Questions tagged [calculus]

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

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28
votes
3answers
4k views

Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$

I'd love your help with finding the following limit: $$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$$ I was asked to find this limit, but honestly I believe that it doesn't exist. According to Heine ...
7
votes
2answers
192 views

$f$ is differentiable and $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0$

This is quiet a simple question, but still I'm not sure that I am correct. Let $f$ be a differentiable function in $\mathbb{R}$ such that: $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0.$ We ...
3
votes
3answers
737 views

Mills' Ratio for Gaussian Q Function

Suppose I have the following lower and upper bound for the Gaussian Q Function: $$ \frac{x}{x^2 + 1} \varphi(x) < Q(x) < \frac{1}{x} \varphi(x), $$ where $Q(x) = \int_x^\infty \frac{1}{\sqrt{2\...
4
votes
2answers
317 views

Parametrization of curve length in D dimensional space. How is it done?

Sorry, its been a while and my calculus was never good. This is really a very elementary question which I am unable to un -complicate from its shroud of notation. My difficulty is how does this ...
0
votes
1answer
127 views

Need help with a proof involving nonlinear differential equations

I'm trying to solve a problem that stated: If $ae \neq bd$ prove that you can choose 2 constants, h and k, so that the substitution $t= s - h$ , $ x = y - k $ reduce the following equation to a ...
4
votes
2answers
356 views

Summation Formula for Product of Two Distinct Integers within a Range (Derivation of Covariance)

I am trying to follow the derivation for the covariance of two discrete random variables. The joint distribution of the two random variables is known: $$ P(x=a, y=b) = \frac{1}{(m+n)(m+n-1)},$$ when $...
10
votes
4answers
812 views

Integral with $\sqrt{2x^4 - 2x^2 + 1}$ in the denominator

$$\int\frac{x^{2}-1}{x^{3}\sqrt{2x^{4}-2x^{2}+1}} \: \text{d}x$$ I tried to substitute $x^2=t$ but I am unable to solve it and I also tried to divide numerator and denominator by $x^2$ and do ...
11
votes
3answers
317 views

Integrating $\int\nolimits_0^{\infty}{\frac{e^{-ax}-e^{-bx}}{x}\sin{mx} \, dx} \quad (a > 0 \, , b >0)$

I have this integral $$\int\nolimits_0^{\infty}{\frac{e^{-ax}-e^{-bx}}{x}\sin{mx} \, dx} \quad (a > 0 \, , b >0)$$ What I did was this $$ \begin{align} \int_0^{\infty}{\frac{e^{-ax}-e^{-bx}}{x}\...
1
vote
3answers
8k views

Relation between the tangent to a curve and the first derivative of a function

There is a relation between the tangent to a curve of a function and the first derivative of that function. However, how do I show that connection? How can you explain it to someone so that it becomes ...
8
votes
6answers
3k views

Why is $\infty^0$ indeterminate?

In a recent test question I was required to us L'Hopital's rule to evaluate: $$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$ I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...
2
votes
4answers
7k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
81
votes
1answer
8k views

Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$

I need your help with evaluating this limit: $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$ i.e. we apply the $\sin$ function $n$ times. Thank you.
4
votes
1answer
222 views

Show the correctness of a logarithmic inequality

Let $p_1>p_2$ and $n_1>n_2$ be positive numbers. I want to show that, $$ \frac{\log \left(\frac{p_1}{n_1}+1\right)}{\log \left(\frac{p_2}{n_2}+1\right)}\leq \frac{\log \left(\frac{p_1}{c+n_1}+...
2
votes
2answers
150 views

Stuck on a homework question: if $t = \frac{1}{x}$

If $\displaystyle t = \frac{1}{x}$ then a) Explain why $\displaystyle\lim_{x \to 0^-}f(x)$ is equivalent to $\displaystyle\lim_{t \to -\infty}f\left(\frac{1}{t}\right)$ b) Using that, ...
2
votes
2answers
2k views

Approaching from the right or from the left?

Simple question: If I have $\lim_{x \to -3^-}f{(x)}$ and I'm looking at a graph, am I approching -3 from the direction of +$\infty$ to -3 (as in going in the negative direction)? Or am I approaching -...
1
vote
3answers
314 views

Question on convergence of improper integral

For what values of $\alpha$ is the following integral convergent? $$\int\limits_{-\infty}^{\infty}\frac{|x|^\alpha}{(1+x^2)^m}dx$$ Should the limit comparison theorem be used in this case? I am not ...
4
votes
4answers
407 views

$\tan(\frac{\pi}{2}) = \infty~$?

Evaluate $\displaystyle \int\nolimits^{\pi}_{0} \frac{dx}{5 + 4\cos{x}}$ by using the substitution $t = \tan{\frac{x}{2}}$ For the question above, by changing variables, the integral can be rewritten ...
28
votes
11answers
34k views

What is the length of a sine wave from $0$ to $2\pi$?

What is the length of a sine wave from $0$ to $2\pi$? Physically I would plot $$y=\sin(x),\quad 0\le x\le {2\pi}$$ and measure line length. I think part of the answer is to integrate this: $$ \int_0^...
2
votes
1answer
292 views

Solving an integral with Laplace method

I'm trying to approximate the sum $$\sum_{\alpha=1}^{\mu} \Big(1-\frac{(\alpha(2 \mu-\alpha))^2 \gamma_1 \gamma_2}{2n^2 \mu^4}\Big)^{\frac{\lambda}{2}}$$ with an integral $$\int_{a}^{\infty}\exp\left(\...
1
vote
0answers
593 views

Golomb sequence

How to compute $n$-th element in golomb's sequence? I've found that $A_n$ is approximately: $$\phi^{2-\phi} * n^{\phi-1},\;\; \text{ where }\;\; \phi \;\;\text {is golden ratio}.$$ As far as I ...
3
votes
1answer
227 views

Minimum value for this function

Consider the following function: $$F(A_1,\dots,A_n,\lambda_1,\dots,\lambda_n)=\sum_{i=1}A_i\Big(\frac{1}{(n-2)^2}\sum_{k,l\neq i}(\lambda_k-\lambda_l)^2-\frac{2n}{n-2}\lambda_i^2\Big),$$ where $A_i\...
2
votes
2answers
2k views

How to prove correctness of a formula (differential calculus, integral)?

How do I prove the correctness of the following formula relating to the fundamental theorem of calculus? $$\int \! x\cos{3x} \, \mathrm{d}x = \frac{\cos{3x}}{9}+\frac{x\sin{3x}}{3}+C$$
13
votes
2answers
4k views

Is there an analogue to the “Delta” symbol for ratios?

A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$ My question is: is there an analogue of this notation ...
7
votes
6answers
474 views

Question about Holomorphic functions

I try to show: Let $f: \mathbb{C} \longrightarrow \mathbb{C} $ be holomorphic with $$\Re(f)+\Im(f)=1 $$ then $ f $ is constant. ($\Re$ = Real Part, $\Im$ = Imaginary Part) I have certain ideas ...
1
vote
1answer
207 views

What's the minimum value of this function?

I am considering these functions: For $i=1,\ldots,n$, define $$f_i(\lambda_1,\dots,\lambda_n)=\frac{1}{(n-2)^2}\sum_{k,l\neq i}(\lambda_k-\lambda_l)^2-\frac{2n}{n-2}\sum_{k=1}^n\lambda_k^2.$$ Suppose ...
3
votes
1answer
143 views

help to prove an inequality

Suppose $f\in C^2[a,b]$, $f(a)=f(b)=0$. Then for any $x\in [a, b]$: $$\frac{f(x)}{(x-a)(b-x)}\le \frac{1}{b-a} \int_a^b{|f^{\prime\prime}(t)|dt}.$$ Any help is appreciated.
3
votes
3answers
19k views

Integral of $\sqrt{1+\tan^2x}$ [duplicate]

Possible Duplicate: Ways to evaluate $\int \sec \theta d \theta$ I'm having a bit of a problem with an integral. The original problem was the length of a curve given parametrically. I've managed ...
4
votes
2answers
202 views

Exercise: manufacture a bound on $f'$ from $f$ and $f''$

Exercise Let $f\colon \mathbb{R} \to \mathbb{R}$ be $C^2$ and nonnegative. Prove that $$\big( f'(x)\big)^2 \le 2f(x) \lVert f''\rVert_{\infty}.$$ I have found this innocent-looking little ...
5
votes
3answers
4k views

Rationalizing the denominator [duplicate]

So I feel stupid for asking this, but I can't figure this out. I haven't taken algebra for about 8 years, so doing this is kind of fuzzy. Just started Calc 1 and we're finding limits. $$\lim_{x \to 9}...
0
votes
1answer
423 views

On the pointwise limit of $\sqrt[n]{p_n(x)}$ when $n\to\infty$, for some polynomials $(p_n)$

For every $n$, I have a polynomial $p_n(x)=a^{(n)}_{n-1}x^{n-1}+a^{(n)}_{n-2}x^{n-2}+\dots+a^{(n)}_0$ (the $n$ in the exponent of the coefficients is merely an index). I can show that $\lim_{n\to\...
3
votes
2answers
706 views

Looking for a good precalculus/algebra reference

I'm working my way through Calculus, 9th ed. by Larson and Edwards in independent study. The problem is that many of the exercises and examples use a lot of algebra tricks that are glossed over and ...
5
votes
7answers
2k views

How many points in the xy-plane do the graphs of $y=x^{12}$ and $y=2^x$ intersect?

The question in the title is equivalent to find the number of the zeros of the function $$f(x)=x^{12}-2^x$$ Geometrically, it is not hard to determine that there is one intersect in the second ...
2
votes
2answers
165 views

How can I show $\log(1+\frac{X}{A})\log(1+\frac{Y}{B})\ge \log(1+\frac{X}{B})\log(1+\frac{Y}{A})$?

How can I show $$\log(1+\frac{X}{A})\log(1+\frac{Y}{B})\ge \log(1+\frac{X}{B})\log(1+\frac{Y}{A})$$ if $X\ge Y>0$ and $A\ge B>0$?
1
vote
2answers
268 views

Examine the Maximum and Minimum Value

I have to do the following problem, and I need help. Examine the function $f(x,y) = \dfrac{-3x}{x^2+y^2+1}$ with respect to maximum and minimum.
1
vote
1answer
268 views

What formal mathematical models exist for digital hardware?

What formal mathematical models exist for digital hardware? I am familiar with several non-formal models that are used as the basis of several hardware description language simulators and ...
37
votes
8answers
3k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
9
votes
3answers
870 views

Does finiteness of $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ imply $\lim\limits_{x\to\infty}f'(x)=0$?

Assume that $f:{\bf R}\to{\bf R}$ is differentiable on ${\bf R}$, and both of $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ are finite. Geometrically, one may have $$\lim_{x\...
13
votes
6answers
2k views

For an integrable function $f$, do continuity conditions on its integral affect continuity of $f$?

In this question, whenever I say "integrable" I mean "Riemann-integrable". The (first) fundamental theorem of calculus states: if $f\in C[a,b]$, then the function $F(x)=\int_a^x f(t)\, dt$ is ...
5
votes
4answers
273 views

Why $\lim_{n\to\infty}\int_{0}^1f_n(x)dx=\int_{0}^1(\lim_{n\to\infty}f_n(x))dx$?

Consider the following equality: $$\lim_{n\to\infty}\int_{0}^1f_n(x)dx=\int_{0}^1(\lim_{n\to\infty}f_n(x))dx$$ where $$f_n(x):=\frac{x^n}{1+x^n}\qquad x\in [0,1]$$ Since the sequence $(f_n(x))_{n=1}...
9
votes
4answers
285 views

Bounding ${(2d-1)n-1\choose n-1}$

Claim: ${3n-1\choose n-1}\le 6.25^n$. Why? Can the proof be extended to obtain a bound on ${(2d-1)n-1\choose n-1}$, with the bound being $f(d)^n$ for some function $f$? (These numbers describe ...
0
votes
1answer
104 views

Eigenfunction of $(a(x) f^{II})^{II}= - \lambda^2f$

I need the eigenfunctions $f$ and eigenvalues $\lambda$ of $(a(x) f^{II}(x))^{II}= - \lambda^2f$ for a given $a(x)$. For $a(x)$ constant the solution is a combination of sin, cos, sinh and cosh. ...
4
votes
4answers
2k views

First order logic and higher order logics?

I hear that Prolog is based in first-order logic. This makes me wonder, C/C++ are based on which higher order logics? If this question is incorrect, please point out that. and how are these logics ...
1
vote
1answer
659 views

Bézier Curve and Gravitational Pull

Okay, so I'm trying to manipulate an object programmatically. When it gets near another object, lets say a globe. I want the globe to have a gravitational pull on the original object. This isn't so ...
2
votes
1answer
136 views

Need help proving the following relationship

$$ \frac{\partial \phi}{\partial \xi} ( \parallel \mathbf{x} - \xi_i \parallel) = - \frac{\partial \phi}{\partial x} ( \parallel \mathbf{x} - \xi_i \parallel) $$ from page 14 of these lecture ...
8
votes
3answers
305 views

Evaluating $\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)}$

On a lark (as a followup to this question), I was playing around with Wolfram alpha, and it seems that $$\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)} = \frac{\...
3
votes
3answers
764 views

Implicit differentiation misunderstanding

I'm trying to see why my textbook's solution is correct and mine isn't. "Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$ First, the ...
1
vote
2answers
177 views

Does this equality always hold?

Is it true in general that $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x} f(u,x) \mathrm{d}u = \int_0^{x} \left( \frac{\mathrm{d}}{\mathrm{d}x} f(u,x) \right)\mathrm{d}u +f(x,x )$ ? Thank ...
6
votes
3answers
321 views

the approximation of $\log(266)$?

Consider the following exercise: Of the following, which is the best approximation of $\sqrt{1.5}(266)^{1.5}$? A 1,000 B 2,700 C 3,200 D 4,100 E 5,300 The direct idea is using the "...
61
votes
15answers
25k views

How to convince a layperson that the $\pi = 4$ proof is wrong?

The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi = 4$? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing ...
2
votes
2answers
2k views

Partial derivative with respect to a function

Let $p$ be a function of the form $\mathbb{R}^2 \to \mathbb{R}$. How do i find the derivatives of the following expressions with respect to p(x,y) : a. $\int_\mathbb{R^2} p$ b. $ \dfrac{\partial p}{...