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

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0
votes
3answers
44 views

Work done to fill up a spherical tank

A spherical tank of radius $12$ feet is $40$ feet above the ground. How much work is done in pumping water into the tank until it is full? I obtained $$ w= \int_{16}^{40}[12^2-(40-y)^2y] \, dy. ...
0
votes
1answer
46 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
1
vote
2answers
18 views

Volume of solid of revolution by shell method

consider the region bounded by $ \displaystyle y=4{{x}^{2}}$ and $ \displaystyle 2x+y=6$. What is the volume of solid of revolution about $\displaystyle x$-axis. What is thought about setting the ...
14
votes
3answers
168 views

Sum of $k$-th powers

Given: $$ P_k(n)=\sum_{i=1}^n i^k $$ and $P_k(0)=0$, $P_k(x)-P_k(x-1) = x^k$ show that: $$ P_{k+1}(x)=(k+1) \int^x_0P_k(t) \, dt + C_{k+1} \cdot x $$ For $C_{k+1}$ constant. I believe a proof by ...
0
votes
3answers
24 views

Derivative with Logarithm Problem

I'm not sure how to approach this problem and solve it. $$y=\log_5\ln(x^3+6)^4$$
5
votes
4answers
48 views

Proving limit through definition

Prove $$\lim_{x\to 2}\frac{x^2+4}{x+2}=2$$ through definition. My solution: Fix $\epsilon >0$ and find $\delta$ \begin{align} 0<|x-2|<\delta &\Rightarrow \left| \frac{x^2+4}{x+2}-2 ...
1
vote
2answers
64 views

Intuitive but hard question about an integral?

Let $f \colon [0,1]\rightarrow \mathbb{R}$ be a function with continuous derivative such that $f(1)=1$. Evaluate $$\lim_{y\rightarrow \infty}\int_0^1yx^yf(x)dx$$
5
votes
1answer
59 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
0
votes
1answer
9 views

Is there a closed form expression for the Taylor series of (1- a X - b Y - c XY )^ (-1)?

Is there a closed form expression for the Taylor series of f(X , Y ) = (1- a X - b Y - c XY )^ (-1) ? a, b and c are constants X and Y are thank you
1
vote
1answer
27 views

If a function has asymptote and the derivative does not, then its second derivative is not bounded

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be a function with second derivative everywhere in is domain. Prove that if $\lim_{x\rightarrow\infty}f(x)=b \in \mathbb{R}$ and ...
0
votes
1answer
12 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
2
votes
1answer
42 views

Logarithmic Differentiation - Always possible?

Logarithm functions in basic calculus classes are defined only for positive real numbers. But whenever we find an expression of the form $f(x)^{g(x)}$ we try to use logarithmic differentiation, we ...
1
vote
0answers
27 views

Second derivative of the position vector in a spherical coordinate system

In a spherical coordinate system my unit vectors are: $\vec{e_r}=\begin{pmatrix}\sin\theta\cdot \cos\phi \\ \sin\theta \cdot \sin\phi \\ \cos\theta \end{pmatrix}$; ...
1
vote
2answers
45 views

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $. This is supposed to be question about continuity, but I’m not sure exactly what they mean, ...
1
vote
1answer
18 views

Double solutions and plotting transcendental equations

I have the following transcendental equation: $y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$ and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x ...
23
votes
3answers
991 views

Is it OK to evaluate improper integrals this way?

Today in class we learned that when you have an improper integral like this one: $$\int_{-\infty}^\infty {f(x)} \: dx$$ you must split it before you do the limits (like so): $$\lim_{a \to \infty} ...
0
votes
0answers
23 views

To check the differentiability of following functions and my attempt

Hello i am posting my attempt .kindly please confirm that it is correct or incorrect .Kindly please suggest . I Need to check differentiability at origin of following
1
vote
2answers
65 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
0
votes
3answers
31 views

Optimization Problem multivariable calculus or single variable

Problem is that a right circular cylinder is inscribed in a sphere of radius a .What is height of cylinder when its volume is maximal ? As per suggested by answer i attempted Any hints please ? ...
0
votes
1answer
36 views

Is it possible to find the closed-form expression for $\int_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt$?

Is it possible find the closed-form expression or represent it in other function for this integral: \begin{align} I= \int \limits_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt ...
5
votes
2answers
85 views

Antiderivative of $\frac{1}{\ln(x)}$?

I was looking on wikipedia, and found that the following expression cannot be expressed in terms of elementary functions: $$\int\frac{1}{\ln(x)}\text{d}x$$ Although the function looks simple, why is ...
0
votes
0answers
22 views

Theorem proving skills in calculus, clearer idea to read in reverse order; linear-reading with writing down helps little

It is said that theorem proving skills are better trained via reproducing proofs from sketch rather than passive reading. Here we need more precise extension. e.g. in Multivariable Calculus, there ...
4
votes
1answer
45 views

Euler Complexification Help.

We have$$e^{ix}=\cos x+i\sin x$$ I want $$\begin{align} I&=\large\int \underbrace{\sin x}_{\large\Im e^{ix}}\quad e^{\pi x}\,\mathrm dx \\ &=\int e^{ix}e^{\pi x}dx\\ &=\int e^{(i+\pi) ...
9
votes
3answers
671 views

Pretty Simple Integral

I am trying to find the following indefinite integral: $$ \int \sqrt{x^2+x^4}dx $$ At first, it seems like an easy u-substitution after we factor out an $x^2$ from the square root, but when we do ...
1
vote
3answers
36 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
0
votes
1answer
66 views

How to prove this number is not rational?

Consider $$x\:=\:\sum _{k=0}^t\frac{1}{k!}+\frac{\alpha }{t\cdot t!}$$ with $t\in \mathbb{N},\:\alpha \in \left(0,1\right)$. How can I prove that $x \notin \mathbb{Q}$ ? So far I assume in negative ...
1
vote
0answers
64 views

Meaning of $dx$ [duplicate]

If I remember correctly, we use $ Δx$ for changes in $x$ and when $Δx \rightarrow 0$ then $ Δx$ takes the form of $dx$?
3
votes
3answers
97 views

Compute the fourier coefficients, and series for $\log(\sin(x))$

I posted a similar question with a bad response, so I am retrying with hopes of better knowledge. The fourier series is in the form: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + ...
0
votes
1answer
69 views

Can I demonstrate that this final sum is natural number? [on hold]

Let $m$ be some natural number. How can i say that $\sum _{k=0}^m\:\frac{1}{k!}$ is also natural or at least complete number?
3
votes
4answers
76 views

Calculate $\lim_{x \to 0} (e^x-1)/x$ without using L'Hôpital's rule

Any ideas on how to calculate the limit of $(e^x -1)/{x}$ as $x$ goes to zero without applying L'Hôpital's rule?
0
votes
1answer
46 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
-1
votes
2answers
40 views

Determine the value of this limit [L'hospital Rule] [on hold]

$$\lim_{x\rightarrow\infty}(e^x+1)^\frac{-2}x $$ $$\text{Domain}=(0,\infty)$$
0
votes
1answer
12 views

Seemingly basic integration by parts question

I am having a bit of a mind block with the most elementary calculus technique. Trying to integrate $\int_0^\infty x\Phi(x) \mathrm{d}x$ by parts by differentiating $x$ and integrating $\Phi$, how ...
0
votes
0answers
20 views

Factoring a Polynomial to Find Tangent Line

I have a polynomial equation $ x^n + a x^{n-1} + bx^{n-2} ... + z =0$ for which the coefficients depend on a parameter $ t $. The equation has one real root that I am interested in. For this real ...
-1
votes
2answers
71 views

How to evaluate $\int_0^1 \frac{dt}{(t-\frac{1}{2})^2+ \frac{3}{4}}$ [on hold]

How to evaluate $$\int_0^1 \frac{dt}{(t-\frac{1}{2})^2+ \frac{3}{4}}$$
1
vote
2answers
66 views

Solving $ \int_{0}^{2\pi} e^{-x} \lvert \sin x\rvert,dx $

Since it involves an absolute value, I assume I need to split it into two cases? For $ 0 \le x \le \pi $ $$ \int_{0}^{\pi} e^{-x} \sin x\,dx $$ and for $ \pi \le x \le 2\pi $ $$ \int_{\pi}^{2\pi} ...
0
votes
2answers
28 views

How many times does the derivative of this function meet the x=0 line?

How many times does the derivative of the following function meet the x=0 line? f(x)=$\left(x+1\right)\left(x-0.7\right)\left(x-e\right)\left(x-\pi \right)$. I know they meet exactly 3 times, and I ...
0
votes
2answers
63 views

Prove limit doesn't exist using $\delta$-$\varepsilon$

Prove in the $\delta$-$\varepsilon$ definition that the limit $$\lim_{x \to \infty} \frac{3}{2+\sin(x)}$$ does not exist. I know that $\sin x$ gets different values as $x$ approaches infinity but ...
0
votes
0answers
63 views

Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$

Given: $x,y\in R$ : $x^4+y^4+4=\frac{6}{xy}$ Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$ Please help me !
-2
votes
0answers
24 views

Equation for intersection of two solids. [on hold]

Parametric equation $C$ for intersection of $r=2sin \theta $ and $4=x^2+y^2+z^2$
0
votes
3answers
35 views

if $f'(x) = -8xe^{x^2}$ and $f(0)=3$ then $f(x) =$?

if $f'(x) = -8xe^{x^2}$ and $f(0)=3$ then $f(x) =$? Not really sure how to solve this, is it possible to get the steps and a correct solution in the end so I'd be able to check my answers?
15
votes
6answers
126 views

What is the importance of $\sinh(x)$?

I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a ...
4
votes
4answers
55 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
6
votes
1answer
84 views

Second order differential equation with a variable coefficient. Show |f(x)| is bounded.

Was given this question as extra credit on an ODE exam. Didn't have time during the exam to consider it, but I have since then, and I'm stumped. $$ f''(x) + f(x) = -f'(x)x^{2015}$$ $f(x)$ is twice ...
1
vote
3answers
71 views

Help evaluating $\int e^x \sqrt{1+e^{2x}}dx$ [duplicate]

$\int e^x \sqrt{1+e^{2x}}dx$ It's probably been answered somewhere, but I havent found it so far so I decided to post it as a question (if it has been answered point me in the right direction and I ...
0
votes
3answers
56 views

Evaluation of $\lim_{n\rightarrow \infty}\left\{\left(2+\sqrt{3}\right)^{2n}\right\}\;,$ Where $n\in \mathbb{N}.$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\left\{\left(2+\sqrt{3}\right)^{2n}\right\}\;,$ Where $n\in \mathbb{N}.$ $\bf{My\; Try::}$ Let $$\left(2+\sqrt{3}\right)^{2n} = I +f\;,$$ where ...
0
votes
0answers
15 views

Derivative Nullity for nonpolynomial spaces

One thing has been bothering me about derivatives, it's easy to explain nullity of a polynomial, since a term that is constant after n many derivatives will become zero at n+1 many derivatives. How ...
2
votes
2answers
65 views

Evaluation of $\int\frac{1}{1+(x+1)^{{1}/{n}}}dx$ for $n\in \mathbb{N},$

Evaluate $$\int\frac{1}{1+(x+1)^{{1}/{n}}}\,\mathrm dx$$ for $n\in \mathbb{N}$ $\bf{My\; Try::}$ Let $$(x+1)=t^n\;,$$ Then $$dx = nt^{n-1}dt$$ So $$\displaystyle I = ...
0
votes
2answers
36 views

Mclaurins with $e^{\sin(x)}$

To evaluate $e^{\sin(x)}$ I use the standard series $e^t$ and $\sin(t)$, combining them gives me: $e^t = 1+t+\dfrac{t^2}{2!}+\dfrac{t^3}{3!}+\dfrac{t^4}{4!}+O(t^5)$ $\sin(t) = ...
3
votes
1answer
48 views

Alternating infinite sum

I have the following infinite sum: $$ \sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}} $$ Because there is a $(-1)^n$ I deduce that it is a alternating series. Therefore I use the alternating series ...