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

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2answers
136 views

$e^{\ln(-2)} = -2$ but $\ln(-2) = \ln 2+i\pi$. How does this work?

I'm messing with exponential growth functions. I noticed that I can write $y(t)=y(0)\alpha^t$ as $y(t)=y(0)e^{\ln(\alpha)t}$ (and then I can go ahead and replace $\ln(\alpha)$ with $\lambda$.) ...
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2answers
161 views

Evaluating $\int{e^{x^{1/3}}dx}$

How can I get $$\int{e^{x^{1/3}}dx}$$ I think integrating by parts may work, but I can't figure out the exact way.
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2answers
779 views

A difficult differential equation $ y(2x^4+y)\frac{dy}{dx} = (1-4xy^2)x^2$

How to solve the following differential equation? $$ y(2x^4+y)\dfrac{dy}{dx} = (1-4xy^2)x^2$$ No clue as to how to even begin. Hints?
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2answers
1k views

Find the derivative of this function at a specific point [closed]

What is the derivative of $$f(t)= \frac{t^3 +2} t$$ at point $(-2,3)$.
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1answer
82 views

Can we have $\displaystyle \lim_{x \to f(x)}$?

Can we have a limit where $x$ approaches a variable like this one:$\displaystyle \lim_{x \to f(x)}$ or $\displaystyle \lim_{x \to \cos(x)}$ ? And why? Thank you!
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5answers
94 views

Need help with a limit

I'm trying to determine $$\lim_{x \to 0} {x^2 \over \cos (3x) - 1}$$ My guess is that using the fact that $\lim_{x \to 0} {\sin x \over x} = 1$ or perhaps $\lim_{x \to 0} {\tan x \over x} = 1$ could ...
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1answer
74 views

Doeas $ \int_0^{0.5} \frac{\mathrm dx}{\sin x\ln x} $ exist?

So I have a test next week and I found myself struggled in a question. Does $ \ \int_{0}^{0.5} \frac{\mathrm dx}{\sin x\ln x} \ $ exist ? So I saw an answer to that question which I do not ...
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1answer
205 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
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1answer
95 views

Differential of an integral

I have this problem where I need to do a differentiation on a integration. The question is like this: $$ \frac{d}{dt}\int_0^{t} \exp(-z^2) dz$$
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2answers
82 views

Help verify $\lim_{x\to 7} \frac {x^2+7x+49}{x^2+7x-98}$ .

So my question is "Evaluate the limit" $\displaystyle \lim_{x\to 7} \frac {x^2+7x+49}{x^2+7x-98}$ I know you can't factor the numerator but you can for denominator. But either way you can't divide ...
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2answers
85 views

Determine the constants $m$ and $k$ such that $f'(0)$ exists?

Determine the constants $m$ and $k$ such that $$f(x)=\begin{cases} x^2+kx+m & \text{for }x<0 \\ 6\tan(7x)+10\cos(9x) & \text{for }x\ge 0 \end{cases}$$ is differentiable at $x=0$.
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3answers
97 views

Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$

let a,b be real numbers and $\varepsilon>0$. prove the above inequality. I dont know how to even start. please, I need your help.
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3answers
174 views

Cantor Set and ternary expansions.

Does there exists any systematic way to represent a number in ternary expansion? Also, I have hard time trying to figure it out this: Let $x = \sum \frac{a_k}{3^k} = 0.a_1a_2.... $ why is it that ...
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2answers
195 views

Solving i for annuities equation without financial calculator

I would like to know if there was a way to approximate i here without a financial calculator, in the following equation: $\displaystyle -50000 + \frac{12992}{1+i} + \frac{12992}{(1+i)^2} + ⋯ + ...
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2answers
79 views

Common mistakes in concluding that series is convergent or not

Assuming $(na_{n})_{n=1}^{\infty}$ convergent to $0$ then $\sum_{n=1}^{\infty}a_{n}$ is convergent, true or false ? wel' I say it is true because: $n$ convergent to infinity and $na_{n}$ is ...
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1answer
115 views

A way to teach Archimedean property

A student asked me how to understand the Archimedean property, I tried to re-read with him what he has already done in class (well, actually copy from the blackboard in class). However I think I'm not ...
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1answer
69 views

How can we prove that ${\sqrt{1+x}+\sqrt[3]{(1+x)^2}\over{1+x+\sqrt{1+x}}}=1-{x\over 6}$ for small $x$?

x is so small that its squares and higher powers are neglegable then prove that $${\sqrt{1+x}+\sqrt[3]{(1+x)^2}\over{1+x+\sqrt{1+x}}}=1-{x\over 6}$$ This might be a problem of Binomial Series
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2answers
92 views

Proving a derivative exists

If f is differentiable at $x$ then for $\alpha\neq1$ $$f'(x)=\lim_{c\to 0} {{f(x+c)-f(x+\alpha c)}\over{c-\alpha c}}.$$ I am not really sure what it is I need to even show to prove the statement.
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2answers
333 views

Is the following convergent or divergent: $\sum_{n=1}^{\infty} (n+2)/(n^3 + n +5)^{1/2}$?

"Is the following convergent or divergent: $\sum_{n=1}^{\infty} (n+2)/(n^3 + n +5)^{1/2}$ ?" I tried using the Root Test, but got pretty mixed up along the process. $|(n+2)/(n^3 + n + 5)|^{1/n}$ ...
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2answers
181 views

Stuck in definite integral of a function

$$I=\int_{0}^{\pi}\frac{x\tan (x)}{\tan(x)+\sec(x)}dx $$ I was given this problem now using property of definate integral i then equated this expression to $$\int_{0}^{\pi}\frac{\pi-x\tan ...
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2answers
97 views

For which value of r does the following equality hold: $\sum_{n=0}^{\infty}r^n = \sum_{k=1}^{\infty}180/3^{4k}$

For which value of r does the following equality hold: $$\sum_{n=0}^{\infty}r^n = \sum_{k=1}^{\infty}180/3^{4k}$$ I'm just not making any progress with this. I know from memorization that ...
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2answers
84 views

How do I improve my approach to solving integrals to get this and similar ones in the future correct?

$$\int \sqrt{ 8 (\cos t \sin t)^2 } dt = \sqrt{2} \int 2\sin t\cos t dt = \sqrt{2} (\sin t)^2 + C$$ Which seems correct to me, but if I take the definite integral from $0$ to $\pi$, then: $$\sqrt{2} ...
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2answers
88 views

Simple Ordinary Differential Equation [duplicate]

As simple as this should be, I can not seem to solve it. I can't classify its type and thus figure out how to solve it. It's the only ODE in my problem sheet that I can't solve (embarrassing). ...
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3answers
74 views

$f(0) = 65$, $f(20) = 60$, Possible to determine $f(x)$?

My question is pretty much what the title says. I have the following info: $f(0) = 65$ and $f(20) = 60$. The curve is also exponential, it says. Is it really possible to determine $f(x)$ from this ...
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2answers
141 views

Why is $\frac{6\sin(2\pi/3)}2 = \frac{3\sqrt3}{2}$

The source of my problem is that I have an integral: $$\int_{\pi/3}^\pi{(6\cos 2x)dx}$$ For anyone having the same problem as me, to see the boundaries, They are $\pi/3$ and $\pi$ The primitive ...
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3answers
68 views

Taylor Expansion of Function $x^2e^x$

What is the Taylor expansion of the function $x^2e^x$? I know the Taylor expansion of $e^x$, but I don't get how to form the Taylor expansion for $x^2e^x$ Taylor expansion.....can you help me?? ...
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1answer
81 views

Very basic derivative question

Why is the derivative of $e^x + e^{-x}$ equal to $e^x - e^{-x}$ ?
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3answers
5k views

Polar curve $r = 2\cos \theta -1$

$$r = 2\cos \theta -1$$ I am suppose to find the polar curve of the inner loop. Here is its graph, courtesy of Wolfram|Alpha, I am having trouble working out this polar function on a cartesian ...
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2answers
378 views

Integrate $\int{\frac{\sin(2x)+\cos(2x)}{(\sin(2x)-\cos(2x))^{5/2}}}\ dx$

Integrate $$\int{\frac{\sin(2x)+\cos(2x)}{(\sin(2x)-\cos(2x))^{5/2}}}\ dx$$ How do I integrate such an integral? u-substitution, no idea by parts, no idea this is very confusing! please ...
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3answers
150 views

Initial value problem differential equation $y' = (x-1)(y-2)$

$$y' = (x-1)(y-2)$$ $y(2)= 4$ $$\frac{1}{y-2}dy = (x-1)dx$$ $$\int \frac{1}{y-2}dy =\int (x-1)dx$$ $$\ln(y-2) = \frac{x^2}{2} - x + c$$ $$y - 2 = e^{\frac{x^2}{2} - x + c} $$ $$y = ...
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2answers
600 views

Why does a global extremum on an open interval not exist?

Say for example the function $f(x)=x^2$ on the interval $(0,\infty)$. I was taught that there is no global minimum on this interval. But I still can't quite wrap my head around why. ...
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3answers
110 views

A non-constant, increasing function $f$ such that $f(b)=\int_a^bf$

Is there a non-constant, increasing function $f\colon A\to B$, where $A,B\subset\mathbf{R}$ such that $$f(b)=\int_a^bf(x)\;\mathrm{d}x$$ for $a,b\in{A}$ with $a<b$.
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1answer
85 views

Fourth degree Maclaurin $f(x) = \dfrac{\sin x}{1-x}$

$$f(x) = \frac{\sin x}{1-x}$$ I tried to differentiate it but it is very difficult, what is the trick?
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2answers
64 views

Rotate shape on a graph

I have some shape built out of points. The coordinates given for each point . (see pic below) I need to rotate this shape by a particular angle (for example 60° see pic below) Is there some formula ...
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2answers
66 views

The value of a real series

What is the value of $$ \sum_{n=1}^\infty\frac{\sum_{k=1}^n \frac{1}{k^2}}{(n+1)(n+2)} $$
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3answers
77 views

$\lim_{n\to\infty} \sum_{k=1}^{n}\frac{k^{p-1}}{n^p + k^p}$

A couple of series I've been trying to solve: $$1. \lim_{n\to\infty} \sum_{k=1}^{n}\frac{k^{p-1}}{n^p + k^p}\\ 2. \text { For which $x$ does the following converge: }\sum_{n=1}^\infty \frac {1} {n^x} ...
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3answers
3k views

Proving Every open set in $\Bbb R$ is a countable union of open intervals. [duplicate]

This question is from William R. Wade's Introduction to Analysis book: Prove that every open set in $\Bbb R$ is a countable union of open intervals. I have no ideas honestly. Thank you.
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2answers
44 views

Easiest way to show $ \lim \limits_{x\to 0} \frac { |x_1|^{\alpha_1} \dotsb |x_n|^{\alpha_n} } {\| x\|^p} \text { exists } \iff \sum{\alpha_i} > p$

What is the easiest way to show $$ \lim \limits_{x\to0} \frac { |x_1|^{\alpha_1} \dotsb |x_n|^{\alpha_n} } {\| x\|^p} \text { exists } \iff \sum{\alpha_i} > p, \,\,\,\,\text { for } \alpha_i ...
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2answers
61 views

Series of product

Assuming that you have a series of a product $\sum_{l=0}^{\infty} f(l) g(l)$ and you know what $\sum_{l=0}^{\infty} f(l) $ is. Does this help, finding an approximate form for the whole series?
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1answer
86 views

2 Different integrals of $\int \left ( \tan{x}\right ) ^3 dx $.

My friend asked me why this function has 2 different integrals. I'm very confused. \begin{align} \int \left ( \tan{x}\right ) ^3 dx &=\int \left ( \tan{x} \right )^2 \tan{x}dx \\ &=\int \left ...
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1answer
123 views

Taylor Polynomial $\;y = e^{\sin x},\,$ $a = \frac{\pi}{2},\;$ $x = 1.5$

$y = e^{\sin x};\;$ $a = \frac{\pi}{2};\;$ $x = 1.5$ So first things get the derivative. I need to calculate the error and the $T_2$ $$f'(x) = \cos x e^{\sin x}$$ $$f''(x) = \cos ^2x e^{\sin x} - ...
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2answers
1k views

Compute limit with the help of MacLaurin series expansion

Compute $$\lim _{x \to 0} \dfrac{e^{2x^2}-1}{x^2}$$ with the aid of a MacLaurin series expansion.
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3answers
94 views

Prove limit of function

I need to prove this limit: Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.
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2answers
127 views

Area enclosed between $x$-axis from $a$ to $b$ and the curve $f(x)$ is finite

Area enclosed between $x$-axis from $a$ to $b$ and the curve $f(x)$ is finite when $a=0, \quad b=\infty,\quad f(x)=e^{-5x^5}$ $a=-\infty,\quad b=\infty,\quad f(x)=e^{-5x^5}$ $a=-7,\quad ...
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3answers
244 views

Show that $f$ is continuous at 0.

EDIT: Fixed the limit. This is a question from Spivak's Calculus, Ch.6, ex. 3. Suppose that $f$ is a function satisfying $$|f(x)|\leq |x| \forall x$$ Show that $f$ is continuous at 0. (Notice ...
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1answer
102 views

How to show that this limit is identical to..

Suppose we have expressions $$f_1=\frac{x(y-1)}{x(2y-1)-y}$$ and $$f_2=\frac{xw/y+(1-x)w/(1-y)+4}{8}-\sqrt{\left(\frac{xw/y+(1-x)w/(1-y)+4}{8}\right)^2-1/4(1+xw/y)}.$$ Moreover, we can assume ...
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3answers
141 views

Limit of ${x^{x^x}}$ as $x\to 0^+$

Can you please explain why \begin{align*} \lim_{x\to 0^+}{x^{x^x}} &= 0 \end{align*}
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2answers
64 views

Help understanding a derivation in a book

I'm reading about Pattern recognition and when I read the appendix on my book I came across with the following derivation: $J(\theta)$ is cost a function with parameter $\theta = (\theta_1, ..., ...
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1answer
176 views

Checking my proof related to directional derivatives

Please can somebody check my answer? Tell me and explain me my mistakes and so on if there is. Thank you for helping :) Question: Suppose that the function $f:\Bbb R^n \to \Bbb R$ is continuously ...
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1answer
78 views

non differentiable, integrable function

Can a function be unable to be differentiated, but is integrable? By unable to be differentiated, I mean at any arbitrary x coordinate. Thank you