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

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-2
votes
0answers
11 views

Find the volume of the following room [1]

I was working on a project which required me to calculate the volume of the room. The picture of the room is given below: I tried splitting the shape across the diagnols but each time end up getting ...
0
votes
0answers
8 views

Second order total derivative

Suppose we have a function $g: \mathbb R^2 \to \mathbb R$ and $$\nabla g(u,v)=(5v^4-2u\exp(v-u^2), \exp(v-u^2)+20uv^3), (u,v)\in\mathbb R^2$$ Can the function $g$ be twice differentiable, i.e. does ...
-4
votes
0answers
24 views

A question on polunomial

Let $m\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(m) = 0$ and $x\in \mathbb{C}$ $f(m) $ is continuous decreasing function of $m$. $a_i\ge0$ for all $i$. $k(m)$ is positive zero of first ...
0
votes
0answers
10 views

Differential? equation in car dashboard problem

I stumbled upon this question while I was driving my car. On my dashboard I have fuel gauge and engine temperature gauge next to each other, look at the pic: http://i.stack.imgur.com/aDgKj.png Fuel ...
5
votes
1answer
62 views

Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$ \int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}. $$ ...
0
votes
0answers
17 views

3rd degree polynomial fraction decomposition

Was solving some differential equations and came upon this integral: $\int\frac 1{x(x+1)^2} dx$ Looked it up on wolframalpha and it can be decomposed to: $\frac 1x-\frac{1}{(x+1)^2}-\frac{1}{x+1}$ ...
1
vote
1answer
18 views

Can every separable differential equation be rewritten to potentially be exact (or NOT exact)?

Let's say an ordinary linear DE is separable. Then $$\frac{dy}{dx} = P(y)Q(x) \Leftrightarrow \frac{1}{P(y)}dy = Q(x)dx \Leftrightarrow Q(x)dx + R(y)dy = 0$$ is in exact form where ...
-1
votes
3answers
83 views

Solving $x \ln x=25$ [on hold]

Can someone help me solve the following equation? $$x \ln x = 25$$
0
votes
0answers
22 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
votes
1answer
11 views

Orthogonal trajectories - why is it necessary to isolate the parameter

For orthogonal trajectory, I realized that I need to express the parameter of the given family of curves in terms of x and y, in order to get the right answer. e.g. in $y = kx$, $k$ is the parameter ...
1
vote
0answers
44 views

I've tried to work out a calculation in my physics book, but the book is telling me one answer and my calculator is telling me another one. [on hold]

I'm trying to work out a physics equation. The question is $\sin(30)/1.3$. My book tells me the answer is $22.6$ degrees, but when I type that equation into my calculator it gives me $0.384615$. I've ...
1
vote
2answers
30 views

adding a “negative area” and a positive area when the two are infinite

If we take the integral of, say, $\sin x$ from $x=0$ to $x= 2\pi$, we get $0$, which is adding the area between the curve and the $x$-axis between $0$ to $\pi$, plus the so-called "negative area" from ...
0
votes
0answers
30 views

Receiving different answers

Ok, so im following a tutorial on how to calculate a limit numerically and when the tutor plug'd in the number $(-1.1)$ into the equation $\frac{(t^6 -1)} {(t^3 + 1)}$ HE gets −2.331 as the ...
0
votes
0answers
39 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
2
votes
0answers
29 views

what exactly is arc length element $ds$ or area element $dA$

I am reading a book on complex analysis and it has something like: The spherical arc length element on the Riemann sphere ($S^2$) works out to be $ds=\frac{|dz|}{1+|z|^2},$ and the spherical area ...
2
votes
0answers
27 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
1
vote
2answers
28 views

Relationship between derivatives of two functions and between anti-derivatives of two functions

Given that $f,g: [a,b] \to \mathbb{R}$ are differentiable with $f(x)\geq g(x),a\leq x \leq b$. What kind of relationship can we observe between derivatives $f'(x)$ and $g'(x)$? Furthermore, what can ...
12
votes
3answers
166 views

How can I find the limit without using a closed form expression [duplicate]

I am trying to evaluate this limit without using the closed form expression for the sum of natural numbers raised to $k$th power. $$\lim_{n \to \infty} \dfrac{ 1^n +2^n+\cdots +n^n}{n^n}$$ So far I ...
0
votes
2answers
38 views

Unsure why ODE non-exact equation solution is wrong?

The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. ...
0
votes
2answers
31 views

Need help with a second-degree Taylor polynomial

It says to let T2(x) be the second degree polynomial for the functionf(x) = 6 + xe4x where a=0. I need to find T2(1). I thought it was just a taylor expansion and look at the second term, which I ...
2
votes
1answer
49 views

f: R → R and $|f'(x)| ≤ |f(x)|$ [duplicate]

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
0
votes
1answer
17 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
1
vote
0answers
68 views

Evaluating a Difficult limit!

I have to evaluate a very complicated limit, I've done this task already but I wanna make sure I did it right. The function I have in my hands is $$ F(\omega)= \tanh \Big[a\cdot ...
0
votes
5answers
72 views

How to find a simplified expression for $\binom{1/2}{n}$? [on hold]

How to find a simplified expression for this specific binomial coefficient? $$\binom{\frac{1}{2}}{n}$$
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
1
vote
1answer
26 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
1
vote
1answer
15 views

Proving that if the sequence $\{s_n-L\}$ converges to zero, then a sequence $\{s_n\}$ converges to a limit $L$

I am having trouble proving this statement without using the limit rules. I know I start by assuming that the sequence $\{s_n-L\}$ converges to zero, therefore, for every number $ ϵ > 0 $, there ...
-2
votes
1answer
40 views

If given the limit that is a derivative, how do I find it's function and the point? [duplicate]

How would I solve for something like this?? $$\lim_{x\to 5} \frac{2^x - 32}{x-5}$$ using the definition of derivatives.
2
votes
1answer
49 views

To show the $\epsilon-\delta$ definition for limits holds.

Question: Check if the following limit exists, if so show that the $\epsilon$ $\delta$ definition for limits holds. $$\lim_{(x,y) \to (1,2)} \frac{(x-1)^2(y-2)^2}{x^2+y^2-2xy-4y+5}$$ My answer: So ...
1
vote
2answers
34 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
2
votes
5answers
54 views

Prove that $f$ has a minimum

Let $f$ be a positive and continuous function in $[0,\infty)$, such that $\lim\limits_{x\to \infty} f(x)=2$. Prove that if $f(0)<2$, $f$ has a minimum in $[0,\infty)$. I am stuck in the ...
0
votes
1answer
29 views

How to do this rather basic Surface area question

I am having a bit of difficulty evaluating the surface area of the region that consists of the part of the sphere $$x^2+y^2+z^2=3c^2$$, within the paraboloid $$2cz=x^2+y^2$$, where $c \gt 0$ I know ...
-8
votes
0answers
45 views

Real Analysis-Find the limit of this series [on hold]

Problem # 3 enter image description here Find the limit of this series.
3
votes
0answers
55 views

Prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$

For a beginning calculus student, prove $\lim_{n \to \infty} \frac{4n^3}{2n^2+1} \sin(\frac{\pi}{n}) = 2\pi$ I'm guessing this means something like Allowed: Pre-university maths, precalculus, ...
2
votes
0answers
23 views

The sum of two subspaces

Let $V_{1}$ and $V_{2}$ be two subspaces of V. Define the sum of $V_{1}$ and $V_{2}$ to be the subset of V $V_{1}+V_{2}=${$\overrightarrow v_{1} + \overrightarrow v_{2}:\overrightarrow v_{1} \in ...
0
votes
1answer
25 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
1
vote
0answers
16 views

Integral of least squares and general rules of integration to solve the integral.

My calculus is very rusty and I am interested to know if the following is solvable: $$ \int_0^{\pi}( \log( \frac {(x_0 + e^{-i\omega})(x_0 + e^{i\omega})(x_1 + ...
0
votes
2answers
55 views

How would you calculate $(200\int_0^\infty e^{-0.8t}-e^{-1.8t}\,dt)/(250\int_0^\infty e^{-0.8t} \,dt)$?

$$\frac{200\int_0^\infty e^{-0.8t}-e^{-1.8t} \, dt}{250\int_0^\infty e^{-0.8t} \, dt}$$ I am confused as to how you would integrate the e's from zero to infinity. What steps would you take? By the ...
5
votes
2answers
36 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
3
votes
2answers
27 views

Show that $f(x):=\frac{2x^3+x^2+x\sin(x)}{(\exp(x)-1)^2}$ is continuously extendable to $x_0=0$.

What I know If $\lim\limits_{x \to x_0}f(x) := r$ exists, we can create a new function $\tilde f(x) = \begin{cases} f(x) &\text{if }x\in\mathbb{D}\setminus x_0 \\ r & \text{if }x = x_0 ...
0
votes
1answer
8 views

Scale series of number up by uniform amount

I have a series of numbers associated with a grid that determine the hue of each cell. Some of these cells are too dark and I'd like to scale them up slightly yet not to exceed the max value of $1$. ...
0
votes
0answers
21 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
1
vote
0answers
31 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
0
votes
4answers
76 views

How to find $\lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2}$

By plotting $\dfrac{1-\cos x \sqrt{\cos 2x}}{x^2}$, we find that in sufficiently small domain near $x = 0$, $f(x)\to 0$ as $x\to 0$. So the limit seems to be $0$. Now I tried to evaluate it using ...
1
vote
1answer
62 views

What type of discontinuity is found in this graph?

$$ f(x) = \begin{cases} \dfrac{1}{x} && \text{when $x > 0$}\\ 4 && \text{when $x < 0$} \end{cases} $$ What type of discontinuity is present when $f(0)$ ? ...
1
vote
0answers
11 views

Induced Riemmanian metric and Differential of embedding

Suppose I have a manifold $M$ which is defined as the image of a 1-1 smooth map $G:\mathbb{R}^d\rightarrow H$ into a Hilbert space $H$. I want to understand the Riemmanian metric on $M$ concretely, ...
2
votes
2answers
57 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
0
votes
0answers
22 views

Unit normal vector at inflection point for any curve: Defined or Undefined?

Consider an arbitrary parametric planar (for simplicity) curve: $ \vec{r}(t) = f(t) \,\hat{i} \, + \, g(t) \, \hat{j}$ Differentiable twice over its domain. $ \vec{r'}(t) = f'(t) \,\hat{i} \, + \, ...
2
votes
2answers
70 views

Taylor expanding $\frac{e^x}{x}$?

How can you taylor expand $$\frac{e^x}{x}$$ Can it be expanded at $x = 0$? Can it be expanded as $x \to 0$?
1
vote
3answers
63 views

Is $\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} = \infty$?

It was asked in our test, and below is what I did: $$\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5} $$ $$=\lim_{x\to -3}\frac{x^2+9}{\sqrt{x^2+16}-5}\times\frac{\sqrt{x^2+16}+5}{\sqrt{x^2+16}+5} $$ ...