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

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2answers
136 views

Do Fourier coefficients (and series) of $f(x):\mathbb{R} \to \mathbb{C}$ and $f(x+2012)$ are the same?

I'd like to know whether Fourier coefficients (and series) of $f(x):\mathbb{R} \to \mathbb{C}$ and $f(x+c)$ for $c \in \mathbb{R}$ are the same and if so, why? I have this question where $f$ is a $ ...
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1answer
155 views

Analytically solving limits

I read the theory of limits and i have some misunderstanding. For example we have simple limit expression: $$\lim _{x\rightarrow \infty}{\frac{1}{x}}$$ I see that this limit is 0 and if build graph ...
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1answer
607 views

Integral involving sgn function

I am having trouble with calculating or approximation the following integral: $$ \int_{-\infty}^{\infty}(t^2-1)^{pn}(\mathrm{sgn}(t-1)-\mathrm{sgn}(t+1))^pdt, $$ where $1\leq p<\infty$ and $n \in ...
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1answer
78 views

Graph of $ax+f(x)$

Could someone help me in this problem: I have a nondecreasing function $f(x)$ on $\mathbb R$ (so we know how the graph will looks like). How can we graph the functions $ax$, $f(x)$, and $ax+f(x)$, ...
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1answer
408 views

Change of variables of PDE

I have a particle of mass $m$ that moves in 2-d in the potential $V(x,y)=\frac{1}{2}m\omega^2(6x^2-2xy+6y^2)$. I have to use the coordinates $u=\frac{x+y}{\sqrt 2}$ and $w=\frac{x-y}{\sqrt2}$ to show ...
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1answer
1k views

Question about the limits of integration using polar coordinates

I haven't been able to find an answer to something I've been thinking about. If you are taking the integral of a circle in polar coordinates you always use the limits for theta as $0$ to $2\pi$. ...
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2answers
144 views

$f,g: \mathbb{R} \to \mathbb{C}, 2\pi $ periodic. Prove: if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$

Let $f,g: \mathbb{R} \to \mathbb{C}$ $2\pi$ periodic , Riemann integrable in $[0, 2\pi]$. I need to prove that if $f(x)=0$ for every $x$ around $x_0$ so $S_nf(x_0) \to 0$ when $n \to \infty$. We ...
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1answer
163 views

average value theorem and calculus integration

i have sat for 2 hours trying to understand how the area of R1+R3=R2 cant really get this becuz R2 has a negative section and i think f(c) should be lower. Once again, how is R1 and R3= R2? R1 and ...
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2answers
81 views

Calculus problem

If $\lambda=\lambda\left(r\right)$ , how do you get from $$\left(1-2r\lambda'\right)e^{-2\lambda}=1$$ (the textbook then says, “integrates to”)$$\frac{d\left(re^{-2\lambda}\right)}{dr}=1$$ ...
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1answer
141 views

Convergence of limit with integer part of $ x $

Good day! I tried to solve this problem;the process is correct? The problem si: Let $x\in\mathbb{R}$. With $[x]$ denote the integer part of $ x $. Calculate $$\lim_{x\to 0^+} \Biggr(x^2 ...
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2answers
525 views

Use Lagrange Remainder Theorem to Prove Inequality

I'm supposed to use Lagrange Remainder Theorem to prove that $$1 + \frac{x}{2} - \frac{x^2}{8} < \sqrt{1+x} < 1 + \frac{x}{2} \text{ } \text{ if } x>0$$ Obviously, the left and right hand ...
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3answers
631 views

find the area of the largest rectangle that can fit inside a semi circle of radius 2 cm

find the area of the largest rectangle that can fit inside a semi circle of radius 2 cm I have absolutely no idea where to get started on this...What I did do is $A=(\pi r^2)/2$ (its a semi circle) ...
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1answer
101 views

Derivative $= 0$ at Extremum Necessary and sufficient?

The condition that the derivative at the extremum of an uniformly differentiable function is zero. is this necessary and sufficient or both?
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1answer
76 views

Differential of this

Define a operation $(\partial \cdot x)*f(x)=\frac{d(xf(x))}{dx}$ In the notes it has that $(\partial^n \cdot x)*f(x)=(n \partial^{n-1 }+ x \cdot \partial^n)*f(x)$ Is this wrong? I can't get this ...
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1answer
38 views

Finding points at which $f(x)=\frac{{{\sin }^{2}}x}{x|x(\pi -x)|}$ is not continuous

I have a homework question to find the points at which $$f(x)=\frac{{{\sin }^{2}}x}{x|x(\pi -x)|}$$ is not continuous. I can't seem to figure it out for this function. Can someone help me find ...
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1answer
3k views

Fourier transform of $f(x) = e^{-x^2}$ [duplicate]

Possible Duplicate: Fourier Transform of complicated product: $(1+x)^2 e^{-x^2/2}$ I calculate the Fourier Transform of $f(x)$ by $$\mathbb{F}(t) =\int_{-\infty}^{\infty}e^{-x^2} \cdot ...
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2answers
698 views

Maximum Area of a Rectangular Trough

I didn't learn how to do this in my class, and the examples in my book do not apply to this type of problem. To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to ...
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1answer
192 views

Simple Differentiation

I saw this question and with my basic knowledge of differentiation I don't know what it means. $\frac{d}{dx}(x^2)$ where $x=3$ Where would I start to solve this?
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2answers
204 views

Find function which contains two points

A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation $$y-y_1=m(x-x_1)$$ but could we ...
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1answer
300 views

Fourier transform of exp(exp(x))

I am interested in the Fourier transform of a function of the form $f(x) = \exp(g(\exp(x)))$, where g has a "simple" form, for example $g(y) = \frac{(y-1)^2}{y^2 - 1}$. Has anyone a starting point ...
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2answers
131 views

Need help on integrating a trigonometric function

Is there any way of integrating this trigonometric function $$\int \cos(x^2)dx$$ ? Wolfram alpha straight away gives this $$\sqrt{\frac{\pi }{2}}C\left ( \sqrt{\frac{2}{\pi }} x\right ...
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1answer
247 views

Continuity and Differentiability of a function

Would like some guidance. What I've done so far is included. Given, $$f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0) \end{cases}$$ ...
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2answers
367 views

Concavity and critical numbers

I am attempting to study for a test, but I forgot how to do all the stuff from earlier in the chapter. I am attempting to find the intervals where $f$ is increasing and decreasing, min and max values ...
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1answer
51 views

Uniform convergence for a summation involving exponentials

For $y > 0$, why does the series $\sum_{m = 1}^{\infty} e^{-2\pi my}$ converge uniformly in $y$? Can I just use the following argument? For $m$ large enough, there exists a constant $C$ independent ...
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2answers
76 views

Values of parameters for which a function is differentiable

I am given the piecewise function $G(t)$, which is written below, and I am asked to find all values of the parameters $\alpha$ and $\beta$ for which the function $G$ is differentiable at $t=1$. ...
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2answers
87 views

Evaluating integral $A(x)=\int_0^x f(t)\mathrm dt$

If $f:\mathbb{R}\to \mathbb{R}$ and $f(x)=0$ if $x\in \mathbb{Z}$ and $f(x)=x-\lfloor x\rfloor-\frac12$ if $x\in \mathbb{R}-\mathbb{Z}$. Let $A(x)=\int_0^x f(t)\mathrm dt$. Show that ...
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1answer
632 views

Isolate a variable in a polynomial function

How would I go about isolating $y$ in this function? I'm going crazy right now because I can't figure this out. The purpose of this is to allow me to derive $f(x)$ afterwards. $$ x = \frac{y^2}{4} + ...
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1answer
116 views

Implicit Differentiation and Tangent Lines

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $x^2 + y^2 = (2x^2 + 2y^2 - x)^2$ Point: $(0, 1/2)$ The graph is cardioid.
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1answer
5k views

Finding an equation and parametric description given 3 points

Let m be the plane through (0,1,1), (0,1,0) and (-2,-1,-1). This concept has always confused me: How would I find the equation and parametric description given just these points?? I think the ...
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2answers
113 views

A fencing problem

So the problem is: I have bought a fence 30 meters long and I need to put it around three of my rectangular fields sides. How long should be each of the field sides, to create the biggest ...
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2answers
1k views

Issues Calculating Limits Using Derivatives

I'm pretty bad at math, and Im doing some practice questions for school, I was wondering if someone could help me with this question, I cant get the right answer no matter what I try. Here's the ...
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2answers
68 views

If $ 0< x < y$ and $r > s> 0$ then when is $x^s < y^r$?

The following post is related to If $x \in [0,1]$ then how do you show the inequality $r > s > 0$ implies $x^r < x^s$? and can be thought of as a generalization to the questions posted ...
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2answers
47 views

Finding the derivatives of logarithms

I am attempting to find the derivative of $\log_{10} (x^3 + 1)$ I am not too sure to do with this actually, I know the formula states that it should be $1/ (x\ln a)$ but does that mean just plug it in ...
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2answers
76 views

How to put $y=5$ and $x=(-3)$ into discussion form?

Discuss the slope, $y$-intercept and graphs of the following:   a) $y=5$   b) $x=-3$ How can I answer this in discussion form?
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2answers
348 views

Finding derivatives using the quotient rule

I am attempting to find the derivative of $\frac{(v^3 -2v\sqrt v) } v$ using the quotient rule I set the problem up like this: $v(3v^2 - 3v^{1/2}) - (v^3 - 2v^{3/2})1$ and the denominator is $v^2$ ...
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1answer
312 views

Question in Bartle's elements of real analysis - integration and differentiation

In the exercise section of the integration chapter of Bartles's book it says that if we define the function $$f(x)=\begin{cases}0&x\in [0,1) \\ 1 & x\in [1,2]\;,\end{cases}$$ this ...
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1answer
897 views

Comparing different derivatives of a function with a given graph

I am give $g'$ of a series of numbers and I am suppose to rank them in increasing order. I have no idea what this means. I think it meant to take that as an $x$ point on the graph and then find ...
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1answer
329 views

Limit with trinomials in numerator and denominator

Again, I have a problem in the indeterminate form and I can't find out how to reduce this to something that'll give me the limit. $$\lim_{ x\to-2} ...
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3answers
736 views

Prove that this equation has only one solution

Consider an equation of the form $$x(t) = A_+ e^{-\Gamma_+ \; t}+A_- e^{-\Gamma_{-}\; t}$$ $A_{\pm},\Gamma_{\pm}$ are real constants, both $\Gamma_+$ and $\Gamma_-$ are greater than zero and $t\geq ...
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1answer
96 views

differentiable curve, its polygonal converges?

Let $$ \alpha :\left[ {a,b} \right] \to R^3 $$ be a differentiable curve, and let $ a= t_0 < .... < t_n = b $ a partition consider $$ \eqalign{ & L\left( {\alpha ,P} \right) = ...
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2answers
1k views

How to find a smooth parametrization of a Curve

In order to solve a line integral, I need to establish a smooth parametrization of the curve over which it is supposed to be integrated. The curve, $D$, is the intersection of the surfaces $x^2 + ...
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1answer
11k views

Use WolframAlpha to compute the real Fourier series of a function

How can I use Wolfram|Alpha to compute the Fourier series (with real coefficients $a_0, a_n$ and $b_n$)? (The 'Fourier series' command seems to summon the complex series) I.e. $f(x) = x + \pi$ for ...
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1answer
153 views

Trigonometric Input to an First Order Differential Equation, Exponentials

In an ODE class, the differential equation is given $y' + ky = kq_e(t)$ where the input $q_e(t)$ is given as $cos \ \omega t$. The teacher "complexifies" the problem by using the real part of ...
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1answer
358 views

suggest a textbook on calculus [duplicate]

Possible Duplicate: What are the recommended textbooks for introductory calculus? Hi, i am a software engg. and math was never my favourite. I somehow dragged thru the math i had to do in ...
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1answer
418 views

Maximizing directional derivatives?

Alright, I'm trying to fine the gradient of the function and the maximum value of the directional derivative. $$z = \frac{y}{(x^2 + y^2 )}\text{ at }(1,1)$$ So far, I've gotten the following values ...
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1answer
354 views

Resulting Solid Questions Help?

Q: A ball of radius 17 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid. A: 4500pi I am mostly confused about the question...If we have a bead-like ...
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1answer
227 views

Need help understanding differentiation. Question and Answer provided

Could someone walk me through this derivative? Thanks.
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1answer
2k views

Calculate derivatives on Casio Graphics Calculator?

Is there any way I can calculate derivatives using a Casio Graphing Calculator? Is there a program available for the calculator which can calculate them?
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2answers
52 views

Help with Implicit Differentation and Trigonometry

For an assignment, I've been asked to find the $\frac{dy}{dx}$ for the formula $$2 \cos(4x) \sin(9y)=7$$ The major problem I'm having with understanding implicit differentiation is understanding what ...
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1answer
84 views

Surface given by an arc-length parametrized curve

Given $\alpha(s)$ a smooth arc-length parametrized curve, how can you write the equation for the surface $f(s,t) = \alpha(s) + t \alpha'(s)$ component-wise? That is, I want to write $f = (f_x, ...