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

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0
votes
1answer
26 views

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

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

Find the volume of the solid obtained by rotating the region bounded by the curve

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.
0
votes
1answer
22 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
22 views

Calc 3 Problem about work

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).
0
votes
1answer
12 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.
0
votes
0answers
16 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
23 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
43 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 ...
1
vote
4answers
49 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 acute angled,then prove that ...
1
vote
1answer
38 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
17 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
10 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
24 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
31 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
23 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
22 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
34 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
20 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
44 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
22 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
29 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
49 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
37 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
9 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
39 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 ...
1
vote
1answer
21 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
1
vote
2answers
18 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle ...
3
votes
2answers
64 views

Since $\lim\limits_{x\to0}\frac{\sin kx}{kx}=1$ for constants $k$, is it also true for general arguments?

To be more specific, is it true that $$\lim_{x\to0}\frac{\sin f(x)}{f(x)}=1~~?$$ I'm tempted to say yes at first glance, so long as $f(x)\to0$ as $x\to0$. The reason I ask is to verify this limit ...
1
vote
2answers
95 views

Prove the limit is $\sqrt{e}$.

How do you show $$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$ I know that $$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't ...
0
votes
2answers
23 views

interpreting $(1-t)f(a)+tf(b)$ from $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$

For a convex function $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$ holds. I understand how the graph looks like but why is the equation of the secant line $(1-t)f(a)+tf(b)$? Can anyone pleasae give me a ...
0
votes
1answer
24 views

Consider a function $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all real roots of $f(x)$ and $t=|s|$. Then…

the real number $s$ lies in the interval (A)$(-0.75,-0.5)$ (B)$(-0.5,0)$ (C)$(0,1)$ (D)$(-0.25,0)$ and the area of region bounded by $f(x),y=0,x=0$ lies in the interval (A)$(0.75,3)$ ...
1
vote
0answers
24 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
0
votes
0answers
23 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
8
votes
1answer
73 views

What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?

This was inspired by similar posts like this one. Define the function, $$F(p) = \lim_{n\to\infty}2^n\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}_{n \textrm{ square roots}}}$$ We know that, ...
0
votes
1answer
14 views

Calculating the length of a helix

I have a pipe and I want to put a wire through it in a helix form. I need to calculate how long the wire (wl) has to be. I know the internal diameter (id), and therefore the circumference (c). I know ...
-1
votes
1answer
27 views

simple percentage problem [on hold]

a man sold a watch of rs 2400 at a loss of 25%.at what rate should he sold the watch to earn a profit of 25%
0
votes
0answers
9 views

Solving for the poisson rate

Say I have an equation of the form $$ 0 = -a + \sum_{k=0}^\infty f(k, \lambda)R(k)\\ 0= -a + \exp(-\lambda)\sum_{k=0}^\infty \frac{\lambda^{k}}{k!}R(k) $$ where $f()$ is the Poisson mpf, $a$ is a ...
-2
votes
1answer
33 views

Area under the given curve [on hold]

The area under the curve $\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|}$ , $x$-axis and the ordinates at $x = -3$ and $x = 1$
0
votes
1answer
25 views

Finding the gradient of a function.

A function $f=f(x,y)$ has continuous partial derivatives , and assume that maximal directional derivative of $f$ at $(0,0)$ is equal to $100$ and is attained in the direction towards $(3,-4)$ , we ...
0
votes
1answer
92 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
1
vote
2answers
32 views

Sketching functions $f(x) = \frac{e^x}{x^2} \quad \text{ and } \quad g(x) = \frac{1}{x}$ - First Derivative test and domain restriction

when working on a "Sketching a function" problem, some textbooks have a step-by-step procedure. The first one is usually stating the Domain of a function. When working with functions like $$ f(x) = ...
-2
votes
0answers
58 views

Rolle's theorem question

Let $f(x)=\sin2x/e^{2x}$. Note that $f$ is continuous on $[0,\pi/2]$, and differentiable on $(0,\pi/2)$, with $f(0)=f(\pi/2)=0$. So by Rolle's theorem, there exists a $c\in(0,\pi/2)$ with $f'(c)=0$. ...
2
votes
0answers
58 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
4
votes
4answers
54 views

I'm stuck in this one of trig substitution for fuctions.

I got this: $$\int\frac{dx}{\sqrt{(4x^2-9)^3}}.$$ I know that the answer is: $$\frac{x}{9*\sqrt{4x^2-9}}+c.$$ And with the steps that I know about this type of substitution, I came up here, but.. ...
0
votes
0answers
60 views

Prove √2 exists by Archimedean Axiom [duplicate]

I am trying to prove the existence of the square root of 2. The proof: Let $$S=\{x \in \mathbb{R} ∣x \ge 0, x^2 < 2\}.$$ I understand the proof of LUB, $\alpha$ and so I am at the step where ...
4
votes
1answer
39 views

Integrating $\frac{x^3}{(81-x^2)^2}$

I've been trying to figure out this integral for an hour or so now, but keep failing. I can't figure out where I go wrong: $$I = \int \frac{x^3}{(81-x^2)^2} dx$$ Let $x = 9sin\theta \implies dx = 9 ...
-1
votes
2answers
50 views

What is the error in the following working?

$$\frac{\int_0^1 (1-x^{50})^{100}\mathrm{d}x}{\int_0^1(1-x^{50})^{101}\mathrm{d}x}$$ The question asks us to evaluate 5050 times the above fraction> To solve this i had made the following ...