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

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2answers
45 views

Derivative (Tangent) of a function

What are the steps to constructing the tangent function at the curve $h(x)$? I was looking over this question, Constructing function tangent to $h(x)$, and I got baffled since I did not know how to ...
0
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1answer
45 views

Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
1
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0answers
52 views

A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
2
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1answer
101 views

How to solve the following limit?

$$\lim_{n\to\infty}{\bigg(1-\cfrac{1}{2^2}\bigg)\bigg(1-\cfrac{1}{3^2}\bigg) \cdots \bigg(1-\cfrac{1}{n^2}\bigg)}$$ This simplifies to $\prod_{n=1}^{\infty}{\cfrac{n(n+2)}{(n+1)^2}}$. Besides partial ...
1
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0answers
112 views

Is this a correct way to prove what the derivative of a polynomial function is?

After trying a polynomial long division problem with a lot of wondering how to go about answering it I proceeded by most likely overcomplicating things but the equation derived seems to work at ...
2
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3answers
326 views

Limit of sequence with floor function

$$ \mbox{How do I compute}\quad \lim_{n \to \infty}% \left\lfloor\sqrt[4]{\vphantom{\Large A}n^{4} + 4n^{3}}-n\right\rfloor\ {\large ?} $$ I know that could use squeeze theorem, but for that I would ...
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1answer
117 views

finding all possible values of $a,b,c$ when $f(x)=\frac{\ln^3 x - c \ln x + 30}{(\ln^2 x - b \ln x + 6)(\ln x + a)}$

Given the function $$f(x)=\frac{\ln^3 x - c \ln x + 30}{(\ln^2 x - b \ln x + 6)(\ln x + a)}, \quad (a,b,c)\in\mathbb{R},$$ and vertical asymptote at $x=e^2$, and removable discontinuity at $x=e^3$. ...
1
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1answer
92 views

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity. My initial thoughts were that this is a straight plug in of value at ...
0
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1answer
117 views

Evaluate the Integral $\frac{\sec^2\theta}{\tan^4\theta-1}$

Need Help Evaluating the Integral $\frac{\sec^2\theta}{\tan^4\theta-1}$ Think that you use U substitution and then partial fractions, but I keep getting the wrong answer. Thanks in advance.
0
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2answers
117 views

Calculate the work done.

A circular swimming pool has a diameter of 10 meters, the sides are 1.5 meters high and the depth of the water is 1.2 meters. How much work is required to pump all of the water over the side? (Density ...
2
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2answers
178 views

Apostol Calculus Vol.1 Exercise 9 , Chapter 1.5 (Prove property of polynomial function)

Ok, so I have a huge problem with this exercise. It is a property of polynomial functions that needs proving. Thing is, I can not even get a clue and I put in some numbers and it does not seem to ...
0
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2answers
53 views

Help with Derivative

So I'm trying to find the derivative of $f(x) = 3x + 4\sqrt{x}$ So far I got this far: 1. $\displaystyle \frac{3(a+h)+ 4\sqrt{a+h} - (3a + 4\sqrt{a})}{h}$ 2. $\displaystyle \frac{3a + 3h + ...
0
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2answers
42 views

Confused with a limit

Given the limit as x approaches 0 from the right side of $f(x)$ equals $A$ and the limit as x approaches 0 from the left of $f(x)$ equals $B$. Evaluate the limit as x approaches 0 from the right side ...
4
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2answers
295 views

given positive real numbers x,y how to demonstrate that is defined as the maximum of x and y?

Is it true that given positive real numbers $x,y$, then we have that $$ \sqrt{x^2 + y^2} \geq \max\{ x, y \} $$ I cant find a counter-example although it seems it is true... Any comments?
0
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1answer
78 views

$f(x)$ is periodic with period p.

Suppose $f(x)$ is periodic with period p and $g(x)$ is periodic with period q. Let $r$ be the L.C.M. of p and q, if it exists. Then show that: If $f(x)$ and $g(x)$ cannot be interchanged by adding a ...
0
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1answer
132 views

Problem with friction [closed]

Imagine two boxes stacked on top of each other. The bottom box is larger than the top box and it rests on a surface. So: Between the boxes is a friction force constant $\mu_s$ and between the ...
2
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3answers
65 views

Limits of trigonometric function

I know my answer is correct, but are my steps correct? $$ \begin{align} & \lim_{t \to 0} \frac{\tan(2t)}{t}\\[8pt] & = \lim_{t \to 0} \frac{1}{t} \tan(2t)\\[8pt] & = \lim_{t \to 0} ...
0
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1answer
21 views

Can I use this proof that $\lim_{x\to p}[1/f(x)] = 1/\lim_{x\to p}f(x)$?

My textbook on Calculus use a much wordy proof. Maybe the author didn't want to declare that $f(x)/f(x) = 1$ since we have not prooven it. And maybe there is more in this statement that meets the eye. ...
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1answer
96 views

$f(x)$ is a periodic function with period $T$.

Prove that if $f(x)$ is a periodic function with period T, then the function $f(ax+b)$, where $a>0$, is periodic with period $\frac{T}{a}$. I started with, $$f[(a(x+T/a)+b]=f[(ax+b)+T]=f(ax+b).$$ ...
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1answer
50 views

What is the integral of this equation?

How do you integrate the following statement? $$\int \csc^2\left(x-\frac13\right)\,dx$$
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3answers
129 views

Does $\sum_{n = 1}^\infty\frac{\ln^2(n)}{\sqrt{n}(8n + 9\sqrt{n})}$ Converge?

I've been working with the following series: $$\sum_{n=1}^\infty\frac{\ln^2(n)}{\sqrt{n}(8n+9\sqrt{n})}$$ I know that I must use the comparison test for convergence, but I'm unsure what to compare ...
3
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2answers
120 views

Complex exponential inequality .How to solve it?

Ok so I started learning calculus from Spivaks book and on first chapter exercises I have stuck on one inequality for a few weeks now. Even my math professor does not know how to solve it. I have used ...
5
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3answers
173 views

Let $f:[a,b]\rightarrow \mathbb{R} $ be differentiable with $f'(a) = f'(b)$. There exist a $c\in(a,b)$ such that $f'(c) = \frac{f(c) - f(a) }{c -a}$.

Can someone help me with the following problem? Let $f:[a,b]\rightarrow \mathbb{R} $ be differentiable and suppose that $f'(a) = f'(b)$. Show that there exist a $c\in(a,b)$ such that $f'(c) = ...
0
votes
1answer
20 views

Bounding a function above into a ball of radius $R$

Let $n:[0,T] \to \mathbb{R}$ be a non-negative function. It satisfies $$n(t) \leq C_1n(0) + C_2$$ where $C_1 > 1$ and $C_2 > 0$. Is it possible to find a number $R >0$ such that if $n(0) ...
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4answers
96 views

Prove that $\sum_{k=0}^{\infty} (k-1)/2^k = 0$

How to prove that this series converges, and that the limit is 0 ?
3
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0answers
61 views

Derivatives and the tangent as the “limit of a secant” and as “no other line can be drawn between”

I've read (for example, in Boyer's History of The Calculus) that the definition of tangent as the limit of a secant is due to D'Alembert (1717 – 1783). So I searched for which other definitions were ...
2
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3answers
91 views

Why $\lim_{n \to \infty} \frac{2^nn!}{(2n)!} = 0$

A friend asked me, why \begin{align} \lim_{n \to \infty} \frac{2^nn!}{(2n)!} = 0, && n\in \mathbb{N} \end{align} and I couldn't answer. We already know that the sequence converges and we are ...
2
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4answers
256 views

Evaluate $\int \sqrt{\sin^2 \frac{x}{2}} \, \operatorname{d}\!x $

$$ \int\sqrt{\sin^{2}\left(x \over 2\right)\,}\,{\rm d}x $$ the answer I saw considered $\sqrt{\sin^{2}\left(x\right)} = \sin\left(x\right)$ and not $\left\vert\,\sin\left(x\right)\,\right\vert$ is ...
8
votes
3answers
470 views

A mistake in Stewart's book

I'm making a revision of calculus and I'm using Stewart's book I think he is wrong in case $(a)$: In case $(a)$ the function is not defined in $x=2$, then we can't say that the function is ...
1
vote
3answers
108 views

Critical number $y = \frac{1}{x^2 + 2}$

Seems pretty straight forward but my book seems to be giving an incorrect answer without any explanation to their magic. $$y = \frac{1}{x^2 + 2}$$ I know that this has no 0 so that rule of finding a ...
6
votes
2answers
106 views

Find the derivative in term of $x$ and $y$.

$\frac{\mathrm{d}y}{\mathrm{d}x} = 3x + 2y + 1$ Find $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}$ in term of $x$ and $y$. I get $3+2y^{\prime}$ for the answer. Lost upon how to get the answer in terms ...
0
votes
2answers
80 views

Calculus question velocity

A cricket jumps in the air with a velocity of $3.3\;m/s$. Its height (in meters) after $t$ seconds is given by $h(t) = 3.3 t - 4.9 t^2$. How fast is the cricket moving after $t = 0.1$ seconds?
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1answer
685 views

Finding the equation of a line normal to a curve and perpendicular to a line.

Find the equation(s) of the line(s) normal to the curve $g(x)=4x^2 + 6x + 1$ that are perpendicular to the line $y(x)=2x-5$.
1
vote
1answer
94 views

Please help solve this calculus calculation

If I was trying to design a printed billboard using a minimal area of plywood. The printed area must be $2000$ sq ft.Here are the margins. side margins $10$ ft top margin $8$ ft bottom margin ...
3
votes
1answer
60 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
0
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2answers
113 views

Partial differentiation problem

I have a question that from a past exam(2007S2, UQ, MATH1052), that is as follows: "Let $w = f(x,y)$ and $x = rcos(\theta), y = rsin(\theta)$ Show that $\frac{dw}{dx}=\frac{dw}{dr}*cos(\theta) - ...
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0answers
44 views

Question about some inequality

Let $p(t)>0, \forall t\in\mathbb{R}$; $p(t)$ is continuous function on $\mathbb{R}$ and $\int_{-\infty}^{\infty}\frac{ds}{p(s)}<+\infty,$ $a_1, a_2, b_1,b_2$ nonnegative real numbers satisfying ...
2
votes
1answer
124 views

Let f : R → R be a function such that | f(x) - f(y)| ≤ | sin x - sin y|, for each x,y Є R …

Let f : $\Bbb R$ → $\Bbb R$ be a function with the property | f(x) - f(y)| ≤ | sin x - sin y| , for every real numbers x,y. Prove that that there exists an unique real number c such that f(c) = c. ...
0
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1answer
33 views

Prob $n$-th arrival of Poisson Process ($\lambda_{1}$) occurs before $m$-th arrival of P.P. ($\lambda_{2}$)?

Define $ T_{1} $: time until $n$-th arrival of $ \{N_{1}(t)\} $; $T_{2} $: time until $m$-th arrival of $ \{N_{2}(t)\} $. $ \{N_{1}(t)\}, \{N_{2}(t)\} $ independent. Then \begin{align} ...
0
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2answers
91 views

Is $\log(1+x)\sim x$ correct?

This is my problem if $\log(1+x)\sim x$ when we are given $x\to 0$? The assignment doesn't mention if $\log$ is $\log_{10}$ or $\log_{e}$? I assumed the first version and saw the claim is wrong. Am I ...
0
votes
1answer
58 views

Double integration question

How can i solve this double integration? $$I=\int_{0}^\infty \int_{x}^\infty \frac{1}{y} \mathrm e^\frac{-y}{2}dydx$$.No idea.Please help me with a hint.
2
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3answers
111 views

Inequality for Integral of Vector Function

I am trying to prove $\| \int^{b}_{a} \vec{r}(t) dt\| \leq \int^{b}_{a} \| \vec{r}(t) \|dt$. I am fairly certain that this can be derived from the Cauchy-Schwarz inequality, but I can't quite ...
2
votes
3answers
51 views

I am missing 1/2 in this integral

I am trying to solve a integral for cal 2 using substitution. $$\int x(6+x^2)^{10} dx , u = 6 + x^2$$ I know the answer is $$\frac{(6+x^2)^{11}}{22}+C$$ i can't figure out where the last ...
0
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2answers
63 views

Simplify/Evaluate $y = x + a \frac{dx}{dt}$

I have the following equation which I am totally unable to solve. $$y = x + v$$ $$v = a \frac{dx}{dt}$$ $$ y = x + a \frac{dx}{dt}$$ Nothing is a constant and everything is somehow related to ...
0
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0answers
39 views

Fourier Sine Transform Identity Relation through Integration by Parts

This is purely for my own recreational interest. I've spent the last few days trying to demonstrate to myself that the Fourier Sine Transform and the inverse Fourier Sine Transform return their ...
0
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1answer
149 views

Gauss divergence theorem: verification in first octant

Please help me. I have checked several times, and resulted wrong in verification of this particular problem. Please solve for me. Verify Gauss divergence theorem for $F = (3xz)i + (y + zx)j + (xyz + ...
4
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1answer
144 views

Why is the sum of the numbers $1+2+3+4+\dots+N$ equal to the integral $\int_{0}^{N}x+0.5$, rather than $\int_{0}^{N}x$?

Sorry if this question has been asked in the past, or if it seems silly, but I couldn't find anything on Google or SE, and we didn't cover series in our math classes. So basically, I'm working on ...
1
vote
1answer
81 views

Show that its inverse, denoted by $h^{-1}$, is also a strictly increasing function.

Let h be a strictly increasing continuous function. Show that its inverse, denoted by $h^{-1}$, is also a strictly increasing function. How to approach this question?
3
votes
1answer
154 views

Important results of calculus before Newton and Leibniz?

We have all come to know that calculus was invented by Newton and Leibniz, right? But many calculus results were already proven by the time. I have read that Fermat already found how to calculate ...