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

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1
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1answer
22 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
1
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3answers
61 views

How to find such limits : $\lim_{m \to \infty} \lim_{n \to \infty} cos^{2m}(n! \pi x)$ [duplicate]

How to find such limits : $$\lim_{m \to \infty} \lim_{n \to \infty} cos^{2m}(n! \pi x)$$ Please suggest not getting any idea how to approach such problems, will be of great help thanks.
2
votes
2answers
113 views

Compute $\int\frac{1}{3+\cos^3{x}}\mathrm{d} x$

I have an integral which seems hard for me: $$\int\frac{1}{3+\cos^3{x}}\,\mathrm{d}x.$$ If I use Weierstrass substitution I get $I=\int{\frac{(1+t^2)^2}{3(1+t^2)^3+(1-t^2)^3}dt} $ I was stuck here
-3
votes
2answers
19 views

Slope of the tangent line from limit definition [on hold]

I really have no idea where to even begin with a problem like this, someone please explain the steps! a. $f(x)=7x-3$, at the point $(3,18)$ b. $f(x)=-7x^2 + 20x$, at the point $(-2,-68)$
0
votes
1answer
48 views

Proving $x_ky_k\to ab $

Prove that the sequence $x_ky_k\to ab $ if $x_k\to a$ and $y_k\to b$. I wanted to try and do this with the epsilon definition but i am having a few technical issues. Proof: $$ |x_ky_k - ab| = |x_k ...
-3
votes
1answer
27 views

Max and min of two variable function [on hold]

What is the max/min of $f(x,y)=x^3-y^3+3xy$ in the set $K=[0,4]\times[-4,0]$
-1
votes
1answer
59 views

Radius of convergence of sum [on hold]

what is the radius of convergence of the following sum? $$\sum_{n=0}^{\infty}\frac{6n^3+7}{16^n(3n^3+4n+2)}x^n$$
-1
votes
1answer
25 views

Understanding Composition Function (fg)(-1) for f(x)=x-3 & g(x)=x^2-8x+15?

Can someone help explain how to do the following composition function to me? (or at least get me started) Find the value of (fg)(-1) if ...
0
votes
0answers
5 views

To be a quasi concavity or quasi convexity at some interval

To be a quasi concavity or quasi convexity at some interval at given interval the function values should not create oscillation in the given interval. Would this be good definition?
0
votes
2answers
78 views

How do I find the exact value of $\cos^2\left(\frac{5\pi}{12}\right)$?

I'm having trouble finding the exact value of $\cos^2\left(\frac{5\pi}{12}\right)$ in radians. I was able to figure out that: $$ \begin{align} \cos\left(\frac{7\pi}{12}\right) &= ...
2
votes
3answers
36 views

Find tangent to trigonometric function

I want to find the tangent to the curve: $x\sin{y} + y\sin{x} = \frac{\pi}{4}(1+\sqrt{2})$ through the point $(\frac{\pi}{2}, \frac{\pi}{4})$ Now I know I can fill certain information into this ...
0
votes
0answers
30 views

Stoke's formula for a sphere

I have a question, raised form kinetic theory (pure mathematical). Imagine, that $\Psi (\overrightarrow{r},\overrightarrow{p},t)$ - sufficiently smooth function, where $\overrightarrow{r}$ - radius ...
0
votes
1answer
29 views

Confused About Definition of a Limit Proof

I'm working on $\epsilon-\delta$ limit proofs, and there's something about the proof I don't get. Currently doing a proof for $\lim_{x\to 3} (2x-1) = 5$. The first part of the definition says "if ...
2
votes
2answers
33 views

Binomial expansion derivative limit definition

Can someone help me with this? I am supposed to use a binomial expansion to calculate $\sqrt x$ directly from the limit definition of a derivative.
0
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1answer
25 views

Quick way of writing equation of a tangent

The quick method and the more elaborate one do not reconcile. I must be making a mistake somewhere, mind pointing out? Equation: $x^2-3y^2=4y$ Tangent to a general point $(x_1,y_1)$ can be written ...
0
votes
2answers
37 views

Question involving gradient of a function.

We are given any arbitrary ellipse with focii $F1$ and $F2$ , $T$ is the unit tangent to the ellipse through a point $P$. Let $f$ be the sum of the distances of the of $F1$ and $F2$ from $P$ , we ...
0
votes
1answer
15 views

what does it mean for a curve to be the union of two curves?

http://i.stack.imgur.com/xNXw4.png Question 2a asks for the length of C, but what does it mean for a curve to be the union of two curves? Is it just a curve defined by $cos(t)=1+t-2\pi$ and ...
0
votes
0answers
22 views

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $.

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $. I have no idea where to start with this one, any help would be ...
-2
votes
3answers
62 views

What does $2\cos^2(\theta)−1$ equal to in radians? [on hold]

$$2\cos^2(\theta)−1$$ How would I go about simplifying an expression like this?
0
votes
0answers
8 views

Finding the Requested Value of Composition Functions?

I seem to still be having difficulty with understanding Composition Functions in Calculus and wondered if someone might be able to help explain in a way that will make the light bulb "turn on"? For ...
1
vote
1answer
52 views

How do I numerically solve this type of differential equation? (Wave Equation)

I'm trying to solve the wave equation numerically. I'm brand new to this and what I'm basically trying to accomplish is simulating a plucked string with fixed endpoints. How do I find the $h(x,t)$ ...
0
votes
1answer
96 views

How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test?

How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test? I tried using the comparison test but I could not come up with an inequality that helps ...
0
votes
1answer
21 views

Find the volume of the solid obtained by rotating the region bounded by the curve [on hold]

Find the volume of the solid obtained by rotating the region bounded by the curve $y=\sin(10 x^2)$ and the $x$-axis, $0 \le x \le \sqrt{\frac{\pi}{10}}$, about the $y$-axis.
1
vote
1answer
44 views

A Taylor series given by an integral: how to compute the radius of convergence?

I have to compute the radius of convergence for the Taylor series of $f(x)$ around $x=0$, where $$ f(x)=\int_0^1 \log\left(x+\sqrt{t^2+1}\right)dt.$$ Any hints?
-2
votes
1answer
25 views

Calc 3 Problem about work [on hold]

Find the work done by the force → F = −2k to move an object from the point P(2, 1, 1) to the point Q(−1, −1, 1).
1
vote
1answer
17 views

Continuous $n$-th order derivative

Does a continuous $n$-th order derivative imply that all previous order derivatives are also continuous? I intuitively believe this is the case, but I can't entirely convince myself.
-1
votes
0answers
18 views

Integral parameter convergency

For which $p$ is the following integral convergent? $$ \int_0^1 \frac{(\arctan3x)^p}{\sin x-x-\frac{x^3}{6}}\mathrm dx$$
1
vote
1answer
32 views

Sum with parameter convergence

$$ \sum_{n=2}^{\infty}\frac{1}{(\sqrt{n}+\sqrt{n+1})^p}\ln\left(\frac{n-1}{n+1}\right). $$ For which values of the parameter p is it convergent?
0
votes
2answers
46 views

Why is it that differentiating x would give me 1 when 1 appears to be multiplied by 0

If I am to differentiate $x$, I would do $$\frac{d}{dx} = nx^{n-1}$$ and $n$ is $1$ so I would get $$\frac{d}{dx} = 1(0)$$ and so $0$. But this isn't so and I would end up with $1$. Why is it ...
0
votes
0answers
56 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1 times} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 = ...
1
vote
4answers
63 views

Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$

I have two similar looking questions. $(1)$Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$ $(2)$If $\Delta ABC$ is an acute angled,then prove that ...
1
vote
1answer
42 views

Jensen inequality conceptual doubt

Prove that in a triangle $ABC$,$\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\geq\frac{3}{4}$. I tried to solve it by Jensen's inequality.I let $f(x)=\sin^2\frac{x}{2}$ ...
0
votes
0answers
23 views

Implementing FizzBuzz game

I need to build an electrical-circuit for the FizzBuzz game. There's a signal, called next which increment the current number by one. The rules are simple - You ...
0
votes
1answer
36 views

Calculate the flux through a surface S from a field described by vectors

I have encountered yet another example which is not that typical. I need to calculate: $$\iint\limits_{S} \vec{F} \vec{ds} =\text{ ?}$$ Where the $F$ and $S$ are as follows ($S$ is oriented ...
3
votes
1answer
33 views

Why does max. increase have to be along the x,y,z axis in gradient?

$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$ These components are the rate of increase along the $x$, $y$ and $z$ directions ...
-1
votes
1answer
32 views

Solving a trigonometric equation

Let $n=3m$ and $k=3t$ be positive integers. Does the following equation have any solutions for $0 \leq j \leq n-1$ $$\cos \left ( \frac{2\pi j (k+1)}{n}\right )+\cos\left (\frac{2\pi j (k-1)}{n} ...
0
votes
1answer
28 views

Can single variable function be represented by field? [on hold]

Is field concept in mathematics directly related to multi variable functions? Can single variable function be represented by field?
0
votes
1answer
24 views

Which are the good books,resources,extensive question banks to learn real analysis,calculus

Which are the good books,resources,extensive question banks to learn real analysis,calculus(indefinite,definite,area under curves),differential equations for IIT plus plus level.Foreign authors are ...
0
votes
0answers
7 views

Does semi positive definite function imply quasi convexity? [on hold]

Alternatively does semi negative definite function imply quasi concavity?
2
votes
0answers
21 views

Coming up with a function or a single graph, given its characteristics (pre-calculus)

Give an example of a function or a single graph which has the following characteristics: Hole at $(3,-1)$ Domain is all real numbers except $3$ Local minimum at $(-1,-2)$ Local ...
1
vote
2answers
35 views

Finite series identity [duplicate]

How would I prove this statement? I know that it's a finite series. I don't know how to approach this at all. $$\sum_{i=1}^N i^3 = \left(\sum_{i=1}^N i \right)^2$$
0
votes
0answers
24 views

Trig substitution triangle restrictions

I apologize if this is a dumb question, or if I am a little slow, but I've been thinking about this for all of yesterday and today and I just can't figure it out, despite googling it. I am confused ...
3
votes
3answers
51 views

Sum of infinite geometric series

How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus. $$\sum_{i=0}^\infty \frac{i}{4^i}$$
0
votes
1answer
24 views

Area under a parabolic trajectory

I have this problem: "prove that the area under the trajectory described by a parabolic shot that has: $f(x)=\tan(\theta)x - (\frac{g}{2v^2\cos^2(\theta)})x^2$ and $x=v\cos(\theta)t$ is defined ...
1
vote
1answer
31 views

Identifying the formula for a quartic graphic

I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't ...
1
vote
2answers
52 views

Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
0
votes
0answers
15 views

Fiding the most general antiderivative of a function bounded by two x's.

At first I thought this problem would simply become a definite integral since it appears two be bounded by two x's. However, I feel as though I may be wrong and I'm curious as to how I would approach ...
1
vote
1answer
43 views

Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
-3
votes
1answer
11 views

Calculus :Work of an inverted right circular cone [on hold]

A tank in the shape of an inverted right circular cone has height 6 meters and radius 4 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot ...
3
votes
2answers
43 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...