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

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10
votes
5answers
432 views

How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$

Well,this is a homework problem. I need to calculate the differential entropy of random variable $X\sim f(x)=\sqrt{a^2-x^2},\quad -a<x<a$ and $0$ otherwise. Just how to calculate $$ ...
2
votes
1answer
165 views

Find the functions family that satisfies the inequality $\int_0^1 \frac{dx}{1+f^{2}(x)} <\frac{f(1)}{f'{(1)}}$

I wonder what is the functions family that satisfies the following inequality: $$\int_0^1 \frac{dx}{1+f^{2}(x)} <\frac{f(1)}{f'{(1)}}$$ This inequality seems to be a very interesting inequality, ...
3
votes
3answers
889 views

Orthogonal Trajectories

I am asked to show that the given families of curves are orthogonal trajectories of each other. $$x^2+y^2=ax$$ $$x^2+y^2=by$$ I know that two functions are called orthogonal if at every point their ...
1
vote
2answers
101 views

Integration of coordinates giving different answers

Let's consider a two-sided an unit ice cream cone defined by $$E = \left \{ (x,y,z): x^2 + y^2 \leq z^2 \leq 1 - x^2 - y^2 \right \}$$ What is volume of this icecream cone (I only want one side ...
2
votes
2answers
137 views

Two different characterization of “differentiable function”

In a calculus class we were given the following definition of "differentiable function" (working with 2 variables): Definition: Let $A \in \mathbb{R^2}$, and $f : A \to \mathbb{R}$. We say that $f$ ...
6
votes
1answer
366 views

How find the value of this integral

How can I compute $$\int_{-\pi}^\pi\frac{\sin(13x)}{\sin x}\cdot\frac1{1+2^x}\mathrm dx?$$
3
votes
2answers
314 views

A limit question related to the nth derivative of a function

This evening I thought of the following question that isn't related to homework, but it's a question that seems very challenging to me, and I take some interest in it. Let's consider the following ...
15
votes
2answers
457 views

Does existence of anti-derivative imply integrability?

If $f$ has an anti-derivative in $[a,b]$ does it imply that $f$ is Riemann integrable in $[a,b]$?
7
votes
2answers
200 views

Generalized PNT in limit as numbers get large

If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
1
vote
3answers
814 views

Finding the critical points of $\sin(x)/x$ and $\cosh(x^2)$

Could someone help me solve this: What are all critical points of $f(x)=\sin(x)/x$ and $f(x)=\cosh(x^2)$? Mathematica solutions are also accepted.
1
vote
3answers
53 views

Given that $x=\dfrac 1y$, show that $∫\frac {dx}{x \sqrt{(x^2-1)}} = -∫\frac {dy}{\sqrt{1-y^2}}$

Given that $x=\dfrac 1y$, show that $\displaystyle \int \frac 1{x\sqrt{x^2-1}}\,dx = -\int \frac 1{\sqrt{1-y^2}}\,dy$ Have no idea how to prove it. here is a link to wolframalpha showing how to ...
2
votes
4answers
149 views

Show that $f'' > 0$, $\lim_{x \to b^-} = \infty$ implies that $\lim_{x \to b^-} f'(x) = \infty$

Let $f$ be a continuous function on $[a,b)$, $f$ twice differentiable in $(a,b)$ so that $f''(x)>0$ for each $x \in (a,b)$. Prove that if $$\lim_{x\to b-}f(x) =\infty $$ then $$ \lim _{x\to ...
17
votes
3answers
478 views

Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$

I wonder what kind of functions satisfy $$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$ I suppose all functions must be continuous.
2
votes
2answers
117 views

Solve the differential equation $y'=|x|$, $y(-1)=2$

Given the differential equation, $y'=|x|$, $y(-1)=2$ I believe I understand how to solve the implicit solution but have questions about using the initial condition to solve the explicit solution. I ...
0
votes
3answers
281 views

Prove $ \oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V $

I need help proving the following vector calculus identity: $$ \oint_{\partial V} (\mathbf{\hat{n}} \times \mathbf{A}) \; \mathrm{d}S = \int_V (\nabla \times \mathbf{A}) \; \mathrm{d}V $$ the ...
2
votes
2answers
199 views

How can I properly isolate the variables for this differential equation?

I'm attempting to solve a problem involving this differential equation: $$\frac{dy}{dx} = x^2y^2 + x^2 - y^2 - 1$$ Because this is a separable differential equation, I tend to split the equation ...
2
votes
2answers
156 views

Positive twice differential decreasing function, is it convex?

If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
1
vote
3answers
177 views

Please help me for a substitution method to evaluate $\int\frac{dx}{(x+a)^2(x+b)^2}$

Please help me evaluate: $$\int\frac{dx}{(x+a)^2(x+b)^2}$$
2
votes
1answer
132 views

Is there easier way to calculate the limit of this function?

$$ \lim_{K\rightarrow\infty}\frac{(1-\epsilon)^K}{1+(1-\epsilon)^K}\frac{\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ i ...
5
votes
1answer
91 views

$(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$

It seems that $(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$. However, I could not prove this nor the same result for a general concave and increasing function (for which it ...
1
vote
2answers
133 views

Stuck on space curves for vector valued functions

I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point. How would you sketch the graph for the parametric equations: $x = \cos t$, $y = \sin t$, ...
4
votes
4answers
1k views

Evaluate $\lim_{x\to 0} (\ln(1-x)-\sin x)/(1-\cos^2 x)$

I've got this limit: $$\displaystyle\lim_{x\to 0} \frac{\ln(1-x)-\sin x}{1-\cos^2 x}$$ and the problem is that it doesn't exist. But I am not very perceptive and I didn't avoid catching in a trap and ...
2
votes
1answer
396 views

What's the difference between a curve and a graph?

My book states "... if $S$ is a level curve and $C$ is a curve in $S$ passing through a point $a$ ..." What is a curve in $S$? Is it simply a subset of $S$? From what I comprehend, a graph is ...
2
votes
2answers
170 views

Compute $\lim_{n\to\infty} \frac{\int_{0}^{1} f(x) \sin^{2n} (2 \pi x) \space dx}{\int_{0}^{1} e^{x^2}\sin^{2n} (2 \pi x) \space dx}$

Suppose that $f$ is continuous on $[0, 1]$. Then calculate the following limit: $$\lim_{n\to\infty} \frac{\displaystyle\int_{0}^{1} f(x) \sin^{2n} (2 \pi x) \space dx}{\displaystyle\int_{0}^{1} ...
2
votes
3answers
298 views

Given that $\tan^{-1}(x)+\tan^{-1}(y)+\tan^{-1}(xy)=11/12π$, prove that when $x=1, dy/dx=-1-\sqrt{3}/2$

Given that $x$ and $y$ satisfy the equation: $$\arctan(x)+\arctan(y)+\arctan(xy)=11/12π$$ Prove that, when $x=1, dy/dx=-1-\sqrt{3}/2$. I tried to differentiate both sides: ...
2
votes
2answers
73 views

What's $T\left(n\right)$?

If $T\left( n \right) = 8T\left( n-1 \right) - 15T\left( n-2 \right); T\left(1\right) = 1; T\left( 2 \right) = 4$, What's $T\left(n\right)$ ? I use this method: Let $c(T(n) - aT(n-1)) = T(n-1) - ...
4
votes
2answers
152 views

Method to solve $xx'-x=f(t)$

I would like to resolve this differential equation: $xx'-x=f(t)$ any suggestions (or any online texts on similar differential equation) please? Thanks.
4
votes
3answers
647 views

Differentiation from first principles of specific form.

I've been posed a question in which I'm to differentiate with respect to $x$ a function of the form $(x+a)^k$. I've successfully completed (matches the book's answer) the question by using the chain ...
3
votes
1answer
193 views

How to differentiate trace of a quadratic function?

Some basic question about matrix calculus. Let $X$, $A$, $B$ be real matrices. Let $\operatorname{Tr}$ denote trace. Is \begin{equation} \frac{d }{dX} \operatorname{Tr}(X^T A XB) \end{equation} equal ...
1
vote
1answer
130 views

Find $\lim \ a_n$ if $a_n = \frac{1}{\sqrt[3]{n^3+1}} + \frac{1}{\sqrt[3]{n^3+2}}+\cdots+\frac{1}{\sqrt[3]{n^3+n}}.$

I would greatly appreciate some help in finding $\lim \ a_n$ if $$a_n = \frac{1}{\sqrt[3]{n^3+1}} + \frac{1}{\sqrt[3]{n^3+2}}+\cdots+\frac{1}{\sqrt[3]{n^3+n}}.$$
1
vote
1answer
77 views

Limit of the sequence of rational numbers above a given real

I need to prove that for any $a \in\mathbb{R}^+$ the sequence $S[a]_n=a+b_n$, where $b_n = \min\{|{x \over n}-a|\;\colon\;x \in \mathbb{N}\}$, converges to $a$. The only way I know how to prove ...
5
votes
3answers
800 views

Could someone remind me why is incorrect to switch an infinite sum and an integral?

Could someone jog my memory on this? The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum Why is it that $\int ...
1
vote
1answer
224 views

maximum of the function at limit

I have a simple question. Let $$P(\theta;K) = \left(1-\theta\right)^K\left[\frac{1-(1-\theta)^K-\theta^K}{(1-\theta)^K+\theta^K}-\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ ...
1
vote
3answers
525 views

What does “increases in proportion to” mean?

I came across a multiple choice problem where a function $f(x) = \frac{x^2 - 1}{x+1} - x$ is given. One has to choose the statement that is correct about the function. The different statements about ...
2
votes
2answers
102 views

The limit of hazard rate $h(x)=A/(1-B)$ as $x$ approaches $\pm \infty$

Can we tell what happens to the limit as $x$ approaches $\pm \infty$ of a hazard rate $h(x)$ defined for unspecified or generalized density as: $$ h(x)=A/(1-B) $$ where $A=f(x)$ is the density ...
1
vote
1answer
230 views

Finding the equation of a curve that has a perpendicular distance of $d$ from another curve

Let's say we have an equation of a curve as $y = f(x)$. I want to find the curve $y = g(x)$ where $(x_1, f(x_1))$ has a perpendicular distance of $d$ from that curve. Doing this with straight lines ...
2
votes
2answers
149 views

Directional Derivative and Level Sets Orthogonality

My book states without proof that the directional derivative at any point is orthogonal to the tangent to the level set at the same point. I don't even know where to get started. All I contribute ...
0
votes
1answer
83 views

How to solve $a\ddot{u}+b\ \left(\dot{u}\right)^2 +\dot{u}+\dot{u}\ c\ e^{u}+e^u-e^{2u}+1=0$

I already asked a similar question on another post: Solving non linear differential equation. \begin{align} &a \ \ddot{u}+b\ \left(\dot{u}\right)^2 +\dot{u}+\dot{u}\ c\ e^{u}+e^u-e^{2u}+1=0 ...
0
votes
3answers
557 views

Implicit differentiation question

Differentiate given $$\frac{y}{x-y}=x^2+1$$ Initially I wanted to use the quotient rule to solve this, but then I tried differentiating it as it is: $$\frac ...
4
votes
1answer
225 views

Value of $n$ for which an improper integral is convergent.

A question from the Calculus book that I'm self-studying is asking me to determine the value of $n$ for which the improper integral below is convergent: $$\int_1^{+\infty}\left( \frac{n}{x+1} - ...
8
votes
5answers
909 views

Calculate integrals involving gamma function

What are the usual ways to follow in order to solve the integrals given below? $$\begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}$$
1
vote
1answer
753 views

Jacobian physical meaning [duplicate]

Possible Duplicate: What is Jacobian Matrix? Is there any physical intuition for the Jacobian? I understand that it is the matrix of partial derivatives and how to construct it. What I ...
8
votes
5answers
512 views

the sum: $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$ using Riemann Integral and other methods

I need to prove the following: $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+(-1)^{n+1}\frac{1}{n}+\cdots=\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$$ Method 1:) The series $\sum_{n=1}^\infty ...
3
votes
3answers
901 views

Power (Laurent) Series of $\coth(x)$

I need some help to prove that the power series of $\coth x$ is: $$\frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) \ \ \ \ \ $$ I don't know how to derive this, should I divide the expansion of ...
3
votes
2answers
1k views

How to explain what it means to say a function is “defined” on an interval?

I am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval ...
0
votes
2answers
134 views

Largest domain for function

Consider the function $f =\sqrt{xy}$? What is the largest domain for the function? My first instinct was that x and y both be positive reals. But this is not the largest since we have not accounted ...
0
votes
0answers
483 views

Conditions for differentiation under integral sign

I can evaluate the following integral $$ ...
1
vote
1answer
121 views

Help with binomial theorem related proof

I'm currently working through Spivak on my own. I'm stuck on this proof, and the answer key is extremely vague on this problem. I think I'm missing a manipulation involving sums. Prove that ...
1
vote
4answers
1k views

Series convergence, finding values that cause convergence.

I am trying to find the $x$ values that make this series converge: $$\sum_{n = 1}^\infty (x+2)^n.$$ To me it seems like $x = -2$ would make the series converge but that is a wrong answer, I am not ...
0
votes
1answer
89 views

How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?

In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or ...