Questions tagged [calculus]

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

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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}{...
1
vote
1answer
66 views

Derivative Line Equation question

I am not sure how to go about solving this problem: I know the derivative of $f$ would be $2x$ but I am not sure where to go from there. If anyone could help out that would be great. Thanks! Edit: ...
2
votes
3answers
309 views

Issue with textbook exercise on vectors

the following is a question from my textbook on vectors: EDIT: Added text, so that the post is self-contained even without the picture. The points $A$ and $B$ have position vectors $\begin{pmatrix}2\...
4
votes
1answer
274 views

How do you prove $| \int (f)| \leq \int (|f|)$?

For an integrable function $f$ on $(a,b)$, how would you prove $| \int (f)| \leq \int (|f|)?$
2
votes
2answers
875 views

Limits of function compositions

Is it possible to evaluate limits involving sequences of function compositions? For example, given the expression $$g(x, n) = \sin(x)_1 \circ \sin(x)_2 \circ [...] \circ \sin(x)_n$$ is it possible ...
1
vote
2answers
145 views

$p_n = 1- \left( 1-\frac{1}{365}\right)\left( 1-\frac{2}{365}\right)\cdots \left( 1-\frac{n-1}{365}\right)$ then $p_n>\frac{1}{2}$ for $n>?$

$$p_n = 1- \left( 1-\frac{1}{365}\right)\left( 1-\frac{2}{365}\right)\cdots \left( 1-\frac{n-1}{365}\right)$$ Then $p_n>\frac{1}{2}$ for $n>?$ This occured in a probability problem. The result ...
1
vote
3answers
587 views

Expanded concept of elementary function?

After searching about why $\int e^{x^2}$ is not an elementary function, I was disappointed that I should understand about Galois theory, but then I started to think about a concept that treats ...
0
votes
2answers
313 views

Can the derivative be defined game-theoretically?

It is often said, by way of intuitive explanation, that the derivative of a function at a point is the slope of the line that “best fits” the function through that point. Can this be pressed into a ...
5
votes
2answers
231 views

Asking for general form of Integral Inequality of this kind

Let $f\in C^1[0,a]$ and $f(0)=0$. Is it true that $$\int_0^a \left(\sqrt{x}f(x)\right)^{\prime} \left(\frac{f(x)}{\sqrt{x}}\right)^{\prime}\, dx\geq 0\;\;?$$ What is the general form of this type ...
16
votes
7answers
17k views

Proof of $ f(x) = (e^x-1)/x = 1 \text{ as } x\to 0$ using epsilon-delta definition of a limit

I am in calc 1 and we have just learned the epsilon-delta definition of a limit and I (on my own) wanted to try and use this methodology in order to prove $(e^x-1)/x = 1$ (one of the equivalencies), ...
2
votes
0answers
258 views

Convexity of a Set

Consider the following function, $$ f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right) $$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is quasi-...
6
votes
1answer
523 views

Area vs Volume Paradox [duplicate]

Possible Duplicate: How can a structure have infinite length and infinite surface area, but have finite volume? Hi, I have this question that I quite cant explain why. So the area under the ...
11
votes
2answers
380 views

Is $f(2x)/f(x)$ nonincreasing for concave functions with $f(0)=0$?

I have a question about concave functions. Let $f:R_+\rightarrow R_+$ be any nonidentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that the ratio function $f(2x)/...
4
votes
3answers
651 views

What is $\lim\limits_{n \to \infty}\sum\limits_{k=1}^n \frac{1}{k}$ [duplicate]

Possible Duplicate: Why does 1/x diverge? I'm a math tutor. This is a high school level problem. I'm unable to solve this. What is the value of: $\lim\limits_{n \to \infty}\sum\limits_{k=1}^n \...
5
votes
1answer
143 views

Sum of derivative of integrals

For all $x$ in $\mathbb R$ define $\displaystyle f(x)=\left(\int_0 ^{x} e^{-t^2}dt\right)^2$ and $\displaystyle g(x)=\int_{0}^{1}\frac{e^{-x^2(t^2+1)}}{t^2+1}dt$. Show that for all $x$ in $\mathbb R$ ...
4
votes
3answers
4k views

Power series of $\ln(x+\sqrt{1+x^2})$ without Taylor

The answer is $$x-\frac{ 1}{2}\frac{ x^3}{3}+\frac{ 1\cdot 3}{2\cdot 4}\frac{ x^5}{5}-\frac{ 1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{ x^7}{7}+\cdots$$ But I can't see how. Unfortunately, "how" can't be ...
41
votes
6answers
32k views

Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist

I've been trying to solve the following problem: Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
3
votes
2answers
104 views

Derivative of $x^2e^{-x(y+c)}$

What's the derivative of this function on x $x^2e^{-x(y+c)}$ And why? I'm not sure about the result I have on the book y and c are both constants
4
votes
1answer
139 views

Limit of the functions of two variables at $\infty$

Does the following equality generally hold? $$ \lim_{x\to\infty, y\to\infty} f(x, y) = \lim_{z\to\infty} f(z, z) $$ If not, what are the necessary conditions for the above equation to hold?
5
votes
1answer
164 views

Continuity: Does this need to be proven?

Given a function $f(x) = (x-2)(x-3)(x-4)(x-5) + 1$, I am asked to show that $f'(x) = 0$ has exactly three distinct roots. This is simple enough, it's done with Rolle's theorem: $f(2) = f(3) = f(4) ...
14
votes
1answer
743 views

Is this condition sufficient to ensure monotonicity of a function?

Suppose $f:[a,b]\to\mathbb{R}$ is continuous and $$\limsup_{h\to0}\frac{f(x+h)-f(x)}{h}\geq0$$ for every $x\in(a,b)$. Does it follow that $f$ increases monotonically on $[a,b]$? It is a problem in ...
10
votes
2answers
534 views

Derivative of $f(x) = (x+x)$

I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my ...
3
votes
1answer
182 views

questions about limits and derivatives

I am trying to solve a set of problems, this one is causing my some troubles. For the first one I tried to use the $\epsilon-\delta$ definition but I couldn't solve it, I would appreciate some hints ...
3
votes
2answers
319 views

Calculating derivative using definition for $f(x)=\frac{x - \sin x}{x^2}$

Really stuck on this one.... $\displaystyle f(x) = \frac{x - \sin{x}}{x^{2}}$ for $x \neq 0$ and $0$ when $x = 0$ Using the definition of the derivative, find $f'(0)$ I know the definition is $...
2
votes
2answers
353 views

Integrals Converging

There always seems to be a question about whether or not an integral converges. Can I ask what the best mental method is to pick the right test/process to calculate the integral? Let’s take an example....
5
votes
1answer
208 views

How to Express $\Gamma (x)$ in terms of $\cos$

I'm trying to figure out how to express the integral $$\int\limits_{0}^{\infty} \cos(x) \times x^{-a} \rm{dx}$$ as $$\cos\frac{\pi-a\pi}{2}\times \int\limits_{0}^{\infty} e^{-x} x^{-a} \ \rm{dx} \...
1
vote
0answers
153 views

Change of variables of inverse Jacobi multiplier

I have got an Inverse Jacobi multiplier $$M=x^{3}z-3(1+kz)4-2\frac{c}{k}$$ and I have a change of variable $$Z=\frac{z(1+kz)c}{3k}-\frac{2}{3}$$ and I want to use this theorem: Let $M$ be an inverse ...
36
votes
2answers
4k views

Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility

I was told by a tutor that if $f: \mathbb{R}^n \longrightarrow \mathbb{R}^n$ has an invertible Jacobian Matrix for all $x \in \mathbb{R}^n$ and $\lim_{|x_k| \rightarrow \infty}|f(x_k)|=\infty$ for all ...
5
votes
4answers
1k views

Limit of monotonic functions at infinity

I understand that if a function is monotonic then the limit at infinity is either $\infty$,a finite number or $-\infty$. If I know the derivative is bigger than $0$ for every $x$ in $[0, \infty)$ then ...
3
votes
2answers
334 views

What is the formal definition of $d$, or $\partial$, in differation and integration

This might sound a bit like a silly question, but i'm a second year math student, and so far i've encountered $d$ or $\partial$ in many cases ofcourse (mostly in calculus :)). Those letters or symbols ...
10
votes
2answers
13k views

$\Delta x$ in limit problem?

I was working on some limit homework and everything was going fine until I reached this problem: $$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$ I am understanding limits but the ...
2
votes
2answers
260 views

Stumped by this Maclaurin series question

It says to find the Maclaurin series in sigma form and the interval on which it converges for $$\dfrac{7x^4}{2+3x^2} \qquad \text{if} \qquad \frac{1}{1-x} = \sum_{k=0}^{\infty}\, x^k, \;\; -1 \lt x \...
1
vote
1answer
294 views

Finding Lagrange's form of the remainder with $a = \pi/2$ and $n\to\infty$

The function is $$f(x) = 2\cos x. $$ Again, $a = \pi/2$ and $n \to\infty$. I was to do this for $n = 3$ as well and I had no problems doing that at all, but I'm confused on how you do it for for $n \...
6
votes
1answer
807 views

Calculating arc length of a curve, stuck on dy/dx part (algebra mostly)

The equation is: $$x=\frac{1}{8}y^4 + \frac{1}{4}y^{-2},\qquad 1\leq y\leq 2.$$ I have the formula. I'm not sure how to write it out but this is what it says: Length is equal to the integral (with $...
20
votes
5answers
4k views

The sum of $(-1)^n \frac{\ln n}{n}$

I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$ This is a problem in Calculus by Simmons. It's in the end of chapter review and it's ...
1
vote
2answers
338 views

Trying to plot these points in a polar coordinate system

I started with: inside $r_1=5 \sin(θ)$ and outside $r_2=2+\sin(θ)$ and was told to sketch curve in the same polar coordinate system I first set both equal to $0$ and solved to get $\pi$, $2\pi$, and ...
11
votes
6answers
3k views

How do I find roots of a single-variate polynomials whose integers coefficients are symmetric wrt their respective powers

Given a polynomial such as $X^4 + 4X^3 + 6X^2 + 4X + 1,$ where the coefficients are symmetrical, I know there's a trick to quickly find the zeros. Could someone please refresh my memory?
2
votes
2answers
393 views

Two ways to find the derivative, which is correct?

I was asked to do this problem $\displaystyle \frac{d}{dx} |3x-x^2|=\frac{d}{dx} y$ I used the fact that $\displaystyle \frac{d}{dx} |x|= \frac{|x|}{x}$ so, $\displaystyle \frac{|3x-x^2|(3-2x)}{3x-x^...
4
votes
1answer
2k views

How to integrate by parts in spherical coordinates

I'm running into some troubles with the integration of a spherically symmetric 3D function. I'm having the following expression to evaluate : $$ I=\int_0^{2\pi} d\phi \int_0^\pi \sin\theta d\theta\...
5
votes
3answers
283 views

An elementary integral inequality

I was playing around with some integrals and decided that $$ a\int_a^b t^{n−2}\alpha(t) dt \le \int_a^b t^{n−1}\alpha(t) dt $$ for $a,b>0$, $\alpha \ge 0$, $\alpha$ nice enough so that the ...
6
votes
4answers
923 views

Problem: Sum of absolute values of polynomial roots

Can you please give me some hints as to how I might approach this problem? Thanks! Given the polynomial $f = 2X^3 - aX^2 - aX + 2, a \mathbb \in R$ and roots $x_1, x_2$ and $x_3,$ find $a$ such ...
2
votes
2answers
728 views

Did I sketch this polar curve correctly?

The equation is: $r^2=-4 \sin(2\theta)$ I first made a reference graph in cartesian coordinates using values $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{3 \pi}{...
4
votes
2answers
1k views

Unbounded second derivative of continuous bounded function

Does a continuous function $f$ exist where: $f$ is continuous, $f$ is known to be bounded with a codomain of $\left[0,1\right]$ $f''$ (second derivative) is unbounded with a codomain of $(-\infty,+\...
1
vote
2answers
330 views

Sketching a polar curve

Continued off the question I asked earlier, I also have to sketch the curve. $r^2=−4\sin(2\theta)$ So I have to set up a table of values I'm assuming. How do I know what values to choose for $\theta$...
4
votes
3answers
17k views

How do I explain why $dA/dr = 2 \pi r$ geometrically?

There's this question in my calculus book that goes something like this: The derivative of the area of a circle with respect to its radius is equal to the circle's circumference ($dA/dr = 2 \pi r$). ...
1
vote
2answers
419 views

How to solve a polar equation when $r$ is $r^2$ instead?

I have $r^2=-4\sinθ$ and I'm asked to set $r=0$, then find θ. If I just set $r^2=0$ then I'll get $\sin(2θ)=0$. That doesn't seem right. Then I'm asked to set $θ=0$ and then find $r$. If I use the $...
0
votes
2answers
630 views

Double Integrals

$(a)$ Sketch the region of integration in the integral $$\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} dx dy$$ By changing the order of integration, or otherwise, evaluate the ...
5
votes
3answers
252 views

What are a , b and c?

$$y = ax^2 + bx + c$$ which is tangent at the origin with the line $y=x$, It is also tangential with the line $y=2x + 3$. Determine the function! Draw a figure! My main question is this solvable? I ...