Questions tagged [calculus]

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

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4
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1answer
2k views

Modified Dirichlet function non-differentiability

I need some ideas to start with this problem. Show that the modified Dirichlet function defined as $D_M(x)=\begin{cases}0&\mbox{if }x \notin \mathbb{Q} , \\ \frac{1}{b}&\mbox{for } x = \frac{a}...
0
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2answers
270 views

Multivariate Chain Rule

Given: $$ \begin{eqnarray} R & = & \ln(u^2 + v^2 + w^2)\\ u & = & x + 6y\\ v & = & 2x - y\\ w & = & 4xy \end{eqnarray} $$ I am trying to determine $$\frac{\...
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3answers
212 views

Having trouble verifying absolute/conditional convergence for these series

Greetings, I'm having trouble applying the tests for convergence on these series; I can never seem to wrap my head around how to determine if they're absolutely convergent, conditionally convergent ...
4
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1answer
6k views

Would it be fine to use Serge Lang's two Calculus books as textbooks for freshman as Maths major? [closed]

I'm a freshman in Maths major, but the recommanded textbook(Calculus:A Complete Course by Robert A. Adams) by Prof. of Calculus course is too much expensive, well, I found there're Serge Lang's two ...
4
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2answers
389 views

$\sin(x) = \sum{a_n \sin(n \log(x))+b_n \cos(n \log(x))}$

Can $f(x)=\sin(x), x>0$ be represented by the series $\sum{a_n \sin(n \log(x))+b_n \cos(n \log(x))}$ ? note: this is in continuation of another question that I asked earlier here
2
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1answer
402 views

Confused about question on iterative method for numerical analysis

This question from my textbook on using the iterative method to find roots of an equation has confused me. I answered the question correctly but part c is the source of my confusion. I know that the ...
1
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2answers
1k views

Scaleless (or self-similar) function: $\sin ( \log x)$

Consider the function $f(x)=\sin( \log x)$ defined over $x>0$. It has the cool feature that when you plot it, and change the x scale, it's overall shape does not change much. For example if you ...
11
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3answers
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Finding the inverse of the arc length function

I'm just a simple high school math student, so please don't eat me =) In my calculus text, I have the formula: $$L(x) = \int_{c}^{x} \sqrt{[f'(t)]^2 + 1}\,dt$$ Where $L(x)$ is the arc length of ...
0
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2answers
122 views

Integral u-substitution Problem [duplicate]

Possible Duplicate: Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$ I'm given, $$\int \sin^2(x) \,dx$$ I'm struggling finding the appropriate value for u to integrate using u-substitution. ...
4
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2answers
111 views

Evaluating a linear integral (Calculus I)

I'm given: $$\int_{5}^{-2} [3f(x) + 1]\,dx$$ with the additional information that: $$\int_{0}^{5} f(x)\,dx = 10$$ and $$\int_{0}^{-2} f(x)\,dx = -4$$ My layman mind looks at it as, since the sum ...
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2answers
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Proof that $t-1-\log t \geq 0$ for $t > 0$

Using basic calculus, I can prove that $f(t)=t-1-\log t \geq 0$ for $t > 0$ by setting the first derivative to zero \begin{align} \frac{df}{dt} = 1 - 1/t = 0 \end{align} And so I have a critical ...
2
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2answers
161 views

How do I prove that this sequence has a limit of zero?

Given the sequence $\displaystyle\left\{\frac{x^n}{n!}\right\}$, how would I prove that its limit as $n\to\infty$ is zero?
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2answers
155 views

Basic chain rule question

Using the chain rule I am led to believe that the following can be differentiated nicely using the chain rule: $f(x)=\dfrac{1}{(1+e^{-x})}$ It has been 3 years since I have used calculus though. If ...
2
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1answer
155 views

Find connected components of a a set Y

Let Y be the union of all the circles of center (1,0) radius 1-1/n in R^2. Then, we have circles of increasing radius, finally reaching r=1 as n goes to infinity. The connected components are the ...
7
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1answer
166 views

Proving all Solutions of a Polynomial Cannot all be Real

If, a, b, c, d and e are all real numbers how could I prove that the 5 solutions of the equation: $$f(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e == 0$$ cannot all be real valued if: $$2a^2 < 5b$$ Any ...
2
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1answer
3k views

Find equation of cubic function with gradient?

I have to find a cubic function, when I know the gradient of that line at its start and finish. The gradient at the start is 0.2, and at the top -0.4. Using the cubic function: $f(x)=ax^3+bx^2+cx+d$ ...
7
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4answers
518 views

Please help me to show, that $(\ln x)'=\frac1 x$

In school, we recently started with derivations. I looked into a list of simple derivations and tried to prove them, in order to practice. Now, I tried to find the derivative of $\ln x$, but I got ...
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4answers
6k views

Prove that the unit circle is path-connected?

I need to show that the unit circle is path connected and connected. I was able to show that it is connected, by $f:[0,2\pi] \to \mathbb{R}^2$, $f(x)=(r\cos x,r \sin x)$ which is a continuous ...
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2answers
1k views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...
1
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1answer
929 views

Unsure about graph translation of $y=3+\ln(x+2)$

I was surprised by the graph of $y=3+\ln(x+2)$: I understand that $x=0 \implies y=3+\ln(2)$ and that $y=0 \implies x= e^{-3} -2$ and I derived this without problem. I was expecting the results to be ...
6
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1answer
755 views

Newton's method for square roots - digit precision (exercise from Cheney's Numerical Analysis)

I would greatly appreciate some help for this exercise in Kincaid and Cheney's Numerical Analysis: "apply Newton's method to $f(x)=x^2-r$ (where $r>0$). Prove that if $x_n$ has $k$ correct digits ...
4
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2answers
394 views

Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...
5
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1answer
9k views

Connection between chain rule, u-substitution and Riemann-Stieltjes integral

I think I understand these concepts ok: chain rule u-substitution Riemann-Stieltjes integral But there seems to be a layer that I miss: They all seem to be connected, alas I don't know how exactly. ...
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1answer
180 views

Finding $\liminf a_{n}$ and $\limsup a_{n}$

Good morning, I would love your help with this: Given a sequence {$ a_{n} $} with this initial data: $a_{2k}= \frac{a_{2k-1}}{2}$ $a_{2k+1}= a_{2k}+\frac{1}{2}$ I need to find $\liminf a_{n}$ and ...
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4answers
10k views

Function example? Continuous everywhere, differentiable nowhere [duplicate]

Possible Duplicate: Are Continuous Functions Always Differentiable? If such a function exists, can anyone give an example of a function $f(x) : \mathbb{R} \longrightarrow \mathbb{R}$ that is ...
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4answers
8k views

Integral of $1/\sinh(x)$

Can you help me find the integral $$ \int{\frac{1}{\sinh(x)}dx}? $$
3
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1answer
139 views

Relating product integrals to indefinite products

The product integral is the multiplicative version of standard integrals. Indefinite products are the discrete counterpart to this integral; they multiply iterations on a function $f(x)$ by each ...
5
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1answer
856 views

Prove $y=\ln(2x-1)/\ln(x)$ is a decreasing function

Given $y=\ln(2x-1)/\ln(x)$, prove $y$ is decreasing for $x>1$. While this is obvious by couple computations, the usual differentiation method to show this is true is not getting me anywhere since ...
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2answers
215 views

Calculus- problem with subsequences

Let $\{a_n\}$ be a sequence and $k$ a natural number so that: $\{a_{nk}\}$ , $\{a_{nk+1}\},\ldots, \{a_{nk+(k-1)}\}$ - every sub-sequence converges to the same limit $L$. I need to show that $\{a_n\}$...
6
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3answers
572 views

How to find the limit of a sum of reciprocals $\lim_{n\to\infty}(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n})$? [duplicate]

There's a limit that I am unable to solve. I think it should be equal to $\infty$. $$\lim_{n\to\infty}\left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$
3
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1answer
506 views

Spivak's Calculus: Chapter 12, Problem 26

Suppose that $f(x) > 0$ for all $x$, and that $f$ is decreasing. Prove that there is a continuous decreasing function $g$ such that $0 < g(x) \le f(x)$ for all $x$. To be quite honest, I have ...
5
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2answers
9k views

Approximations Involving Exponential Functions

I am reading a text and I am curious to know how certain approximations were reached. The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) ...
3
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4answers
2k views

Limit involving exponential functions

I am trying to figure out why this is true: $$ \lim_{p \to 0}\frac{1}{2p}\left((1+p)e^{-\frac{y}{1+p}} - (1-p)e^{-\frac{y}{1-p}}\right) = e^{-y} + ye^{-y}$$ I have already tried L'Hopital's Rule, ...
9
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2answers
3k views

Integral involving Modified Bessel Function of the First Kind

Why is this true? $$ \int_0^\infty e^{-\frac{1}{2}(b^2+x^2)} I_0(bx) x \,dx = 1 $$ Note that $I_0(x)$ is a modified bessel function of the first kind. The difficulty for me lies in a) translating ...
6
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1answer
2k views

Prove the reduction formula

The question is to "prove the reduction formula" $$ \int{ \frac{ x^2 }{ \left(a^2 + x^2\right)^n } dx } = \frac{ 1 }{ 2n-2 } \left( -\frac{x}{ \left( a^2+x^2 \right)^{n-1} } + \int{ \frac{dx}{ \...
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2answers
5k views

limit comparison test for alternating series

I am trying to understand why does the limit comprasion test doesn't work for alternating series, is it even true? or is there a counter example? I can't find one. can you help me please? thanks. ...
4
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2answers
1k views

Divergence Theorem, Laplacian, Energy Minimization

I am trying to understand a proof for critical points of certain energy functions being harmonic functions. It goes as follows: For a function $u(x_1,..,x_n)$, a functional E(u) is defined as $E(u) =...
3
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3answers
186 views

compose two functions to make them differentiable

Are there two continuous functions ( $f:R^n\to R$, $g:R\to R$ ) which are not differentiable at point $0$, but if you compose them, $h=g(f(x))$ is differentiable at $0$? ($f(0)=0$)
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1answer
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Euler-Maclaurin Summation Formula for Multiple Sums

The Euler-Maclaurin summation formula is \begin{eqnarray} \sum_{k = a}^{b} f(k) = \int_{a}^{b} f(t) \, dt + B_1 (f(a) + f(b)) + \sum_{n = 1}^{N} \frac{B_{2n}}{(2n)!} ( f^{(2n-1)}(b) - f^{(2n-1)}(a) ...
2
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3answers
639 views

How to show $\arcsin{x} = \frac{\pi}{2} + i \ln{(x+\sqrt{x^2-1})}$?

Is the following identity correct $\arcsin{x} = \frac{\pi}{2} + i \ln{(x+\sqrt{x^2-1})}?$ Here, $x < 1$. How can we show that it is true? One way to see it is by differentiating, since $\frac{d}{...
3
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3answers
5k views

Find max/minima of $x\sqrt{16-x^{2}}$

This function has a closed interval of (-4, 0), (4, 0), while passing through the origin. I'm struggling to find the maxima and minima of the function, since this finction doesn't have a standalone ...
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2answers
2k views

A question about infinite derivative

There is an assertion that if f and g are both differentiable at x, then so is f + g at x. It is safe if $f'(x)$ and $g'(x)$ are both finite, I wonder if it still holds for infinite derivative....
7
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2answers
592 views

How can I find $\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$?

My question is, how can I evaluate the following integral? $$\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$$ Thanks.
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vote
1answer
2k views

Transforming Trig Function for Easier Integration

$$\int_0^\frac{\pi}{4} \! \frac{1+\cos^2\theta}{\cos^2\theta} d\theta$$ I've been attempting to mix and match identities to make this equation easier to integrate. Mathematica has given me an ...
7
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2answers
1k views

Question on conservative fields

I'm hoping to really knock out several questions I have in my mind with just this one. I've been doing a lot of practice problems on this topic, and although I get the right answers, I really don't ...
2
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2answers
169 views

CalcI — Fundamental Theorem of Calculus Question

So I'm given: $h(1) = -2$ $h'(1) = 2$ $h''(1) = 3$ $h(2) = 6$ $h'(2) = 5$ $h''(2) = 13$ The question is find $$\int_{1}^{2} h''(u) \text{d}u$$ So based on the ...
3
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3answers
709 views

Curl and Divergence

This is an online solution to one of the problems I'm working on. The question is to analyze the statement at the beginning of each sentence and determine whether its meaningful and if so then is it a ...
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2answers
351 views

Antiderivative of sec(x) [duplicate]

Possible Duplicates: Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$ Ways to evaluate $\int \sec \theta \, \mathrm d \theta$ Using Mathematica to get the antiderivative for sec(x), I get $$-\...
18
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2answers
4k views

Delta function integrated from zero

I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a 2-...
6
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3answers
318 views

Integral yielding part of a harmonic series

Why is this true? $$\int_0^\infty x \frac{M}{c} e^{(\frac{-x}{c})} (1-e^{\frac{-x}{c}})^{M-1} \,dx = c \sum_{k=1}^M \frac{1}{k}.$$ I already tried substituting $u = \frac{-x}{c}$. Thus, $du = \frac{-...