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

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9 views

Prove that f is differentiable

Prove that if $f$ satisfies $|f(x,y)| \leq (x^2 + y^2)$ then f is differentiable at $(x,y)$. I understand how to prove this: one can deduce that $f(0,0)=0$, and then we can assume that $L_{(0,0)} ...
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2answers
43 views

What is valid and what is not in limits.

I have been looking for $$ \lim_\limits{x\to 0}{\left({\sin x\over x}\right)}^{1\over x^2}. $$ So I took $$ \lim_\limits{x\to 0}\left({\left({1+{\sin x-x\over x}}\right)}^{({x\over \sin ...
2
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1answer
21 views

Related Rates Problem Involving Airplanes

I took a test yesterday, and would like to know how to answer this specific question on the exam: One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane ...
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0answers
8 views

Polar and Spherical Coordinates Conversion

I learnt $dxdy = rdrd\theta$ from Calculus, but why the following doesn't work as expected? \begin{align} dxdy &= (drcos\theta)(drsin\theta)\\ &= ((dr)cos\theta-rsin\theta ...
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1answer
29 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
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0answers
12 views

Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$. I know this can be done by the sv-calc method of volumes of revolution but I ...
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0answers
9 views

find angular velocity for so that: $\exp(jt) = \exp( j(3t+\pi/3) )$

I have a fourier series in which there are two different arguments on the exponential function: $jt$ and $j(3t+\pi/3)$ and I have to "choose" a fitting angular velocity. It it probably easy yet it ...
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1answer
23 views

Evaluate an integral quickly

Evaluate the integral $$\int \sqrt{x} \ln(1+x)dx $$ so we should start with the substitution: $t=\sqrt{x}$ $$ \int t\ln(1+t)dt2t = 2\int t^2\ln(1+t)dt $$ From here, it seems reasonable to ...
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0answers
16 views

Calculus - Vector Functions

Consider the following vector function. $r(t) = ({5\sqrt2}{t}, e^{5t}, e^{-5t})$: Find the unit tangent and unit normal vectors $T(t)$ and $N(t)$. Find the curvature $\kappa(t)$
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2answers
68 views

Solving functional equation $f(x)f(y) = f(x+y)$

I'm having some trouble solving the following equation for $f: A \rightarrow B$ where $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{C}$ such as: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
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5answers
68 views

One integral, two solutions?

I ran into an interesting integral problem: the indefinite integral of $\int \frac{dx}{a^2 x^2 - b^2}$. I can do a hyperbolic trig substitution and get that the result is $- ...
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1answer
16 views

Find the derivative of the question mentioned

Question: find the derivative of $$ x^3\cdot y^7+\frac{x}{y}-x^2+y^4=3 $$ I got $$ \frac{dy}{dx}=\frac{2xy^2-y-3x^2y^9}{4y^5+7x^3y^8-x} $$
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0answers
9 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
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3answers
22 views

Multivariable: Show the limit is $0$.

I already proved that for $\alpha =1$ the limit doesn't exists. Now I need to show that for $\alpha > 1$ the limit does exists and equals $0$. $$\lim_{(x,y)\to (0,0)} \frac{\left|x\right|^\alpha ...
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0answers
14 views

When should we use $f'(c)=\lim_{h\to0}\frac{[f(c+h)-f(c)}{h}$ and $\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$?

When should we use $f'(c)=\lim\limits_{h\to0}\dfrac{[f(c+h)-f(c)}{h}$ and $\lim\limits_{x\to c}\dfrac{f(x)-f(c)}{x-c}$? I don't really understand when should I use them. If the question by ...
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2answers
20 views

What are the derivative, differentiable and differentiation? [on hold]

If the question is $f(x)=\frac{2x+1}{1-x}$? The derivative is $f'(x)=\frac{(1-x)(2)-(2x+1)(-1)}{(1-x)^2}$?
-1
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2answers
38 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
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1answer
14 views

Rate of change question involving velocity, displacement and acceleration

I have been having trouble understanding questions c)-e) and am in need of some help: An object is moving in a straight line from a fixed point. The displacement $s$ in metres is given by ...
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4answers
54 views

Find $y^{(n)}(0)$ for every $n$.

Let $y(x)$ fulfill $y''-xy=0$. Furthermore: $y(0)=0,y'(0)=1$. Find $y^{(n)}(0)$ for every $n$. I tried different forms of recurrence relations but I couldn't do much with it without it becoming a ...
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0answers
18 views

A tank of the shape of a right circular cylinder $5$ feet across the top and $9$ feet deep is full of water. [on hold]

A tank of the shape of a right circular cylinder $5$ feet across the top and $9$ feet deep is full of water. How much work is done by pumping the water out of the tank, over the top edge? I need your ...
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3answers
67 views

Help with $\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1})$ [duplicate]

$\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1}) = ?$ I don't know how to solve the indetermination there... is it possible to rearrange the expression in brackets in order to use ...
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2answers
31 views

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly

Prove that $f_n(x) = \left(\frac{x}{n}\right)^ne^{-x}$ converges uniformly at $[1,\infty)$. So for every $x$, there's $N\in\mathbb{N}$, such that for all $n>N$: $\frac{x}{n} < 1$. ...
2
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0answers
14 views

How do I integrate this master equation from a time-continuous Markov chain?

I hope the question is not too vague. My calculus courses are way in the past and I can't remember how to do it :-). I have this master equation for a time-continuous Markov chain I have a two ...
1
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0answers
34 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
2
votes
1answer
52 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
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0answers
27 views

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r$ is a smooth real-valued function?

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r: \mathbb R \rightarrow \mathbb R$ is a smooth real-valued function ? In Calculus I've seen linear (higher-order) ...
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1answer
13 views

Diffeomorphism between open sets of half-space

Let $\mathbb{H}^{m}=\left\{(x_{1},...,x_{m})|x_{m}\geq0\right\}$. How can i prove that if $A$ and $B$ are respectively open set of $\mathbb{H}^{m}$ and of $\mathbb{H}^{n}$, with $n\ne m$, then they ...
0
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1answer
25 views

Generalized angle sum identity for $\arctan$?

The angle sum identity for arctan is: $$\arctan (\alpha)+\arctan(\beta)=\arctan\left(\frac{\alpha+\beta}{1-\alpha\beta}\right)$$ I was wondering if there exists a relationship for any linear ...
0
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0answers
15 views

Partial integration and substition rule.

Well in my day to day usage I now came upon an example of using the substition rule where I can't see how it works, and I wonder how one could handle such an equation with ease. The set of equations ...
1
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2answers
56 views

Find all planes which are tangent to a surface

I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes ...
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3answers
25 views

Identify the Differential Equations from the given problem [on hold]

Dear Math expert, Please solve the above problem. Thanks in advance for your support!
2
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0answers
30 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
0
votes
3answers
36 views

Integral $ \int \frac{1}{x^{1+a} (1-x)^{1-a}} dx~,~a \gt 0$

The following integral is part of a large problem I'm trying to solve and I'm stuck. I'd appreciate some guidance. I would like to know how to compute integrals of the form $$ \int ...
2
votes
1answer
44 views

Can this special case happen when working with L'Hopitals rule?

I am using this version of L'Hopital's rule Assume that $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=0$, and that the limit-value $\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ exists (could ...
2
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4answers
60 views

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$.

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$. I have been trying for hours using the continuity of $e$ and using L'Hopital rule but it gets really scattered and ugly. I am in despaire. ...
2
votes
2answers
254 views

Find $\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
0
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2answers
59 views

How to evaluate the integral $\int x^2/\sqrt{4-x^2}\,dx$?

How to compute this integral? $$\int \frac{x^2}{\sqrt{4-x^2}}dx$$ If there were $x$ instead of $x^2$ in the numerator I know how to do a substitution $y=4-x^2$. But this doesn't help with the $x^2$.
2
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2answers
26 views

Can every differentiable scalar function be written as a divergence of some vector field?

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field ...
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1answer
23 views

Solution of nonhomogenious differential equations

Kindly help me regarding below math problem. How can I prove? Show that if $y_1(x)$ is a solution of $$y'' + ay' + by = f_1(x)$$ and if $y_2(x)$ is a solution of $$y'' + ay' + by = f_2(x)$$ ...
1
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1answer
23 views

is split function derivable

$ f(x) = \begin{cases} \frac{sin(x)}{x}, & x \ne0 \\ x+1, & x=0 \end{cases}$ I know that the function is a continuous function in R. But is this function derivable at x=0? I am not sure.. ...
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0answers
6 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...
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1answer
31 views

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. What has gone wrong?

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. Using L'Hopital's rule I get that $$\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}=\lim_\limits{x\to 1}{\alpha x^{\alpha-1}\over \beta ...
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votes
1answer
46 views

continous function s.t $ \int_{0}^{\infty} f(x) dx$ exist [on hold]

Let $f:R \to R$ be a continuous function s.t. $ \int_0^\infty f(x) dx$ exists. Which of following is correct. If $\lim_{x \to \infty} f(x)$ exists then $\lim f(x) = 0$ The $\lim f(x)$ must exist and ...
0
votes
1answer
28 views

Conditions for a supremum of a set.

Suppose a function $f(x)$ is continuous on $[a, b]$ and there exists, $x_0 \in (a, b)$ such that $f(x_0) > 0$. And then define a set, $$A = \{ a \le x < x_0 \space | \space f(x) = 0 \}$$ We ...
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0answers
67 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
0
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0answers
8 views

Approximating the Arc Length of a Regular Curve with a Broken Line

Question: Suppose $\alpha:[a,b]\to\mathbb{R}^3$ is a regular curve segment. Prove that, for every $\epsilon>0$, there exists $\delta>0$ such that, for any partition ...
0
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0answers
9 views

Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
0
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0answers
9 views

Differentiable vector space valued functions doesn't depend on basis?

Differentiable vector space valued functions. Let $V$ be a vector space over $\mathbb F^n$ ($\mathbb R$ or $\mathbb C$) and let $v_1, \ldots, v_n$ be a basis for $V$. Define the linear isomorphism ...
0
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0answers
13 views

What is a parametrized function?

I am readimg this article: Stochastic Gradient Descent Tricks and I would like some precisions: Each example $z$ is a pair $(x, y)$ composed of an arbitrary input $x$ and a scalar output $y$. We ...
1
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2answers
25 views

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types $$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$ I already showed, that $f$ has poles at points $z=\pi n$ where ...