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

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0
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3answers
64 views

Find global minimum

for the course Non-Linear Optimization I faced the following problem I couldn't solve: $x^2 + xy + 2y^2 + 3 \rightarrow min$ where $x,y \in \mathbb{R}$ Now I first computed the boundary points ...
0
votes
1answer
139 views

On the fundamental theorem of Calculus (Finding the derivative of a definite, trigonometric integral)

While working on a subchapter in the chapter about integration named "The Fundamental theorem of Calculus", I am presented with the following task: "Find the indicated derivative: ...
2
votes
2answers
109 views

A Cauchy sequence in $L^2$

Problem Is $f_n = 1_{(n, \infty)} \frac{1}{x^2} $ Cauchy in $L^2(0, \infty ) $ ?? $$ ||f||_2 = (\int\limits_E |f|^2 )^{1/2} $$ Atttempt We start as follows: $$ || f_n - f_m ||^2_2 = ...
1
vote
3answers
98 views

Minimize the area of a triangle

Let $A \neq B$ be fixed points outside a fixed circle with centre $C$. The point $D$ can be chosen freely on the circle. The goal is to minimise the area of triangle $ABD$. Degenerate triangles ...
3
votes
0answers
46 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
2
votes
2answers
76 views

Intermediate value theorem for mean

Suppose the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous. For a natural number $k$, let $x_1,x_2,...,x_k$ be points in $[a,b]$. Prove there is a point $z$ in $[a,b]$ at which ...
0
votes
1answer
382 views

Solve $A = \displaystyle \int_{0}^{3} \left[- \frac {x^2}{4} - 8\right] \,dx$

Consider the following definite integral $A = \displaystyle \int_{0}^{3} \left[- \frac {x^2}{4} - 8\right] \,dx$ Calculate the Riemann sum that approximates the value of A as a closed-form formula ...
0
votes
1answer
65 views

Calculus $3$: Vertex, equation for tangent plane, normal line at $P_0$?

If $P_0(x_0,y_0,z_0)$ is a point on the cone $z^2=a(x^2+y^2)$.What is the equation of the tangent plane and normal line at $P_0$?
4
votes
4answers
263 views

How would I differentiate $\sin{x}^{\cos{x}}?$

How can I differentiate $\displaystyle \sin{x}^{\cos{x}}$? I know the power rule will not work in this case, but logarithmic differentiation would. I'm not sure how to start the problem though and I'm ...
2
votes
1answer
39 views

integration using substitution: symbols $\frac{dy}{dx}$ are used as variables?

I'm learning integration using substitution and the symbols $\frac{dy}{dx}$ are used as variables which is confusing me as I thought they weren't normal variables. So if I'm integrating something and ...
1
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2answers
57 views

Chain rule method doesn't result with same answer as u-sub. Why?

$\int \ln \left(2x\right)\,\mathrm{d}x$ is the integral in question. I know how to solve it with the chain rule. $\frac{1}{2x}\times 2x = \frac{1}{x}$ But, because I know $u$-sub method, I wish to ...
0
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2answers
74 views

The Derivative of $\cos(x-2)$

I would think that the solution would be $-\sin(x-2)$, but when i use WolframAlpha it says that the answer is $\sin(2-x)$. Are these $2$ answers equivalent or I am missing some fact here? Thanks in ...
3
votes
1answer
47 views

Different real roots polynomial, roots of $P'+aP$

Let $P$ be a polynomial of degree $n$ with real roots $t_1<t_2\ldots<t_n$. Show that $P' + aP$, with $a\in\mathbb{R}$, has only real roots. Is easy to conclude if $a=0$, by Rolle's theorem. ...
0
votes
1answer
77 views

How do you find the equation of motion in the absence of damping (x(complementary) and x(particular) solved)?

Let $x(0) = 0$, $x'(0) = 0$, and take a particular solution $x_p = e^{-2t}(\frac{1}{2}\cos(4t) - 2\sin (4t))$ and homogeneous solution $x_c = c_1 e^{8t} + c_2 e^{8t}$. So I put those two together to ...
1
vote
1answer
63 views

How many solution with the equation $f_{2013}(x)=\frac{x}{2013}$

let $f(x)=f_{1}(x)=\mid \cos{(2\pi x)}\mid,f_{2}(x)=f(f_{1}(x))=\mid \cos{(2\pi (\mid\cos{(2\pi x)}\mid)}\mid$ $f_{n}(x)=f(f_{n-1}(x))$, Question: How many solution with the equation ...
1
vote
3answers
170 views

Prove f(x) <= x for all x>=0 if f ' (x) <= -2 for all x and f(0) = 0

The title basically states the whole question..I was trying to invoke the Mean Value Theorem on it but it hasn't worked..I was wondering if I'm supposed to solve it some other way. I just need hints, ...
-1
votes
1answer
79 views

Limit Comparison Test - ex. prob wrong?

I'm doing some practice problems on the Limit Comparison test from this site: http://archives.math.utk.edu/visual.calculus/6/series.14/index.html But I'm a bit confused on this problem: ...
-1
votes
2answers
522 views

Increasing/Decreasing Test with Exponential Function

The goal is to find the intervals by which the function $f(x) = e^{x} - e^{2x}$ is increasing and decreasing, as well as any local maxima/minima, intervals of concavity, inflection points, asymptotes, ...
1
vote
2answers
227 views

Find this limit without L'hopital Rule : $\lim_{x\rightarrow +\infty}\frac{x(1+\sin(x))}{x-\sqrt{(1+x^2)}}$.

Find this limit without L'hopital Rule : $\lim_{x\rightarrow +\infty}\frac{x(1+ \sin(x))}{x-\sqrt{1+x^2}}$. I tried much! but can't get any progress!
1
vote
3answers
57 views

prove that there is no supremum for mn/1+m+n

D = { mn/(1+m+n) } for m,n natural numbers. To simplify the expression, I presumed m=n, which means: D = { n^2 / (2n + 1) } Now, I know by intuition there is no supremum, for this series ...
6
votes
2answers
433 views

Proof that $\sin(x)$ don't have limit to infinity

I just used the Heine's definition. Let $\alpha,\delta \in \mathbb{R}$ such that $\sin(\alpha)=a$ and $\sin(\delta)=b$. Let $(u_{n})=\alpha+2\pi n$ and $(v_{n})=\delta+2\pi n$ and $f(x)=\sin(x)$. So ...
22
votes
1answer
322 views

Integral $\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$$
3
votes
1answer
84 views

Variable change in integral makes a split domain become conected

Let $D=\{(x,y):0<x,0<y,x^2+y^2<4,x^2-y^2>1\}$. Compute $\iint_D(x^5y+xy^5)dA$. My attempt: Let $u=x^2+y^2$,$v=x^2-y^2$. The Jacobian of the substitution is ...
21
votes
2answers
487 views

Conjecture: $\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\big(\!\frac{2x}{1+x}\!\big)\,dx\stackrel ?=\frac\pi{5\sqrt2}$

$$\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\left(\frac{2x}{1+x}\right)\,dx\stackrel ?=\frac\pi{5\sqrt2}$$ The integral above comes from the evaluation of the integral ...
1
vote
2answers
58 views

proof by negation - there is no supremum

A = { x + 1/x : x > 0 } How do you prove that there is no supremum for this set? I think this is the inequality needed: M - ε > x + 1/x M - the supremum How do you keep from this step? thanks
9
votes
5answers
567 views

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$? I know its answer: 0, -2, -1+i, -1-i. But I dont know how to find? Please show me this. I know this is so trivial, but important ...
0
votes
2answers
58 views

Limit point of a bounded sequence

My textbook says that if $x_n$ is a bounded sequence ($\exists m,M: m \le x_n \le M$) and a is a limit point of this sequence, then $m \le a \le M$. Can somebody explain why is that so? How can I ...
2
votes
1answer
415 views

The conditions of applying L'Hospital's rule

I just finished calculus 1,2 last semester and I am learning calculus 3 now. I saw this question and I post a solution as follow: Prove series convergent,consider the limit: ...
2
votes
2answers
55 views

$ \lim_{x \to 0} a^x = 1 $ using the definition

Given $a > 1$, define $ f: \mathbb{Q} \to \mathbb{R}$ such that for all $ x \in \mathbb{Q}$ we have $ f(x) = a^x$ . Prove that $ \lim_{x \to 0} f(x) = 1$ I'm having issues to find my delta. I ...
6
votes
5answers
243 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
0
votes
2answers
52 views

Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function.

Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function. Using the monotonicity of $f'$, prove that the function $g(x)= ...
0
votes
1answer
52 views

Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin.

Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin. Prove that if $f$ is monotone and $$\lim_{x\to 0} \frac {f(x)-f(-x)}{x}=0,$$ then f is differentiable at $0$. I ...
0
votes
0answers
26 views

Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
19
votes
2answers
585 views

Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
0
votes
1answer
50 views

Monotonicity in $x$ of $\frac{x^\alpha - 1}{x-1}$

I'm trying to show that for fixed $\alpha$, the function $f(x)=\frac{x^\alpha-1}{x-1}$ is monotonic for $x>0$ -- either strictly increasing or strictly decreasing, depending on the choice of ...
1
vote
2answers
597 views

Give an example of closed, disjoint subsets A and B of the plane R^2 for which d(A,B) = 0 [duplicate]

let X,d be a metric space d(x,A) = inf{d(x,a):a of A} d(A,B) = inf{d(a,b):a of A, b of B} (i) prove that d(x,A) = 0 iff x is an element of A bar for this I came to the conclusion that the above ...
0
votes
2answers
117 views

Computing $\displaystyle\int \frac{1}{x^{2}\sqrt{x^{2}+7}}$

Find the primitive function of $$\frac{1}{x^{2}\sqrt{x^{2}+7}}$$ Attempt. And my answer is $$\arcsin \left( \frac{\sqrt{\left( x^{2}+7 \right)}}{\sqrt{7}} \right)$$ Why am I wrong? See link for ...
0
votes
3answers
70 views

Check if a given limit of a sequence is valid

Using the definition of the limit with vicinities (rough translation from my native language) , prove that: $$ \lim_{n\to \infty} \frac {2^n+3}{2^n+4^n}=0 $$ For convenience I will take $a_n=\frac ...
0
votes
1answer
177 views

How to solve an arg max question?

I have no idea how to solve an $\arg \max$ mathematically, for example $\arg \max(x(10-x))$. I know the solution is $x=5$, but how do I get there (for more difficult exercises I will have to solve). ...
2
votes
3answers
2k views

Expected value of a Gaussian

I am trying to find the expected value of a univariate gaussian distribution. After trying to calculate the integral by hand I noticed that my calculus is not good enough for it yet. (I would need ...
1
vote
2answers
341 views

Finding all local maximum and minimum points of function

If $$f(x) = \left\{\begin{array}{lr} x, \ \text {if x is rational}, \\ 0, \ \text {if x is irrational}, \end{array}\right. $$ Find all local maximum and minimum points of $f(x)$. How can I go about ...
0
votes
1answer
76 views

sum of square roots

I was wondering what the estimate for the value $$ S= \sum_{j=1}^N \sqrt{j} $$ is? Is there a way (or a formula) to estimate it well? I am sure it is close to $\int_1^N \sqrt{t} dt$, but I guess I ...
1
vote
1answer
59 views

Is it possible to calculate a minimum initial speed required to travel a distance x, in time t with an acceleration a

Is it possible to work out the minimum initial speed required for a ball to travel a distance x in a time of t, given that a there is a constant acceleration of -friction? I know that given a ...
0
votes
1answer
305 views

Implicit derivitave of a general ellipse

Consider an ellipse centered at the point $(h,k)$. Find all points $P=(x,y)$ on the ellipse for which the tangent line at $P$ is perpendicular to the line through $P$ and $(h,k)$. I know the general ...
-1
votes
2answers
312 views

How to find the domain of $f\left(g\left(y\right)\right)=\sqrt{\left(\frac{x+1}{x-1}\right)^3-27}$

Please help me find the domain of the following equation. \begin{eqnarray} \\f\left(x\right)=\sqrt{x^3-27},\space \space \space g\left(y\right)=\frac{x+1}{x-1},\space \space \space find \space ...
1
vote
0answers
24 views

Product of discontinuous function with zero

This is going to be breathtakingly stupid, but it's one of those things I've forgotten: if we have a function like $\frac{1}{\sqrt{1 -x^2}}$ and we multiply it by zero, i.e. $\frac{1}{\sqrt{1 - ...
1
vote
3answers
49 views

Question regarding simple limit

Why is it, that: $\lim_{x \to \infty} [x (1-\sqrt{1-\frac{c}{x}})] = \frac{c}{2}$ Link: Wolframalpha and not $0$? My (obviously incorrect) reasoning: Since $c$ is an arbitrary constant, and as ...
1
vote
0answers
113 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
0
votes
1answer
38 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
2
votes
2answers
123 views

Show there is a closed interval $[a, b]$ such that the function $f(x) = |x|^{\frac1{2}}$ is continuous but not Lipschitz on on $[a, b]$.

Hi guys I was given this as an "exercise" in my calculus class and we weren't told what a Lipschitz is so i really need some help, heres the question again: Show there is a closed interval $[a, b]$ ...