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

learn more… | top users | synonyms

-3
votes
3answers
33 views

A problem of Schwarz derivative [on hold]

I need help with the following problem analysis: Suppose $f$ is defined on an interval around $x$. The limit $$\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2},$$ if it exists, is called the Schwarz ...
2
votes
2answers
27 views

A function that integrates to zero against a sequence of weights

Fix any $a\in(0,1)$. Is there a nontrivial continuous function $f:[a,1]\to\mathbb R$ so that $$ \int_a^1t^{-2n}f(t)dt=0 $$ for all integers $n\geq0$ and $f(a)=f(1)=0$? I would prefer explicit ...
0
votes
0answers
13 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
-1
votes
1answer
72 views

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ [on hold]

Why $\int _0^{x^2}e^{-t^2}dt$ is positive for $|x|>1$ and negative for $|x|<1$ ? I don't understand .. I can't see.. damn it!
1
vote
1answer
28 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
2
votes
1answer
27 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
0
votes
0answers
14 views

Calculus 1- Derivatives: What Overall Dimmensions Will minimize the Amount of Paper Used? [on hold]

You are designing a poster with 50 sq.in. of printing, a 4in. margin at the top and bottom, and a 2 in. margin on each side. What overall dimensions will minimize the amount of paper used? Explain ...
16
votes
3answers
126 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
1
vote
0answers
17 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
2
votes
2answers
30 views

Convergence: infinite series

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}, n\geq\text{some integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
2
votes
1answer
26 views

How can I find monotonicity intervals? v18

We have $F:\mathbb{R}\rightarrow \mathbb{R}$, $F(x)=x\int _0^x (1+\cos(t)) \, dt$ and we neeed to find monotonicity intervals and I don't know how... Here is what I try to do: $$F'(x)=\int _0^x ...
0
votes
0answers
13 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
0
votes
2answers
21 views

How we can prove that $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n f(t)\:dt$ is convergent?

We have $f:\left(-1,\infty \right)\:\rightarrow \:R,\:f\left(x\right)=\frac{x}{x+1}$ and we need to prove that: $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n\:f\left(x\right)dx$ is convergent.Maybe, in ...
3
votes
3answers
69 views

Evaluation of $ \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos xdx $

Evaluate $$\displaystyle \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos x \,dx .$$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int \left(\cos 2x \right)^{\frac{11}{2}}\cdot ...
3
votes
1answer
32 views

Use implicit differentiation to find an equation of the tangent line to the curve at the given point

Use implicit differentiation to find an equation of the tangent line to the curve $$x^2+xy+y^2=1$$ at $(1,1)$. I am not sure how I should work this out because the given point is not on the ...
0
votes
4answers
47 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
0
votes
2answers
46 views

what is maximum value of $\frac{\sin(n\theta)}{\theta}$?

I differentiated it with respect to theta and equated the equation to zero than the result comes out to be $\tan n\theta={n\theta}$. How to proceed further?
0
votes
2answers
43 views

Why Riemann sum is convergent? [on hold]

Why $\frac{1}{n}\sum _{k=1}^nf\left(\frac{k}{n}\right)$ is convergent? I don't understand how we can prove that is bounded and monotone... For instance: $f:R\rightarrow R,\:\:f=\frac{1+x}{1+x^2}$, ...
-6
votes
2answers
44 views

Two indefinite integral problems. [on hold]

Please help me out in these $$\int \frac{dx}{1-3\sin (x)}.$$ Second $$\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx.$$
0
votes
1answer
24 views

Evaluate $\lim _{n\to \infty }\left(1-f\left(\frac{1}{\sqrt{n}}\right)\right)\cdot \sum _{k=1}^nf\left(\frac{k}{n}\right)$

We have $f:\left[0,\frac{\pi }{2}\right]\rightarrow R,\:f\left(x\right)=cos\left(x\right)$, and we need to evaluate: $\lim _{n\to \infty ...
0
votes
1answer
37 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
2
votes
1answer
35 views

Apply function fractional times

For example, one can apply $\cos x$ to number $a$ one time to get $\cos a$, two times $\cos \cos a$, three times $\cos \cos \cos a$, and so on. Is there a way to define fractional application for ...
0
votes
1answer
69 views

Spivak's calculus: Chapter 7 problem 18 d)

In cases (a) and (c) [where it was proven that such a number exists for a continous $f$ on $\textbf{R}$], let $g(x)$ be the minimum distance from $(x,0)$ to a point on the graph $f$. Prove that ...
2
votes
2answers
52 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluation of Integral $\displaystyle \int\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt\;,$ ...
0
votes
1answer
24 views

Simultaneous equation

Solve that $x-y=a , 2x^2+y^2=9$ and let solution set $(x,y) = (\alpha,\beta)$ Find the maximum value of $|\alpha + \beta|$ where $a$ is a real number
1
vote
0answers
31 views

Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
0
votes
0answers
19 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
0
votes
1answer
8 views

Using step 1 of the FTOC to solve the derivative of the integral.

I was going to say that the derivative would simply be the equation inside the integral, since if you're taking the derivative of the integral, but that's not right. Does anyone know how to go about ...
0
votes
0answers
15 views

Are the signs of total derivatives the same as partial derivatives

I have a set of five equations which I have totally differentiated and solve for the relevant variables. I have another equation that I totally differentiated and have a set of partial derivatives ...
0
votes
0answers
22 views

What is $\frac{\partial}{\partial x}\int_0^t x(\tau)f(\tau)\, d\tau$? [on hold]

If $F(x,t)=\int_0^t x(\tau)f(\tau)\, d\tau$, What is $\frac{\partial}{\partial x}F(x,t)$ ? And what is $\frac{d}{dt}(\frac{\partial}{\partial x}F(x,t))$?
-2
votes
1answer
68 views

Calculate the upper sums Un and lower sums Ln, on a regular partition of the intervals, for this integral:

sorry new to this site. Can someone please help me with this? I have tried for such a long time and have yielded no correct answers. $$\int_1^7 (3−5x)dx$$ We have $n$ rectangles, so what I did first ...
0
votes
1answer
15 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
2
votes
0answers
52 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
3
votes
3answers
153 views

Finding the derivative of the integral using the Fundamental Theorem of Calculus.

I'm still not entirely solid on the concept of the Fundamental Theorem of Calculus, but I believe that the first step of the theorem will give us $$2x-1$$ which is the derivative of F(x). Usually, ...
2
votes
1answer
31 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
0
votes
2answers
29 views

Finding the derivative of the integral using FTOC. [on hold]

Since they're simply asking for the derivative of $h$, would the answer simply be: $$\cos^2 (x) + x$$
1
vote
1answer
37 views

Eigenvectors of derivative

I'm trying to consider how linear algebra relates to calculus. It seems to me that the only eigenvectors of the derivative operator on $\Bbb R$ are the functions $ce^{kx}$ for constants $c$ and $k$. ...
1
vote
3answers
31 views

Finding the local maximum from a definite integral

For this, is the First Derivative Test being used? If that's the case then wouldn't the equation be: $$(x^2 - 4) / (2 + \cos^2 (x))$$ I'm just not sure how to go about starting this problem.
4
votes
3answers
65 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
0
votes
3answers
41 views

Evaluate the integral in terms of areas.

I understand that the first one is 4 from basically adding the squares inside the signed area, but I'm unsure on how to proceed in getting the other integrals. Any help would be appreciated, thank ...
0
votes
1answer
17 views

Differentiability functions

If $f:A\subset \mathbb R^n\rightarrow \mathbb R^m$ and $g:B\subset \mathbb R^n\rightarrow \mathbb R^m$ are differentiable functions on the open sets A, B and $\alpha,\beta$ are constants. Prove that ...
1
vote
1answer
19 views

Solving a limit by recognizing the sum as a Riemann sum for a function defined on [0,1].

I understand that the change in x is represented by $$7/n$$ but how would you go about solving the actual limit here?
2
votes
3answers
70 views

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$?

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$? Source: calculus made easy by S. Thompson. I have looked up the formula for binomial theorem with negative exponents but it is confusing. ...
1
vote
2answers
33 views

Find when the population is growing the fastest, under the logistic model

The population $P$ of an island $y$ years after colonization is given by the function: $\displaystyle P = \frac{250}{1 + 4e^{-0.01y}}$. After how many years was the population growing the fastest? ...
-1
votes
1answer
47 views

Prove that if f is continuous in A then |f| is also continuous. [duplicate]

Prove that if $f$ is continuous in a then $|f|$ is also continuous. I have this exercise for homework of calculus I, and I was thinking that it could be treated by cases when $f>0$ and $f<0$, ...
0
votes
2answers
36 views

Raising/lowering with natural logs

I had a question on a test, and while I have already figured out that I should have done u substitution (I was running out of time and my brain froze), I was wondering if the following would be legal? ...
0
votes
1answer
23 views

Equation for the curve in terms of x,y

we got the equation $$r(t) = (t-2)i + (t^2+4)j$$ I got $$x = 1-2t$$ $$y = 1+4t$$ Would that be correct?
0
votes
1answer
25 views

Find an equation for a sinusoid with minimum and maximum

Here's my problem: Find an equation for a sinusoid that has a minimum at (30°,-1) and an adjacent maximum at (75°,7). Please help! I've tried everything I can think of, but I'm really drawing a ...
0
votes
0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
2
votes
0answers
76 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...