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

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2
votes
1answer
23 views

How do I find the relative extrema of a function in spherical coordinates?

I want to find the relative extrema for the following function. $f(\theta,\phi)=AR\cos\theta\sin\phi + BR\sin\theta\cos\phi + CR\cos\phi $ $A,B,C,R$ are constants In a function $g(x,y)$ using ...
0
votes
1answer
12 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
0
votes
1answer
30 views

Why does the $\tan$ reduction formula have a restriction?

My book says the reduction formula is only valid for an integer $n > 1$. Why? This derivation doesn't require $n$ to be an integer or greater than $1$.
2
votes
1answer
25 views

Could I get a critique of this epsilon-delta limit proof?

$\lim \limits_{x \to 3}$ ${(x^2-2)}$ = 7 So I want to find some $\delta$ > 0 such that for every $\epsilon$ > 0: $\lvert x^2-9\rvert$ < $\epsilon$ $\iff$ 0 < $\lvert x-3\rvert$ < $\delta$ ...
0
votes
1answer
26 views

Calculate the area of the ellipsoid that rotates around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...
0
votes
2answers
37 views

What is the difference quotient for the function? [on hold]

The difference quotient of a function $f(x)$ is defined as $$\frac{f(x+h)-f(x)}{h}.$$ Determine the difference quotient for the function $f(x)= \frac{4}{x}$, in its most simplified form.
1
vote
4answers
54 views

Does $(m+1) + m2 + (m - 1)2^{2} \ldots + 2^{m}$ equal something simpler?

Does $(m + 1) + m2 + (m - 1)2^{2} \ldots + 2^{m}$ equal something simpler, where $m\in \mathbb{N}$? Excuse me if it is too simple, I am bit tired. Thanks.
8
votes
1answer
264 views

Integral$\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \frac{1}{6}\right)$

UPDATED Hi I am trying to prove the following$$ I:=\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \big(\frac{1}{6}\big)\right). ...
1
vote
2answers
40 views

Finding a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$

Let $P(x)=\frac{x^3}{6}+\frac{x^2}{2}+x+1$. I have to find a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$. I tried to use Polynomial long division and solve a system (we need ...
1
vote
6answers
98 views

Proving that $\int \frac{1}{x} dx = \ln(|x|) + c_1$

In all textbooks and online notes, there is always a table of antiderivatives and it always says $\int \frac {1}{x} dx = \ln(|x|)+c_1$ but there is nowhere a proof. I found some proofs online but ...
1
vote
1answer
30 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
0
votes
1answer
18 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
4
votes
3answers
270 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
1
vote
1answer
49 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
1
vote
1answer
62 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? Im thinking of whether of how to remove the modulus sign on the left hand side of the equation ...
0
votes
0answers
16 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
14
votes
2answers
864 views

Definite Integral $\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$

I want to prove that $$\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$$
0
votes
1answer
33 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
18
votes
8answers
937 views

A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$

What is the proof of the following: $$\int_{0}^{1} \left(\frac{\ln t}{1-t}\right)^2 \,\mathrm{d}t=\frac{\pi^2}{3} \>?$$
3
votes
6answers
401 views

What is the limit of the following sum

$$\lim_{n\to\infty}\sum_{k=1}^n \ln\Big(1+\frac{k}{n^2}\Big)$$ According to me, the answer is $0$. I'm curious as to what answers might others come up with, as well as the method of reasoning.
7
votes
4answers
69 views

Integral of $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $dz=1/\sqrt{1+x^2}dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb v}dz=x\int zdz-\int ...
0
votes
3answers
78 views

Convergence or divergence of the integral $\int_0^1 dx/\sin x $

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
0
votes
1answer
28 views

How to prove that an integral converges

Let $(a_n)$, $(M_n)$ be sequences of positive real numbers such that ${a_n} \downarrow 0$, ${M_n} \uparrow \infty$ as $n\to\infty$. Let $\alpha>0$ and $\beta>1$. How to prove the following ...
0
votes
1answer
31 views

How would I use derivatives for suggesting an option to my user?

I was learning derivatives. I understood the theoretical concept behind it. When I was searching for the real-life example in machine learning I came across one of the answers in this question How do ...
4
votes
1answer
33 views

Find $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
2
votes
2answers
56 views

Prove that $\frac{{-\cos(x-y)-\cos(x+y)}}{-\cos(x-y)+\cos(x+y)} = \cot x \cot y$

I solved this from my implicit differentiation, and i end up with this answer, they say it's right but not simplified, I tried to simply it but I get $\cot(x)\cot(y)-\tan(x)\tan(y)$ ...
1
vote
1answer
103 views

Minimum calculus of variation

Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum. it is about the following functional$$ \int_0^b ...
4
votes
4answers
58 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
1
vote
2answers
34 views

Find equation of tangent line to a curve $g(x)$ at $x=4$

So I am trying to find the equation of a tangent line to the curve: $$y = g(x)\text{ at }\,x = 4$$ given $g(4) = -6,\;$ and $\;g'(4) = 2$.
1
vote
4answers
156 views

Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
91
votes
5answers
6k views

Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ You can use, for ...
2
votes
1answer
32 views

Confusing about the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$

What is the domain of $f(x)=(x+|x|)\sqrt{x\sin^2(\pi x)}$? A nice plot of $f(x)$ shows that the domain is $\mathbb{R}$ but we see that $x$ should be non-negative at the first sight. Of course, I ...
0
votes
2answers
21 views

How to show that the points $(0, 0)$ and $(\sqrt{2 \pi},−\sqrt{2 \pi})$ on the curve $e^{x + y} = \cos(xy)$ have a common tangent?

Show that the points $(0, 0)$ and $(\sqrt{2 \pi},−\sqrt{2 \pi})$ on the curve $e^{x + y} = \cos(xy)$ have a common tangent. How do I solve this question? First, I differenciated the curve and I got ...
2
votes
0answers
37 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) \end{equation} ...
0
votes
0answers
34 views

Need help with excel spreadsheet! [on hold]

So I am currently doing an assignment in which I have bought a house using a home-loan. For this part of the question I need to calculate how long it will take to repay the loan. So I have constructed ...
2
votes
0answers
54 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
0
votes
5answers
78 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
0
votes
4answers
43 views

Is intergration and an integral the same thing?

And if not whats the difference? I think the integral is the area under the curve and integration is an anti derivative (what ever that means)
0
votes
3answers
38 views

Minimum Value of graph

I was doing a test and I got this question wrong and I don't know why... What is the minimum value of the function $y=\sqrt{49-x^2}$ on the interval $[−5,2]?$ This is the graph according to ...
1
vote
2answers
47 views

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

I got the following problem: Evaluate the following limits or show that it does not exist: $$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$ and ...
1
vote
1answer
21 views

How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
2
votes
1answer
43 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
0
votes
4answers
50 views

General form for these types of integrals

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables
0
votes
1answer
29 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
1
vote
1answer
27 views

The general solution of first order ODE

How will i get the general solution for this $$y' = {-y^2 \over x} + {2 y \over x}$$ I have tried and come to this far by separating and equating $$\int {1\over-y^2+2y} dy = \int{1\over x} dx$$ which ...
2
votes
2answers
29 views

Interpolation between derivatives [duplicate]

Let $f$ be twice continuously differentiable on $[0,2]$, and $|f(x)|\leq 1$, $|f''(x)|\leq 1$. Prove that $|f'(x)|\leq 2$. If I use Lagrange intermediate value theorem, then ...
-5
votes
0answers
35 views

limits using (ε,δ )-definition of limit

Hi could anyone help me with this proof Prove that lim f(x)=0 whenever lim ||f(x)||=0 For x ε R^(n) x->a x->a f(x) is a ...
1
vote
3answers
49 views

Solve $y' = x^4y+x^4y^4$

Solve the differential equation $$y' = x^4y+x^4y^4.$$ I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one. ...
0
votes
2answers
18 views

proves of parametric curves via parametric equations

Hi could anyone help me with this problem. An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the ...
0
votes
1answer
62 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...