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

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0
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1answer
22 views

Derivatives of Implicit Functions (Abstract Case)

I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on ...
5
votes
0answers
20 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
2
votes
0answers
23 views

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where…

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where $$\mathbb{I_r}=[x_0-r,x_0+r]$$ and $$\mathbb{P}=\{(x,y): |y-y_0|\leq a, |x-x_0|\leq b\}\subset \mathbb G $$ where $\mathbb G-$ ...
1
vote
0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
1
vote
2answers
29 views

Using Lagrange's Method in Finding Extreme Values (New to This Method)

Did I do this hw question correctly (at least in theory, I do not expect anyone to check my algebra work)? In particular, did I solve for lambda and plug lambda back into my equations for x,y, and z ...
2
votes
2answers
74 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a positive integer and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series . My question here is :Is $\lim S_{n,m} <\infty $ for $ n \to \infty$ and ...
2
votes
1answer
46 views

Green's function for Helmholtz equation for the plane with a hole

That is find $G$ which satisfies \begin{align} (\nabla^2+k^2)G(\mathbf{x}, \mathbf{y},\omega) = \delta(\mathbf{x}- \mathbf{y}) \end{align} subject to $$\frac{\partial G}{\partial y_n} = 0 ...
1
vote
1answer
40 views

Decide if the following functions are not continuous on $(-\infty, \infty)$

Suppose $g(x)$ is continuous on $(-\infty, \infty)$. Determine if the following functions are or are not cont. on $(-\infty, \infty)$ and explain. a) $k(x) = \frac{x^2}{4 - (g(x))^2}$ b) $j(x) = ...
3
votes
2answers
122 views

Confused about calculating the area under the curve

What is the area under the curve of the following function? $f(x) = x² + 2x -3$ $x=-4$ $x=2$ Please, I'd like to see an image. Here is the graphic: https://www.desmos.com/calculator/oe0ja17spg
3
votes
3answers
67 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
-2
votes
2answers
70 views

Evaluate the integral: $\int\ x\ 2^{x^2}\ dx$ [on hold]

$\int\ x\ 2^{x^2}\ dx$ using this formula: $\int a^x\ dx =\frac{a^x}{ln(a)}$ I have calculated the answer is $\frac{x^2\ 2^{x^2}}{x^2\ ln\ 2}+C$
0
votes
2answers
23 views

Find the rate of change of the frequency when D, L, σ and T are varied singly.

I'm reading Calculus made easy to learn the notation (I know derivatives with the limit/prime style) and also some integral calculus which I haven't seen at school yet. You can check it here: ...
1
vote
4answers
69 views

Evaluate the Integral: $\int \frac{\log_{10}\ x}{x}\ dx$

$\int \frac{\log_{10}\ x}{x}\ dx$ $du=x\ln10\ dx$ $\log_{10}x\ \ln10+ C$ Is this answer correct? If not what step should I take to convert the log into a term I can manipulate?
2
votes
2answers
58 views

Spivak's 'Calculus', 5-21(b): Is there an easier/shorter way?

Some personal background: I'll be going into my second year as a maths undergraduate in September of this year, and I'm currently working my way through Spivak's Calculus. While $\epsilon$-$\delta$ ...
2
votes
4answers
185 views

Can you help me make sense of this notation?

I am reading through my calculus textbook, and came across an algebra technique that I can't decipher. The author sets up $$e^x \sin(x) = (1 + x/1! + x^2/2! + x^3/3! + ...)(x -x^3/3! + ...)$$ Which ...
1
vote
6answers
85 views

Evaluate the Integral: $\int(x^5+5^x)\ dx$

$\int(x^5+5^x)\ dx$ I made the the terms within the parenthesis u $u=x^5+5^x$ $du=5x^4+5^xln\ 5$ $du=5x^4+5^x\ ln\ 5 dx$ $\frac{u}{5x^4+5^xln\ 5}\ du$ I am stuck at this point. Is there a ...
0
votes
2answers
37 views

Irrational Conjugate

I have irrational number: $\sqrt{3}-\sqrt{2}$ It's has 3 conjugate numbers: $\sqrt{3}+\sqrt{2}$ $-\sqrt{3}-\sqrt{2}$ $-\sqrt{3}+\sqrt{2}$ First variant - it's a standrart form for me. It's ...
0
votes
2answers
29 views

Signum function question

The sum of all values of $a$ for which $$f(x)=\operatorname{sgn}\left((x-a)(x-1)(x+1)\right),$$ $x\in \mathbb{R}$, has exactly two points of discontinuity, is: $(A)-2$ $(B)-1$ $(C)0$ $(D)3$ I ...
0
votes
0answers
72 views

$\int_0^b \ln(\sin(ax))dx$ [duplicate]

Problem: Evaluate $$\int_0^b \ln(\sin(ax))dx$$ Unfortunately I have no idea as to how to proceed with finding a closed form for the above Integral. The $a$ in the integrand made me think of ...
1
vote
1answer
56 views

Find the area using double integral and polar coordinates.

I need to find the area using double integral and polar coordinates. $$y=3-x$$ $$y^2=4x$$ This is what i figured already: $${r\cos{\theta}+r\sin{\theta}} = 3$$ $$r=0, r=3, \theta=0, \theta=\pi/2$$ ...
2
votes
2answers
63 views

Solve $\cos3x - 18\cos x +10 =0$

I want to solve $$\cos 3x - 18\cos x +10 =0 $$ I tried: 1) Replacing $\cos 3x$ to $\cos^3x - 3\cos x$ 2) Replacing $\cos x$ to $t$ we get: $$t^3 - 21t +10 = 0$$ So we get cubic equation. But I ...
-6
votes
1answer
43 views

differentiation [on hold]

Differentiate the following function $\frac{dy}{dx}$ \begin{align} y &= \tan^{\sin x}(x) \tag{1} \\ y &=e^{\tan x}+(\log x)^{\tan x} \tag{2} \\ x^{y} &= y^{x} \tag{3} \\ x^{5} \, y^{5} ...
5
votes
5answers
82 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
-1
votes
1answer
13 views

Find the Laurent's series [on hold]

Find the Laurent's series of $$f (z) = \frac1{z(1-z)^2}$$ $0<|z|<1$ $|z-1|<1$ Please help me.
0
votes
0answers
21 views

Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?

One of my friends asked me this question. "Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?" It is curious how even the most trivial questions ...
-1
votes
0answers
19 views

Fourier Transform of $|x|^\frac{7}{6} K_{-\frac{1}{6}}(|x|)$ [on hold]

What is the Fourier Transform of $|x|^{\frac{7}{6}} K_{-\frac{1}{6}}(|x|)$ with $K_{-\frac{1}{6}}$ the modified bessel function of the second kind?
0
votes
1answer
43 views

Expressing $\cos(\varphi x)$ as a function of $x\sin\varphi,x\cos\varphi$

Let $\varphi,x\in\mathbb{R}$. I wonder if one can explicitly express $\cos(\varphi x)$ as a function of the variables $x\sin\varphi$ and $x\cos\varphi$. Suppose we denote ...
1
vote
1answer
30 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
4
votes
1answer
87 views

How to prove there exist distinct $a_{i}$ such $f'(a_{1})f'(a_{2})f'(a_{3})\cdots f'(a_{n})=1$

Let $f$ be a continuous map from $[0,1]$ to $R$ that is differentiable on $(0,1)$,with $f(0)=0,f(1)=1$, show that for each postive integer $n$ there exist distinct numbers $a_{1},a_{2},\cdots,a_{n}\in ...
1
vote
3answers
84 views

Help me proving $x^{\frac{1}{x}} \geq \frac{1}{3}$

How can I show that $x^{\frac{1}{x}}\ge1/3$ is satisfied for all $x\ge b$ where $b>1/2$. One way of doing this is showing the derivative of $x^{\frac{1}{x}}$ is positive for $x>b$; however I ...
2
votes
1answer
38 views

Double integral - Convert to polar coordinates and find the integration limits by a given domain [on hold]

I need help converting to polar coordinates and find the limits of the integrals by this given domain: $$\iint_{D}{} f(x,y)\, dx\, dy$$ $$D= \left\{ (x,y) \mid \dfrac {x^2}{a} \leq y\leq a, -a\leq 0 ...
1
vote
2answers
36 views

partial fraction derivative question

So I have this partial fraction derivative question. I know how to solve it, but for some reason I keep swapping two numbers. Here is the problem: $$\int\frac{3-4x}{x^2+x}= ...
3
votes
1answer
71 views

$\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n^2+0}}+\frac{1}{\sqrt{n^2+n}}+\frac{1}{\sqrt{n^2+2n}}+\cdots+\frac{1}{\sqrt{n^2+(n-1)n}}$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\frac{1}{\sqrt{n^2+0}}+\frac{1}{\sqrt{n^2+n}}+\frac{1}{\sqrt{n^2+2n}}+\frac{1}{\sqrt{n^2+3n}}+\cdots+\frac{1}{\sqrt{n^2+(n-1)n}}$ $\bf{My\; ...
0
votes
2answers
65 views

Can you tell me if this is a calculus problem?

I have been asked to solve and I do not know where to start. $$|(3\pi+0.58)-11|!$$ I have tried several calculators but seem to come up with different answers.
0
votes
2answers
22 views

Find MacLaurin polynomial of integral

I have not the slightest idea how to begin with the following problem. My first thought is to integrate it before trying to find the MacLaurin polynomial, but I don't know if that is possible. Here is ...
0
votes
2answers
22 views

Help understanding a question

I know this probably isn't the best question to post as far as further use with others, but I literally have no where else to turn to for study assistance. My problem is as follows: Find $T_5(x)$: ...
3
votes
1answer
33 views

An approach to approximating the harmonic series.

I would like to get help on the last step to approximating the harmonic series, here is my work: Consider the equation: $$f(x+1)-f(x)=g(x)$$ Through iteration one can come up with the solution: ...
0
votes
2answers
35 views

I want to know that $r^k \le C(1+r)^k$ holds.

When doe the inequality $$r^k \le C(1+r)^k$$ hold for $r>0$? I want to know the possible value of the real number $k$ so that the inequality holds. Here $C$ is independent of $r$.
1
vote
1answer
82 views

Equivalence of holomorphic functions

Given that $$\left(1-\frac{z}{\zeta_j}\right)^{-z}=\sum\limits_{k=1}^\chi\frac{z^k}{k\zeta_j^k},$$ where $\chi$ is the largest nonnegative integer $k$ for which ...
2
votes
0answers
56 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
3
votes
1answer
58 views

Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$, where $f(x) = 1+x^2$

Given $f(x) = 1+x^2, \alpha(x) = x^3, x \in [-1,1], P = \{-1,\frac{-1}{2},0,\frac{1}{2},1\}$. Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$. Ok, for this problem, we have $\alpha (x) = x^3$. And, I am ...
0
votes
1answer
93 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
3
votes
1answer
123 views

Limit of quotient of two infinite series $\left(\frac{0}{0}\right)$

Let $\sum_{n=1}^\infty a_{n,k}<\infty$. I want to calculate $$L=\lim_{k\to \infty}{\sum_{n=1}^\infty a_{n,k}\over\sum_{n=1}^\infty a_{n,k+1}}$$ if I know that $\lim_{k\to \infty} ...
1
vote
0answers
19 views

differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)} w.r.t$ t?

Here $\phi(u,t)=E\{e^{iut}\} $ is a characteristic function, $x_t$ is Gaussian. Differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)}$ w.r.t t the result is $\phi_u=\phi[i\hat{x}_t-P_tu]$
2
votes
3answers
52 views

kinetic energy approximation

If a pitcher throws a pitch at a velocity $v_0$, then the kinetic energy is $E_0=\frac 12mv_0^2$. If the pitcher releases the pitch from x feet higher, then we will suppose that he can readjust his ...
10
votes
2answers
140 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
1
vote
3answers
64 views

Finding convergence of a series using integral test

The series:$$\sum_{n=1}^{\infty}\left(\frac{\ln(n)}{n}\right)^{2}$$ Question: a) show that it converges b) find the upper bound for the error in approximation $s\approx s_{n}$ Trial: The section ...
-1
votes
1answer
21 views

Compound Interest check [on hold]

A guy has owed me $360.41 from dec. 1, 2007 until today. I compound interest at 18% per anum rounded to the nearest month. How much does he owe me now?
4
votes
6answers
90 views

Differentiate the Function: $y=\sqrt{x^x}$

$y=\sqrt{x^x}$ How do I convert this into a form that is workable and what indicates that I should do so? Anyway, I tried this method of logging both sides of the equation but I don't know if I am ...
2
votes
4answers
36 views

Boundary Value Problem $y''+uy=0$

Consider the boundary value problem $$y''+uy=0 \qquad y(0)=y(\pi/2)=0$$ (a) For what values of $u$ does this problem have the trivial solution $y \equiv 0$? (b) For what values of $u$ does ...