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

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61 views

Proving a function is constant, under certain conditions?

The problem: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant. I believe I have some intuition about why this is the case; ...
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1answer
42 views

Odd extension of $C^\infty$ function

If we have a $C^\infty$ function on the half line $\{x\geq0\}$ which is zero at the origin, and we extend it by odd symmetry, the result should be $C^{\infty}$ at $0$, right? Clearly the first ...
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2answers
86 views

Why does this function have a tangent line at x=0?

$f(x)=\begin{cases} \sqrt x&\text{if $x\ge 0$}\\ -\sqrt{-x}&\text{if $x<0$} \end{cases}$ The graph around $x=0$ looks like this: When I derive $\sqrt x$, I get $\frac12 \sqrt x$. I ...
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1answer
157 views

Show that $f:[0,1] \to [0,1]$ is continuous if $f(x) = x^{1/k}$ for any $k \in \mathbb N$

I'm very confused right now and I want to apply the theorem that says " A mapping f of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for ...
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1answer
50 views

Invertible Functions

In some mathematics texts, a function is invertible iff the function is one-to-one and onto. However, in some calculus texts (thomas's calculus, stewart's calculus, etc.), the only requirement for a ...
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1answer
320 views

FInding the tangent line horizontal to a curve of $\frac{1}{(x^2-16)(x-7)}$.

Determine the values of x where the tangent line is horizontal for the function:$$\frac {1}{(x^2-16)(x-7)}$$ The value(s) of $x$ where the tangent line to the graph of the function is horizontal ...
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1answer
53 views

Finding a minimum value of an expression

I have this question to solve: A city's temperature of y degrees Celsius on a day in February is given by $y = 16 + 8 \sin \left(\dfrac{\pi t}{12}\right)$ where t is the time in hours after 9am. a). ...
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340 views

Cylindrical Shell Volumes Problem

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by $y=3+2x−x^2$ and $x+y=3$ about the y-axis. I have already turned $x+y=3$ into $y=3-x$. However I ...
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1answer
5k views

Find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$, $x=4$ and $y=0$ about the $x$-axis

I do not understand how they want the volume between $y=x^2$ and $y=0$? I don't understand how to do the problem. Please help. I have other similar homework problems and would like to LEARN how to do ...
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1answer
41 views

Deriviative of natural log help finding

$$y=7\ln\frac{11}x$$ I need to use the product rule please $$\frac{d}{dx} (7) \cdot(\ln(11/x) + \frac{d}{dx}\left(\ln\frac{11}x\right) \cdot 7$$ then $$0+\frac{d}{dx}\frac{11}x \cdot7$$ what do ...
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93 views

Calculate volumes using triple integrals

I have to calculate these two volumes using triple integrals: volume of $A = \{(x,y,z) \in \Bbb R^3 : {x^2\over a^2} + {y^2\over b^2} \leq z \leq 1 \}$ volume of $B = \{(x,y,z) \in \Bbb R^3 : ...
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197 views

Show that if f(x) is an n-times differentiable function defined on an interval I

Edit: n is a positive integer. As an extra question, how could I apply L'Hopital's rule to an expresion coming from the definition of a derivative? i.e. $$ \frac{f''(x)}{2} = \lim_{h \to 0}\dfrac ...
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224 views

Optimization question

A rectangular beam will be cut from a cylindrical log of diameter 1m. For part a) I have shown that the beam of maximal cross-sectional area is a square. Then 4 rectangular planks will be cut from ...
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1answer
108 views

Using the Test for Divergence

So I have the problem $$\sum_{n = 1}^\infty (-1)^n \frac{n}{n+5}$$ and I have to figure out if the series converges or diverges. I start out by testing the conditions of the Alternating Serie Test, ...
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2answers
94 views

What *really* are the local maxima and local minima

In math is the local max and local min just any peak ... point where slope of the function changes from positive to negative or vice-versa... Or are the LOCAL max and min just the highest point of the ...
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2answers
30 views

Function Continuity on an Interval.

h is a continuous function on interval [a,b] and h(x) belongs in Q for all x. Which statement is true? (a) h is constant on the interval
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1answer
97 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
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2answers
1k views

Limits of trigonometric functions as $x$ approaches $\infty$

A while back I ran into a problem in which I had to analyze the graph of $f(x) = ( \arctan x )^2$. I was fine until I had to evaluate the limit of the function as is approaches infinity to determine ...
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1answer
49 views

How to formally justify the existence of a limit with two variables?

Problem: Find the limit of the following functions a) $\displaystyle \lim_{x \to \infty, \ y\to \infty}$ $\frac{x+y}{x^2 + y^2} $ b) $\displaystyle \lim_{x \to 0,\ y\to 2} \frac{\sin(xy)}{x} $ I ...
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1answer
64 views

Existence of a global solution to $y = f(y,x)$ when $f$ is continuous

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be continuous. By Peano theorem there exists a local solution to the Cauchy problem $$ \begin{cases} y' = f(y,x),\\ y(0) = y_0. \end{cases} $$ If I ...
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1answer
31 views

Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$

I've been solving a problem in numerical analysis and to finish one of the exercises I need the following result. Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$. Now I ...
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1answer
98 views

Find $\lim\limits_{n→∞}\left(t+\frac{x}{n}\right)^n$

How would one go about finding the limit of $\displaystyle \left(|t|+\frac{x}{n}\right)^n$ as $n\rightarrow\infty$? ($t$ and $x$ are both positive.) Of course $\displaystyle ...
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4answers
2k views

Find the unit vectors that are parallel to the tangent line to the parabola $y=x^2$ at the point $(2,4)$

Can someone explain to me why the parallel vector is $i+4j$? The picture is the answer key. When I did the question I used the point given in the question, which is $2i+4j$.
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1answer
68 views

Two-sided improper integral

What is one way to show that $$\int_{-\infty}^\infty \frac{x^4 e^{x/2}}{e^x+1}\, dx$$ converges? I see it is an even function so it is enough to show that it converges for $[0,1]$. Moreover, it is ...
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53 views

Trigonometric Expression Problem

''Express $sin(4α)$ as a function of $α $ only''? What does it mean? how can one express a trigonometric function using only angles?
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133 views

Sequences and Contraction of a fixed point

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
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342 views

Tangent plane to the surface $\cos(x)\sin(y)e^z = 0$?

The surface is as in the title, $$\cos(x) \sin(y) e^z = 0$$ I'm looking for the tangent plane at the point $(\frac{\pi}{2},1,0)$ I know the equation of a tangent plane for $z = f(x,y)$ is $$z-z_0 = ...
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1answer
50 views

Bounding the integral of the tails of a random variable.

I found an argument like this in a book, but I couldn't understand how we got this bound. Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| ...
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3k views

Find an equation of the tangent line to the curve $y=x^3-3x+1$ at the given point $(2,3)$

The only thing I know is that you must use the formula to find the slope of the tangent line, but I'm not quite sure on the steps to doing so.
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1answer
78 views

convergence sequence

Suppose that $g$ is continuous on an interval $[a,b]$ and that $g(x) ∈ [a,b]$ for all $x ∈ [a,b]$. (a) Use the intermediate value theorem to prove that is at least one number $c ∈ [a,b]$ with ...
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1answer
208 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
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1answer
194 views

Problems in the Ramanujan Class Invariant $G_n$?

In page 290 of his second notebook , Srinivas Ramanujan defines 2 functions $G_n$ and $g_n$. And then proceeds to give a table of $G_n$. But looking at the papers of Bruce C. Berndt, Heng Huat ...
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1answer
743 views

How is $a_n=(1+1/n)^n$ monotonically increasing and bounded by $3$?

I was reading about how completeness is required for limits. And I came across this: the sequence $a_n=(1+1/n)^n$ is monotonically increasing and bounded by 3 and so we expect it to converge, but ...
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1answer
103 views

Supremum and Infimum of $\,S=\{(-1)^k + 2^-k| k≠0 \,\,\text{and}\,\, k>1\}$

Consider $S=\left\{((-1)^k + 2^{-k}|k\in\mathbb{N}k≠0\right\}$ Determine the $Sup(S)$ and $Inf(S)$ and justify. So far I have that that: $-1 \lt (-1)^m + 2^{-m}$ $\forall m=2k+1$ (The odd powers of ...
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1answer
211 views

Operator curl and gradient

Operator curl $\nabla$ x$(\cdot)$ (x is cross product) working on a $ C^1$ vector field and operator gradient $\nabla(\cdot)$ working on scalar fields. And results of these operators is vector ...
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166 views

Elegant or elementary evaluation of $\lim\limits_{x\to 0} \left( \frac{1}{x}-\frac{1}{\sin(x)} \right) $ [duplicate]

I give math tutoring and was wondering about the following limit. I found the answer but I was wondering if someone has a nicer explanation than the one I am giving where I use L'Hôpital's rule twice. ...
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1answer
138 views

Partial Fraction Decomposition of unknown power

I have an equation of the form: $$\frac{1}{(1+as)(1+0.5bs)^m}$$ where $m$ is unknown, and its range is $(1,2,3,...)$ How I can do the partial fraction? I am now reading a paper that derived an ...
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1answer
193 views

Jensen's inequality for frobenius norm

When I was going through a proof, I saw the following step: $$\frac{1}{p}\operatorname{trace}(\Sigma) \leq \frac{1}{\sqrt{p}} \|\Sigma\|$$ where, $p$ is the number of variables, $\sigma$ is the ...
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3answers
648 views

Implicit differentiation of a lemniscate at a point:

So here's the problem: Find the slope of the tangent line of : $2(x^2 +y^2)^2 = 25(x^2 - y^2)$ at the point (3,1) Cool: So here's what I did: Simplification step: $2(x^2 +y^2)^2 = 25(x^2 - ...
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1answer
170 views

Verified $(\ln x)^{\cos x}$

Determine the derivative of $(\ln x)^{\cos x}$ Can anyone verified my answer: $(\ln x)^{\cos x}(-\sin x\ln (\ln x)+\frac{\cos x}{x\ln x})$ Or do I have too many $x$'s in my answer?
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1answer
105 views

$\dot u(t) = - \nabla V(u(t)) $ unique solution if $V$ is convex

I found this statement in a book I am reading: If $V: \mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable and convex, then the differential equation $$\dot u(t) = - \nabla V(u(t)) $$ has a ...
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606 views

Find $x$-coordinates of all horizontal tangents on the graph of $f(x) = \sin^2x + \cos x$

The function is $f(x) = \sin^2x + \cos x$. I found the derivative which was $f'(x) = 2x\cos^2x - \sin x$. I think what you do next is make $f'(x) = 0$ so it becomes: $0 = 2x\cos^2x - \sin x$. ...
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1answer
99 views

Functions with a finite range are bijective?

Suppose we have a function $f : A \subseteq \mathbb{R} \to \{ a_1 , \dots , a_n \} $. So $\operatorname{range} f $ is finite. Can we conclude that $f$ is bijective? Or only can it be said that it is ...
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1answer
71 views

A inequality about infinite product $\prod\limits_{i=1}^\infty(1-\frac1{2^i})$

Show that $$\prod_{i=1}^\infty(1-\frac1{2^i})>0.288$$
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1answer
44 views

How to draw the graph of this function?

I have to draw the graph of $f(a)=\int_{-\infty}^\infty e^{-ax^2}dx$ on $(0, \infty)$. I know the graph of $g(x)=e^{-ax^2}$, which is but I don't know how to graph the integral. Thank you for ...
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2answers
110 views

How to calculate the derivate of this function?

I have to calculate the derivate of this function: $$f(x)=\int_0^x g(s,x)ds $$ They don't specify what is $g$ but it's just another function. I think I could use the Fundamental Theorem of Calculus ...
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2answers
88 views

Elementary question concerning flux integral

I was asked to evaluate the flux integral $\int\int_{S}F\cdot dS$. Here $F=\langle x,y,z^{4}\rangle$ and $S$ is part of the cone $z=\sqrt{x^{2}+y^{2}}$ beneath the plane $z=1$, oriented downward. My ...
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1answer
155 views

Epsilon-delta proof of the existence of the limit of a sequence?

If $\lim_{n\to\infty}a_n \rightarrow L$ and the function $f$ is continuous at $L$, then $$\lim_{n\to\infty}f(a_n) \rightarrow f(L)$$ $\underline{Proof.}$ Let $n, N \in \mathbb{N}$. Let ...
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1answer
234 views

Calculus III Tricky question! (I think so :) )

Can somebody give me the answer to this one? It is very tricky! I have spent countless hours on this one. I am trying to save myself from doing an all-nighter tonight! ALso, dont take countless ...
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1answer
95 views

Zeta question - prime zeta. Basic calculus

In one of the steps to get the prime zeta $$\log \zeta(s) = \sum_{p \in P} p^{-s} + \sum_{p \in P}\sum_{n \geq 2} \frac{p^{-sn}}{n} $$ $$\log \zeta(s) = \sum_{p \in P} p^{-s} + \sum_{n \geq 2} ...