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

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-3
votes
3answers
81 views

How to calculate the following limit?

Calculate the following limit where $n \in \mathbb{Z}$ and log is to the base $e$ $$\lim_{x\to\infty} \log \prod_{n=2}^{x} \Bigg(1+\frac{1}{n}\Bigg)^{1/n}$$
1
vote
2answers
46 views

What is the general definition of a discriminant? (Not just the definition for polynomials)

For example, in regards to the second derivative test for a function of two variables, $D=f_{xx}f_{yy}-(f_{xy})^2$ is refered to as the "second derivative test discriminant." I know that D is the ...
0
votes
1answer
12 views

Volumes of revolution (annular cross sections)

I have been asked to find the volume of the solid of revolution when the region bounded by the curve $y=x(4-x)$ and the $x$ axis is rotated about the $y$ axis. I am confused because the outer and ...
5
votes
3answers
161 views

Closed form for a binomial series

I am wondering if any knows how to compute a closed form for the following two series. $$\sum_{m=1}^{n}\frac{(-1)^m}{m^2}\binom{2n}{n+m}$$ $$\sum_{m=1}^{n}\frac{(-1)^m}{m^4}\binom{2n}{n+m}$$ ...
0
votes
0answers
17 views

Rotation of $y=x(1-x)$ about the $x$-axis

In my calculus book, there was a problem involving the rotation of an area $R$, where $R$ is defined as the area enclosed by the function $y=x(1-x)$ and the coordinate axes, about the $y$-axis. This ...
2
votes
3answers
104 views

$\int_1^\infty 1/\sqrt{1+e^x}dx$

Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or ...
0
votes
1answer
19 views

Total derivative product rule

Definition: Let $U\in \mathbb{R}^n$ be an open set. Let $a\in U$ and $f:U\to \mathbb{R}^m$. We say that $f$ is total differentiable at $a$ if there exists a matrix $T\in \mathbb{R}^{m\times n}$ and a ...
0
votes
1answer
50 views

Prove the function is nondecreasing

Lets take: $A_1,...,A_n$ family of finite, nonempty sets. Define: $$f(t)=\sum_{k=1}^n\left( \sum_{1\le i_1<...<i_k\le n}(-1)^{k-1}t^{|A_{i_1} \cup ... \cup A_{i_k}|} \right)$$ for $t \in [0,1]$. ...
0
votes
0answers
24 views

Use a calculator to compute the error $|e^x-T_2(x)|$ at $x=1.1$

I don't believe i have learned to solve for the error. Any help would be greatly appreciated. I have computed $T_2$ at $x=0.8$ $$T_2=e^.8+e^.8(x-.8)+e^.8/2(x-.8)^2 $$
1
vote
2answers
48 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
0
votes
1answer
75 views

evaluating $ \int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}}dx $

I came across this : I'm trying to evaluate it up to $ o(\epsilon) $ $$ F\left(\varepsilon\right)=\int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}} \, \mathrm{d}x $$ I've trying considering to look ...
1
vote
1answer
79 views

Show that $\lim_{x\to\infty} f(x) = 0$.

Let $f\in C^1$. Let's assume that $\int_0^\infty f(x)\ dx$ converges and $f'(x)$ is bounded. Prove that $\lim_{x\to\infty} f(x) = 0$. Let's assume by contradiction that $\lim_{x\to\infty} f(x) ...
2
votes
1answer
197 views

Integration by substitution - where is the mistake?

I want to integrate $$\int_{-1}^{1} (1-x^2)^{3/2} \, \mathrm{d}x$$ by substituting $x=\cos z$ and $dx = -\sin z \, dz$. $x=-1 \implies z=-\pi $ and $x=1 \implies z=0$. I receive: ...
3
votes
4answers
147 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
0
votes
2answers
45 views

Proving the product rule for n functions

I am trying to prove that the product rule works for $n$ many functions, where $n$ is an integer greater than $2$. I am able to prove it for two functions, where the rule states if $k(x)=f(x)g(x)$ , ...
1
vote
1answer
43 views

Finding the convergence radius of $\sum_{n=1}^\infty n! x^{n!}$ [duplicate]

I need help finding the covergence radius of $\sum_{n=1}^\infty n! x^{n!}$ . The factorials make me think that I need to use derivatives/ integrals but I don't quite know how. I'd love any help, ...
-2
votes
3answers
72 views

Integral of rational function with a squared term in the denominator

I know the integration when in the reciprocal there's only degree $1$, but what about degree $2$? Take an example, $$\int\frac{x \, \mathrm{d}x}{a+bx^2}$$
0
votes
2answers
42 views

Integrating reciprocal of the square of two numbers [on hold]

How do I integrate the reciprocal of the square of 2 numbers? Take an example.. $$\int\frac {dv}{a/b+v^2} $$
0
votes
2answers
33 views

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$.show that $\int_1^3 f = \phi(3) - \phi (1)$

Let $f(x) = [x], x \in [1,3]; \ \phi(x) = x , x \in [1,2]$ and $= 2x -2, x \in (2,3]$. Then to show that $f$ is integrable and evaluating the value of $\int_1^3 f$. I have done upto this. But ...
0
votes
2answers
44 views

Fundamental Theorem of Calculus with 1/lnx

I'm struggling with this problem, because I'm not sure how to integrate $1/\ln(x)$ Suppose that you have the following information about a function $F(x)$: $$F(0)=1, F(1)=2, F(2)=5$$ ...
4
votes
2answers
81 views

Showing $\sum_{n=1}^\infty \sin x \sin nx$ is uniformly bounded

I need to show that for every $x$: $$\sum_{n=1}^\infty \sin x \sin nx \lt M$$ So the first thing came into my mind is applying a well-known trigonometric identity: $$\sum_{n=1}^\infty \sin x \sin nx ...
1
vote
2answers
46 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
2
votes
0answers
37 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
1
vote
2answers
22 views

Can we say that a function is increasing/decreasing on some range if there's a vertical asymptote in that range?

The graph below shows the function $f(x)=\frac{e^x}{x-1}$ Can we say that the function is decreasing for all $x\le2$ (there's a local minimum at $x=2$) or do we have to take the asymptote at $x=1$ ...
0
votes
1answer
21 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
1
vote
4answers
64 views

How to show that this function gets every real value exactly once?

$$f(x) = {e^{\frac{1}{x}}} - \ln x$$ I thought maybe to use the Intermediate value theorem. Thought to create 2 functions from this one. and subtracting. And finally I will get something in the form ...
0
votes
0answers
42 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...
4
votes
1answer
49 views

Limit of equations.

$a_{n}+b_{n}+c_{n}=2n-1 $ $a_{n}b_{n}+b_{n}c_{n}+a_{n}c_{n}=2n+1 $ $a_{n}b_{n}c_{n}=-1 $ $a_{n}<b_{n}<c_{n}.$ Find $\lim\limits_{n\to \infty}na_{n}.$ ATTEMPT: From $ (2). $ ...
3
votes
2answers
206 views

How to prove the non-existence of a polynomial series that uniformly converges to this function?

I was asked to prove the following. There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to ...
3
votes
3answers
72 views

How does parametrization of the intersection of two surfaces induce a space curve?

Given a two surfaces say: $z=1-y$ and $ x^2+y^2+z^2=1$, we find that they intersect at: $$x^2-2yz=0$$ How is the above a space curve? Is it not just another surface? And why do we need to introduce ...
2
votes
3answers
103 views

Derivative of $(-1)^x$

I'm taking a summer calc 2 class and we're getting into alternating patterns. I was interested in seeing the graph of $(-1)^x$ so I typed it into my TI-84 for $y = (-1)^x$. Surprisingly, the graph is ...
1
vote
1answer
48 views

let $\phi (x) =\lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is not differentiable at $1$.

For $x \geq 0$, let $\phi (x) = \lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is continuous at $1$ but not differentiable at $1$. First we calculate $\phi (x) ...
1
vote
2answers
13 views

Function to apply to a linearly increasing positive real number to reach an arbitrary limit

I've got a friend who is making a browser game and he's trying to figure out how to make a function that acts like a logarithm in that it returns higher values quickly but eventually mellows out and ...
1
vote
4answers
31 views

MVT on a logarithmic function and the domain used.

I came across this question in a discussion forum: Use the mean value theorem to show that $ln (\frac{(x +1)}{x}) \lt \frac{1}{x}$ for $x \gt 0.$ I defined f(x) as the product xln(x + 1) and i ...
1
vote
0answers
32 views

How do I figure out a per square inch price?

How do I figure out a per square inch price? For example: If a print is 24x24 inches, and the price is $800.00, what is the price per square inch? Thank you!
0
votes
1answer
38 views

On the Lipschitz continuity of a product of positive function and identity mapping

Let $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive and continuous function and $C\subset\mathbb{R}^n$ be a nonempty compact set. Is the mapping $$ G(x):=\frac{x}{g(x)} $$ is Lipschitz continuous ...
3
votes
2answers
99 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
1
vote
0answers
18 views

formulate brain teaser as calculus rates problem:

I recently was asked the $2$ ropes question: Given two non-homogenous ropes (i.e. length to burn time not constant), with total burn time $=60$, how can you measure $45$ seconds? I'd like to ...
2
votes
2answers
43 views

Why is 1 not a critical point for this function?

For the function $f(x) = \frac{x^2}{x-1}$, why is $1$ not a critical point, along with $0$ and $2$? Don't critical points include discontinuities?
2
votes
1answer
91 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
2
votes
0answers
36 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
1
vote
2answers
51 views

values of sin of multiples of 10? [on hold]

I was in class the other day and the professor was arguing that sin(1), sin(10), and sin(100) are all equal to the same value and that calculators are incorrect due to approximations. This problem has ...
1
vote
0answers
37 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
0
votes
4answers
48 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
3
votes
1answer
84 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
-2
votes
0answers
37 views

Question about relation between Series and Sequences [on hold]

This is a very general question so let me know if doesn't make sense. My teacher gave me a detailed problem of a person eating a piece of pizza and cutting it into one half and then eating 1/4 ...
-4
votes
1answer
64 views

Help finding the limit of the sequence? [duplicate]

I was given the sequence $\frac{1}{2}$, $\frac{1}{4}$,$\frac{1}{8}$, $\frac{1}{16},\ldots$ . I created an equation to represent the sequence $a_n=\dfrac{1}{2^{n+1}}$. Now how do I go about finding the ...
-2
votes
5answers
145 views

Help finding the limit of this series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$

How can I go about finding the limit of $$\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots = \sum_{k = 1}^{\infty} \frac{1}{2^{k+1}}?$$ Could I use the absolute value theorem? I have a ...
2
votes
2answers
37 views

How to solve the integral $\int\tan^{3}x \sec^{3/2}x\; dx$?

How to solve the following indefinite integral $$\int \tan^{3}x \sec^{3/2}x \; dx$$ to get the solution in the form of $$\large\frac{2}{7}\sec^{7/2}x - \frac{2}{3}\sec^{3/2}x +c$$ I tried taking ...
2
votes
1answer
66 views

How to prove that $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $?

If $n \in \mathbb N$ and $n \geq 2$, then we have $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $. My try : Once if we can prove that for all $k \in ...