h\to0}f(x+h)=f(x)$ the same?

An exercise I came across in my calculus text is as follows: Prove that $f$ is continuous at $a$ if and only if $$\lim_{h\to0}f(a+h)=f(a)\tag{1}.$$ Now, I saw a proof of the Product Rule ...
0
votes
2answers
52 views

What is y'' if $\sin y = y + 5x$?

I got $ 5\sin y / (\cos y - 1)^2$ as my answer, but the correct answer was given as $25\sin y / (\cos y - 1)^3$. My thought process: Derive the original equation to get $y'\cos y = y' +5$ $$y'(\cos ...
26
votes
5answers
2k views

Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
0
votes
2answers
60 views

Antiderivative for $\sin(t^2)/2$?

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 ...
6
votes
4answers
10k views

An inflection point where the second derivative doesn't exist?

A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points ...
2
votes
5answers
3k views

Check my workings: Show that $\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$

Let $f''$ be continuous on $\mathbb{R}$. Show that $$\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$$ My workings ...
2
votes
1answer
97 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
2answers
79 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 ...
0
votes
2answers
24 views

Differentiable function and increasing and decreasing problem

I dont get why (I) is not correct. The slope at 3 is positive, meaning the f'(x)>0. Can anyone explain? Correct answer is C
0
votes
1answer
16 views

Maximizing $g(x)$ and monotone transformation $f(g(x)$ is the same?

I have encountered that in some cases maximization of a function had been substituted with a maximization of its monotone transformation. For example, finding the min or max of $f(x,y) = ((x-1)^2 + ...
3
votes
3answers
62 views

Prove that $\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$ for $n\ge0$

Prove that $$\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$$ for $n\ge0$ I am not able to proceed with the integral. For the case $k+1$ please guide me through the problem. ...
3
votes
0answers
30 views

Proving/ Disproving that a set is compact in $l^2$

How can I prove or disprove that the following set in the real sequence space $l^2$ ( equipped with the norm $||(X_1,X_2,...)||_2 = \sqrt {\sum_{i=1}^{\infty} X_i^2}$ ) , is compact? $$ A = ( ...
1
vote
2answers
28 views

Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
0
votes
1answer
807 views

Maximization of sum of two functions

Is there any relationship similar to the following. Let $X$ be the maximum of functions $f_1(x)+f_2(x)$. Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$. Is there any ...
0
votes
2answers
47 views

AP Calculus BC - Related Rates Problem

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
1answer
59 views

HELP to solve - PDE First order

I have this equation $$ u_x+uu_y=0$$ by the book "Handbook of First order Partial Differential Equations - page 290." The general solution is $F(ux-y,u)$, where $F$ is a arbitrary function. I try ...
0
votes
3answers
52 views

How to convert arccos to arctan?

Is this true? $$\arccos\frac{A}{\sqrt{A^2+B^2}}=\arctan\frac{B}{A}$$ If so, how can one show it?
1
vote
1answer
35 views

Find two functions $f$ and $g$ which are integrable, but whose composition $f \circ g$ is not

Find two functions $f$ and $g$ which are integrable, but whose composition $f \circ g$ is not. Hint: Use the fact that the function $f(x) = 0$ for irrational $x$ and $\frac{1}{q}$ if $x = ...
3
votes
1answer
69 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 ...
0
votes
2answers
23 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
0
votes
0answers
4 views

partial order and equivalence relation question [on hold]

Let A = ℤ+ x ℤ+ and R be a relation on A (that is, R ⊆ A xA) defined as follows. (a,b) ~ (x,y) if and only if a + y = b + x. Is R a partial order? Is R an equivalence relation?
2
votes
2answers
60 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 ...
2
votes
0answers
14 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
10
votes
5answers
300 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 ...
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
35 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 ...
-1
votes
0answers
18 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 $-\frac{4}{5}$ ...
0
votes
2answers
63 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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
18 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 ...
-1
votes
1answer
70 views

How can I find $dy/dx$? [closed]

What does $dy/dx$ represent for these questions? $y = x^5$ $y = x+5$ $y = b$, $b$ is a constant Am I supposed to divide the $y$ by $x$? So, $\frac{y}{x^5}$ and $\frac{y-5}{x}$? If so, what do I do ...
1
vote
0answers
56 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
0answers
35 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 ...
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 ...
3
votes
2answers
254 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
5
votes
1answer
43 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 ...