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

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1answer
149 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
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3answers
249 views

Solving initial value problem

Im solving initial value problem $$ \frac {dy}{dx} + xy = xy^2; y(0)=3$$ After applying Bernoulli's equation method i obtained $$ \frac {dv}{dx} -xv = -x$$ So, $$ p(x) = -x, q(x) = -x $$ For finding ...
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1answer
290 views

Is there a continous function which does not have a derivative in any of its points? [duplicate]

Possible Duplicate: Are Continuous Functions Always Differentiable? Is there a continous function (continous in every one of its points) which is not differentiable in any of its points?
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2answers
580 views

How can I find the average y value of a function on a given domain?

Lets say that $f(x) = (10 - x)\ln x$. Over the domain: $1 ≤ x ≤ 10$. How can I find the average value of $y$ over this domain and what is that value?
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2answers
135 views

Which is the easier way to do integration by parts when there is an exponential term?

I am trying to calculate the following integral, and I would like to know if there is a general rule where we set either $u(x)$ equal to the exponential term or $v'(x)$ equal to the exponential term. ...
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1answer
270 views

Checking that a function satisfies a differential equation

Prove that this function proves the relation written next to it. $y=\sin(x)+a\cdot \cos(x)$ ...the relation is $ y\cdot \sin(x) +y'\cdot \cos(x)=1$ I tried using logarithmic differentiation but ...
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1answer
281 views

Find the particular solution using the method of undetermined coefficients

Find the particular solution using the method of undetermined coefficients: $$y''''+y'''=1-x^2e^{-x}.$$
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1answer
128 views

pre calc, sinusoidal equations

If the sea level decreases leaving the seabed exposed (normally $30$ feet below sea level), then it rises a equal distance above sea level. waves have a maximum height of $38.9$ meters. the cycle of ...
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1answer
46 views

Which statement regarding Lipschitz conditions is stronger?

Statement 1: A function $f$ satisfies a Lipschitz condition in the rectangular region $D$ if there is a positive real number $L$ such that $$|f(t, u) - f(t, v)| \leq L|u - v|$$ for all $(t,u) \in D$ ...
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1answer
104 views

Explanation of easy statement regarding derivative and Jacobian needed

Let $\Phi:S \to T$ be a map between surfaces in $\mathbb{R}^n$. What precisely does this mean: Let $\text{det}(\mathbf{D}_S \Phi(.))$ denote the Jacobian determinant of the matrix representation ...
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1answer
45 views

comparision between undefined and complex number

I was asked the following question and asked to evaluate it: $\log(-8)$. Do I answer that the answer will be a complex number and therefore no real answer exists? Is that true to say?
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2answers
71 views

Chain rule application

I want to find $y'$ where $$ y = \frac{\frac{b}{a}}{1+ce^{-bt}}.$$ But I dont want to use quotient rule for differentiation. I want to use chain rule. My solution is: Write $$y=\frac{b}{a}\cdot ...
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2answers
353 views

replace a differential equation by an equivalent system of first order equations:

If a particle of mass $m$ moves in the $x$-$y$ plane, then its equations of motion are $$m\frac{d^2x}{dt^2}=f(t,x,y)\space \space \text{and} \space \space m\frac{d^2y}{dt^2}=g(t,x,y).$$ Here $f$ ...
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4answers
755 views

Use Lagrange multiplier to find absolute maximum and minimum

Use Lagrange multiplier to find absolute maximum and minimum of $f(x,y) =x^2+xy+y^2, x^2+y^2 =8$. What i've done so far.. $f_x = \lambda g_x \Rightarrow 2x+y =\lambda2x, \\f_y = \lambda g_y ...
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1answer
4k views

How do you calculate the rate of change of the volume of a cone at a given height and radius

when the radius is decreasing and the height is increasing? i have to calculate the partial derivatives, right? but then do i add the values? like V = partial derivative height + partial derivative ...
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2answers
4k views

Use the given graph $f$ over the interval (0, 7) to find the following:

(a) The open intervals on which $f$ is increasing. I answered $(0, 1), (3, 5), (5,7)$ (b) The open intervals on which $f$ is concave upward. I answered $(1, 4)$ (c) The open intervals on which ...
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1answer
153 views

Chain Rule for Vectorial Function

Let be $f:\mathbb{R}^N\rightarrow \mathbb{R}$. Let be $r$ a vector function, such that $r:\mathbb{R}\rightarrow \mathbb{R}^N$. Making $r(t)=y$ and $g(t)=f[r(t)]$. My lecture say: $g'(t)=\nabla ...
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1answer
58 views

Extend functions such that they are continuous

Let $f,g:\mathbb{R}\to\mathbb{R}$ denote the functions $$f(x)=\begin{cases}-x+3,&x\leq 1,\\-ax^2,&\text{otherwise}.\end{cases}$$ ...
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1answer
64 views

proof of existence for the Limit of a function

I'm stuck with this proof I just can't get my head around and I would really appreciate any sort of help. The problem is as follows: Problem: Let $f$ be a function defined in the neighborhood of the ...
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1answer
83 views

Convergence of $\int_{0}^{+\infty}f(x)dx$

How can I check convergence of $\int_{0}^{+\infty}f(x)dx$ for the following $f$? $f(x)=\left(\frac{\sin x}{x}\right)^2$ $f(x)=\left(\frac{\cos x}{x}\right)^2$ $f(x)=\frac{1}{(1+x)^2}$ ...
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2answers
45 views

Need a little help with this problem to do with partial derivatives?

Suppose $z = \ln\Big(\frac{x^3+y^3}{x-y}\Big)$. Show that $x\frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=2.$ Any help much appreciated.
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1answer
186 views

Finding lower bound sequence for squeeze theorem

I need to find: $$\lim_{n\to \infty} a_n =\lim_{n\to \infty} \frac{1}{n^2 +n} , for: \forall n \in \Bbb N \setminus \{ 0 \}$$ By using the Sandwich a.k.a. squeeze theorem.My ideas so far: when i ...
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1answer
89 views

Bernoulli Polynomials

I am having a problem with this question. Can someone help me please. We are defining a sequence of polynomials such that: $P_0(x)=1; P_n'(x)=nP_{n-1}(x) \mbox{ and} \int_{0}^1P_n(x)dx=0$ I need to ...
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1answer
32 views

How do I build a DE to model temperature inside something?

This is really a small cluster of questions on the same thing. I was working with this problem to do with the temperature inside a cooler. The idea was that the rate of change of temperature ...
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1answer
107 views

Show $\lim_{x\rightarrow 0^+} x^{-\ln x} = 0$ without L'Hopital

How can I formally show that the following limit is $0$? $$\lim_{x\rightarrow 0^+} x^{-\ln x}$$ (Without using l'Hopital's rule.) I can write it as $$\lim_{x\rightarrow 0^+} x^{-\ln x} = ...
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2answers
88 views

Compute the derivative of $y = 2^{3^{x^{2}}}$

What is the derivative of $$y = 2^{3^{x^{2}}}$$ using the chain rule? Please show step by step solution.
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1answer
83 views

max/min problem with areas

You are given 30 meter of material which you will cut into two pieces. One piece will form an equilateral triangle, the other a rectangle whose length is three times its width. Where should you cut ...
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1answer
82 views

Triple integral in cylindrical coordinates

Solve $$\int\limits_{0}^{3}\int\limits_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int\limits_{0}^{9-3\sqrt{x^2+y^2}}dzdxdy.$$ Update: The problem is setting up the integral. What I have tried was ...
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1answer
127 views

Why $ F(u_\gamma) = \int_{\Omega} | \nabla u (D \tau_{\gamma})^{-1} |^2 \det (D \tau_\gamma) ?$ Is this by change of variables?

I'd like to understand the following: Let \begin{equation} F(u) = \int_{\Omega } |\nabla u|^2 \end{equation} where $ \Omega $ is adomain in $ \mathbb{R}^{n} \cdots $ For $ | \gamma | $ small enough ...
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1answer
103 views

Derivative of a 'weighted average' of decreasing fractions

I'm having some trouble showing the following statement (which intuitively seems to hold): Suppose I have a series of fractions indexed by $i$ , each of them a function of $N:f_{i}\left( N\right) ...
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1answer
54 views

Does the logistic function uniquely satisfy these three conditions?

Given $$r(t)=\frac{f(t)}{1-F(t)} \tag{Eq. 1}$$ where $$f(t)=\frac{dF}{dt} \tag{Eq. 2}$$ and the conditions: $$\lim_{t\rightarrow \infty} r(t)=1 \tag{Eq. 3}$$ $$\lim_{t\rightarrow \infty} F(t)=1 ...
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1answer
93 views

Help with proof of a converging sequence.

I am having a hard time understanding this proof (leading up to Newtons method) about why a sequence must converge. English is not my native language so I will do my best to use the correct ...
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1answer
203 views

Lagrange multipliers word problem

How do i approach this word problem? Say that you are in a pirate ship that is traveling along a curved river (which roughly follows the equation $y_1=x_1(sin(x_1)+1)$) as you travel from the ...
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1answer
459 views

Supremum, infimum

I came across nasty task which includes supremum, infimum and I am confused about it. The question is to find a supremum and infimum of a given set:$$A =(x+y+z:x,y,z>0,xyz=1)$$I tried to eliminate ...
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1answer
42 views

A question related to cosine function

For a fixed $n\in \mathbb{N}$, prove that $\cos(\frac{jr\pi}{n})\neq 1$ if and only if $\gcd(j,2n)=1$, where $1\leq j,r\leq (n-1)$.
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1answer
65 views

Convergent Of Sum of $\frac{k}{n^2}$

We have in real analysis that average of every sequence is convergent to the limit of that sequence. So we assume that $\frac{k}n$ for $k=1,2,\ldots,n$ is a sequence and we want to calculate its ...
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1answer
716 views

Cardioid given by the polar equation $r = 1 − \cos(\theta)$

Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$. (a) Find the intersection points of the curve with the line $\theta = \pi/4$. (b) Find the ...
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1answer
140 views

Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1

Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below. $x'=f(x)=-x$ $x(0)=1$ EDIT / UPDATE: x_n+1=x_n + ...
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2answers
105 views

Inequality with integrals

I am having a problem with the following exercise. Let $f$, $g$ be continuous non-negative functions on $[a,b]$, and let $C$ a positive constant. Suppose that: $f(x) \leq C+ \int_{a}^x f(t)g(t)dt$, ...
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1answer
296 views

Using higher-order derivative tests

I have a function such that $f'''$ is continuous and let's suppose there's some point $a \in \mathbb{R}$ such that $f'(a) = f''(a) = 0$. Does $f'''(a)>0$ tell me anything about whether $a$ is a ...
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1answer
153 views

concavity of function composition

Suppose we have functions $f$ and $g$ that are both concave upward and that both have a continuous and nonzero second derivative at every point. Is there any restriction required on $f$ such that $f ...
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1answer
151 views

Cases when the intermediate value theorem is true

Consider the intermediate value theorem. It says that a continuous function $f(x)$ on a closed interval $[a,b]$ takes on every value between $f(a)$ and $f(b)$ at least once. Excluding trivialities ...
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2answers
387 views

Well-defined integral

How can I show this integral is well defined? Let f be an increasing function on [0,1] and $F(x)= \int _0^x f(t) dt$. My attempt: f is an increasing function so it may only have a countable set of ...
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2answers
36 views

Analyzing a function

I am having a problem with the following function: $f(x)=\sin^2(x)-\cos(3x)$ I need to examine the sign of $f'(x)$ I noticed that f is $2\pi$-periodic, therefore we need to analyze f(x) on ...
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1answer
62 views

simple calculus inquiry

Define $$f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0\\ 1 & \text{if } x = 0\\ \end{cases}$$ Show that $f$ is uniformly continuous (UC) on $\mathbb{R}$. My Approach: ...
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1answer
152 views

Related Rates calculus problem

For some reason I keep getting this question wrong. Suppose a 6 feet tall man is walking away from a 15 foot tall lamp post at 5ft/s. What is the rate at which the man's shadow is moving when he is ...
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1answer
40 views

Conditional Convergents and Absolute Convegent

Is it possible to make an absolutely convergent series from two conditional convergent series? Just to be more clear, if we have an absolutely convergent series, we can rearrange the terms and ...
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1answer
777 views

proof about boundary points and closed sets

Let S be a subset of $R^n$. Show that S is closed if and only if the boundary of S belongs to S. I was thinking about using definitions of closed points and boundary points to proove this, but ...
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1answer
718 views

prove a function is continuously differentiable

$f(x,y) =\begin{cases}\arctan(y/x) & x\neq 0\\ \pi/2 & x=0,y>0\\-\pi/2 & x=0,y<0.\end{cases}$ $f$ is defined on $\Bbb R^2\smallsetminus\{(0,0)\}.$ Show that $f$ is continuously ...
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1answer
123 views

Proving things like $d(a+b)=da+db$

This may sound a bit too elementary but at school, differential equations are taught in a haste . I am quite shaky when it comes to rigorously prove identities like $d(a+b)=da+db$.So, can anyone ...