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

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0
votes
2answers
304 views

Finding the centroid of a polar curve

The curve is $r = e^{-b\theta}$ where $b > 0$ and $θ \in [0, \infty)$. I got that the arc length is $\frac{\sqrt{b^2 + 1}}{b}$ (is this correct?), but computing the centroid $(x, y)$ looks awful. ...
2
votes
3answers
1k views

$x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. Find the condition on k.

The question is: $f(x) = x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. What values can k take? Please drop a hint!
2
votes
2answers
141 views

Infinity divided by infinity and dirac delta?

A dirac delta produces something that's infinitely long, and it could also be seen as infinitely thin. Why do we define the surface of a dirac delta to be 1. If length$\times$width $= \infty \cdot ...
5
votes
1answer
204 views

Weakest hypothesis for integration by parts

I was wondering what are the weakest hypothesis for applying integration by parts to calculate $$\int_a^b fg dm,$$ where $m$ denotes the Lebegue measure on $\mathbb R$. Is it enough that $f$ be ...
4
votes
2answers
119 views

$x^4 + 4rx + 3s = 0$ has no real roots. Relate $r, s$.

It is given that $x^4 + 4rx + 3s = 0$ has no real roots. What can be said about r and s? a) $r^2 < s^3$ b) $r^2 > s^3$ c) $r^4 < s^3$ d) $r^4 > s^3$ How to even begin??
2
votes
4answers
172 views

How do you evaluate the following limit

$$\lim_{x\to\infty} \dfrac {x+2} {\sqrt{64x^2+1}}$$ $$\lim_{x\to-\infty} \dfrac {x+2} {\sqrt{64x^2+1}}$$ I understand how to find the limit as $x$ approaches $\infty$ but I do not understand how to ...
10
votes
2answers
784 views

Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$

$$ 2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x $$ How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
2
votes
1answer
97 views

Represent an Integral by non-elementary functions

I would like to ask everyone about the following integral: $$ I = \int_0^\infty {\frac{{\sqrt {{x^2} + ax + b} }}{x}{e^{\mu x}}dx} $$ where a,b, and $\mu$ are constants. The equation $x^2+ax+b=0$ ...
0
votes
2answers
68 views

Determine if the integral converges: $\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$

Determine if the integral converges: $$\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$$ where $p,q\in\Bbb R$.
0
votes
1answer
67 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
1
vote
1answer
215 views

show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.

If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$ This is a proposition in a book. But I cannot prove this:( ...
1
vote
2answers
68 views

a simple question

I have an inequality: $$2ax^2+by^2\geq0$$that $x^2\geq y^2$. Actually $x^2$ is $c.c$ that is $(\sum c_{ij}^2)$ and $y^2$ is $(tr(c))^2$, where $c$ is $2\times 2$ matrix. Now, I want to show that ...
1
vote
3answers
122 views

Why does definite integral not depend on the variable?

Why does definite integration not depend on the variable ?? If I can use any other variable which is defined in such a way that It lies outside of the given interval, then the expression for intergral ...
0
votes
1answer
43 views

I'm a little bit puzzled by the condition of the second derivative.

Here is the problem. Suppsose that $f$ is a twice-differentiable function of the set of real numbers and that $f(0),f'(0) \text{and} f''(0)$ are all negative. Suppose $f''$ has all three of the ...
3
votes
2answers
80 views

Suppose $f(x)$ is such that $\int_{-\infty}^\infty e^{tx} f(x)dx = \arcsin (t - \sqrt{1/2})$

Suppose $f(x)$ is such that $$\int_{-\infty}^\infty e^{tx} f(x)dx = \arcsin(t - \sqrt{\frac{1}{2}})$$ for all $t$ where the right-side expression is defined. Compute $$\int_{-\infty}^\infty ...
8
votes
3answers
165 views

Proof by induction using Fubini's Theorem

I am asked for the volume of the region $x_1+\cdots+x_n\leq 1$ where $x_1,...,x_n\geq 0$. I am proposing that the volume $V(n)$, is given by $$ V(n) = ...
2
votes
3answers
297 views

Why do we need an open interval to define local maxima?

We say that $f(x)$ has a local maximum at $\tilde{x}$ if for every $x$ in some open interval $(\tilde{x}-\delta,\tilde{x}+\delta)$, $\delta>0$ we have $f(\tilde{x})>f(x)$. Why do we need an ...
0
votes
3answers
115 views

If $a,b$ are extended real numbers, $f$ is differentiable/$f'$ is continuous on $(a,b)$, prove uniform continuity.

Suppose $a<b$ are extended real numbers and that $f$ is differentiable on $(a,b)$. Prove that if $f'$ is bounded on $(a,b)$ then $f$ is uniformly continuous on $(a,b).$
0
votes
1answer
512 views

Probability of two people meeting during a certain time.

I recently read a math problem and, having not yet taken anything beyond calculus 1, was curious about how to solve it correctly. Problem: Calculate the probability of two people meeting at the ...
14
votes
2answers
561 views

Evaluating $\int_0^{\infty}\frac{e^{-x}}{1+x^2}dx$

I'm trying to evaluate $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx$$ By making the substitution $x=\tan\theta$, $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx=\int_0^{\frac \pi 2}\exp(-\tan\theta)d\theta$$ So ...
1
vote
2answers
56 views

Prove that $\ln$ has an inverse function

For $x$ in $(0, \infty)$ let $\ln(x) = \int_{1}^{x}\frac{1}{t}dt$. Prove that $\ln$ has an inverse function My book does not really go into how to prove something has an inverse, besides it needing ...
1
vote
0answers
41 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...
5
votes
5answers
685 views

Solving the improper integral $\int_0^{\infty}\frac{dx}{1+x^3}$

$$\int_0^{\infty} \frac{dx}{1+x^3}$$ So far I have found the indefinite integral, which is: $$-\frac{1}{6} \ln |x^2-x+1|+\frac{1}{\sqrt{3}} ...
5
votes
2answers
55 views

Evaluate $\int_1^\infty (\log(x)/x)^{2011} \; \mathrm dx$

Evaluate $$\int_1^\infty (\log(x)/x)^{2011}\; \mathrm dx$$ I have this question in my book of problems and I'm stumped. I could use some help seeing how this works! Thanks a bunch.
0
votes
3answers
57 views

Find the limit of the following using the definition if possible

Can any one find the following limits for me: 1- $\lim\limits_{x \to 0}\frac{x^2-1}{2x^2-x-1}$ 2- $\lim\limits_{x \to 1} \frac{x^2-1}{2x^2-x-1}$ 3- $\lim\limits_{x \to \infty} ...
0
votes
1answer
61 views

Show that $g$ is continuous using the fact that $f$ is unformily continuous

Working through Advanced Calculus of Several Variables by Edwards. The section is entitled Step Functions and Riemann Sums. Here is the question: Let $A$ and $B$ be contented sets, and $f: A \times ...
4
votes
4answers
2k views

Show that function is strictly monotone increasing

I want to show that $$ f(x)=\dfrac{x-\sin(x)}{1-\cos(x)} $$ is strictly increasing in $(0,2 \pi) $. Unforunately, this is not that easy for me , as the derivative is not very manageable and ...
2
votes
2answers
94 views

Derivative of a natural log to the power.

I'm reviewing for my calc final, and I have this question: $$f(x) = (\ln x)^{x^2}$$ What I did was just take the derivative of the power, which is $2x$, then the derivative of lnx and got: ...
1
vote
5answers
128 views

Differentiation confusion

I've been reading my textbook, and it tells me how to go about differentiating from first principles, it goes something like this: $\eqalign{ & \mathop {\lim }\limits_{h \to 0} {{f(x + h) - ...
2
votes
2answers
58 views

Proving $\sum_{i=1}^nx_i-\prod_{i=1}^nx_i\leq n-1$,using induction.

I want to prove $$\sum_{i=1}^nx_i-\prod_{i=1}^nx_i\leq n-1$$ for $n\in\mathbb N$ and $0\leq x_i\leq 1$ for all $1\leq i\leq n$. I want to prove it by induction, so $n=1$ is clear: $x_1-x_1=0\leq ...
1
vote
1answer
950 views

Parameterization of the surface a torus

For a calculus question I have I need to parameterize the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the z-axis (with $0<a<b$). I've had a go at ...
0
votes
1answer
60 views

Integrating during a least squares approximation, are these partial integrals correct?

I would like to find the least squares approximation: $$g(\alpha_0,\alpha_1) = \int_{0}^{\pi/2} [\sin x-\alpha_0-\alpha_1x]^2dx$$ Taking the derivatives w.r.t. the alphas I end up with: ...
0
votes
2answers
101 views

Doing a least squares approximation, can someone explain this step from the book?

In an example from the book, the author is finding the linear least squares approximation of $e^x$. We have a standard equation for least squares: $$g(\alpha_0,\alpha_1) = \int_{-1}^1 ...
3
votes
3answers
85 views

How to solve $(1+x)^{y+1}=(1-x)^{y-1}$ for $x$?

Suppose $y \in [0,1]$ is some constant, and $x \in [y,1]$. How to solve the following equation for $x$: $\frac{1+y}{2}\log_2(1+x)+\frac{1-y}{2}\log_2(1-x)=0$ ? Or equivalently $1+x = ...
1
vote
4answers
75 views

Let $f$ be continuous on the real numbers. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ on $\mathbb{R}$.

Can someone solve this question? Let $f$ be continuous on $\mathbb{R}$. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ for all $x$ in $\mathbb{R}$. ...
2
votes
0answers
60 views

Calculus beyond common sense [closed]

What is integration actually? Some times we use it to calculate the area under the curve,some times as antiderivative. How could Newton/Leibnitz come up with such a brilliant yet hard to understand ...
0
votes
1answer
41 views

I have done the second direction of the proof. Hopefully, it is true. Please show my mistakes?

Show that two $C^{∞}$ vector fields $X$ and $Y$ on a manifold $M$ are equal if and only if for every $C^{∞}$ function $f$ on $M$,we have $Xf =Yf$. I have sone one direction of the proof. let $p ∈ ...
2
votes
1answer
667 views

Polar Coordinates and Double Integrals

Problem 1: Find the area enclosed by the ellipse $\displaystyle \frac {1} {r} = 1 – 0.6 \cos(\theta)$. We know $0\leq \theta\leq 2\pi$. We know $0\leq r\leq 1/(1-0.6\cos(\theta))$. Questions: ...
2
votes
1answer
58 views

Determine if the integral converges: $\int_{-\infty}^{2} \frac{e^{3x}dx}{1+x^2}$

Determine if the integral converges: $$\int_{-\infty}^{2} \frac{e^{3x}dx}{1+x^2}$$ I've tried this: $$f(x)=\frac{e^{3x}}{1+x^2}>\frac{e^{\ln{3x}}}{1+x^2}=\frac{3x}{1+x^2}=g(x)$$ now since g(x) ...
2
votes
3answers
53 views

Solving $(f'(x))^2 = f(x)f''(x)$ with boundary conditions.

Let $f$ be a continuous real-valued function such that $$(f'(x))^2 = f(x)f''(x).$$ Suppose $f(0) = 1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$. I have this question in my book ...
2
votes
3answers
314 views

Solving the improper integral $\int_0^{\infty} {e^{-ax}\cos{(bx)}} dx$

$$\int_0^{\infty} {e^{-ax}\cos{(bx)}} dx$$ I know I need to use integration by part method. But I'm not sure how does the improper integral take place?
0
votes
1answer
45 views

Factoring negatives out of limits

I had a question about an answer I saw on this website. Here's the link to the full question/answer correspondence (I've also copied the relevant text below): Second derivative "formula ...
2
votes
3answers
54 views

By expressing $y = {u \over v}$ as $y = u{v^{ - 1}}$ prove the quotient rule

How do I go about doing this? I'm clueless.. Thank you. My attempt: Using the product rule and making: $\eqalign{ & u = u \cr & v = {v^{ - 1}} \cr} $ so: ${{du} \over {dx}} = 1$ and ...
3
votes
1answer
166 views

Hessian of a function that takes matrix arguments

I have a function that that takes a matrix and returns a scalar, $f : \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$. I know how to calculate the derivative of this function with respect to the matrix ...
1
vote
1answer
181 views

Computation Of full range fourier series

Question: Given that $f(x)=(x−4)^2\forall x\in[0,4]$. For each of the following questions, define a periodic extension function of $f(x)$ and sketch its graph on the interval $[−8,8]$. Determine the ...
0
votes
2answers
118 views

Use the fundamental theorem of calculus to evaluate the definite integral

Use the fundamental theorem of calculus to evaluate the definite integral $\displaystyle\int_0^3 \frac{1}{\sqrt{1+x}}dx$ I dont get what they want here is it just to take the $F'(x) ...
1
vote
3answers
179 views

Implicit Differentiation Problem???

PROBLEM: Heat flows normal to isotherms, curves along which the temperature is constant. Find the line along which heat flows through the point $(2,5)$ when the isotherm is along the graph of ...
2
votes
0answers
32 views

Using the definition of derivatives.

Let $f$ be a function such that $f(x) = f(1-x)$, for all real numbers $x$. If $f$ is differentiable everywhere, then $f'(0) =$ ? In order to solve this, I thought using the definition of ...
1
vote
4answers
3k views

MacLaurin series of $\ln(1-x^2)$

The MacLaurin series for $\ln(1 + x)$ is obtained from the series for $\frac{1}{1 + x}$ by integration. Use this and appropriate substitutions to obtain the MacLaurin series for $\ln(1-x^2)$. ...
0
votes
1answer
30 views

Figuring out parameters

I need to figure out a parameter to satisfy the following conditions: $H(\frac{1}{2}, 0) = 0$ $H(1,0) = 1$ $H(0,1) = 0$ $H(1,1) = 1$ for $H(s,t)$. I have been at it for hours and can not figure ...