# Questions tagged [calculus]

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

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### Evaluation of the limit, $\lim \limits_{x\rightarrow\infty} \left(\frac{20x}{20x+4}\right)^{8x}$, using only elementary methods

I was assisting a TA for an introductory calculus class with the following limit, $$\lim_{x \rightarrow \infty} \left(\frac{20x}{20x+4}\right)^{8x}$$ and I came to simple solution which involved ...
1answer
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### Traveling up and down a mountain

I am reminded of this question that appeared in a regional Physics Olympiad I had appeared. I was wondering if there is a "mathematical" way of doing it. If you start from a point $A$ at midnight ...
2answers
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### Higher derivatives of an exponential function

Let $$p_n(x)e^{-x^2}$$ be the $n$th derivative of $$e^{-x^2}.$$ Find a formula for $p_n(x)$. We have $p_1(x)=-2x, p_2(x)=4x^2-2$, etc. But what is the general formula for $p_n$?
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### Difference between one-variable calculus and multi-variable calculus?

Is the essential difference between one-variable calculus and multi-variable calculus exterior differential or something else?
5answers
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### Differentiable function - Easy Calc

So I am studying for my calculus midterm and I had the following question: If $f_{x}(a, b)$ and $f_{y}(a,b)$ both exist then $f$ is differentiable at $(a,b)$ I answered true, but the answer is ...
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### Differential Equations: Calculating Barometric Pressure at an Altitude

I'm trying to solve a differential equation problem that has to do with barometric pressure at a certain altitude. The rate at which barometric pressure decreases with altitude is proportional to the ...
3answers
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### Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$

If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots =$ ? My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series (...
2answers
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### For every $y\in \mathbb{R}$ there is some $c$ in $(a,b)$ with $f'(c) = yf(c)$

Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous and has a finite derivative $f'$ everywhere on $(a,b)$. If $f(a)=f(b)=0$ prove that for every $y\in\mathbb{R}$ there is some $c$ in $(a,b)$ ...
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### Calculating The Slope Of A Line At A Specific Point

Let's say I have an "S" shaped curve made up of 50 points. I am only given the data points, and there is no simple or obvious equation for each point on the curve. So given only the data points, ...
9answers
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### How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?

Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2.$$ Thank you
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### How to solve this differential equation? $y''-\frac{y'}{x}=4x^{2}y$

How to solve $y''-\displaystyle\frac{y'}{x}=4x^{2}y$ ? I know that the solution of this equation is: $y = e^{x^{2}}$, but I cannot resolve. First I thought that $z=y'$ could be, ...
3answers
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### how to find bounds of the series $\sum_{i=1}^n \frac1{4i-1}$

$$\sum_{i=1}^n \frac1{4i-1}$$ I know I have to integrate the function but from what to find lower and upper bound.
4answers
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### Another limit task, x over ln x, L'Hôpital won't do it?

$$\lim_{x\to 0 } \frac{x}{\ln x}.$$ This was wrong, I got a big red wrong! Why doesn't L'Hôpital work on this one? The problem is that $\ln$ is not defined for 0. It needs to be rewritten? (Thanks ...
3answers
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### Is there a continuous Taylor/MacLaurin transform (like the Fourier transform)?

My consideration might be total nonsense (as a high school student, I lack the mathematical knowledge to really check my idea), but I was just wondering whether one could find a continuous ...
4answers
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### How can I show the sequence $a_{n+1}=\frac{4a_{n}+2}{a_{n}+3}$ is bounded for every $a_{1}\in \mathbb{R}$?

Investigating this sequence led me to the following conclusions: if $a_{1}\geq 2$ then $a_{n}\geq 2$ and $a_{n}$ is decreasing, thus it converges and is bounded. if $-1\lt a_{1} \lt 2$ then $a_{n}$ ...
1answer
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### Hydrostatic Force and Integration

A homework problem states: An aquarium 5 ft long, 2ft wide, and 3ft deep is full of water. Part C: find the hydrostatic force on one end of the aquarium. Questions (I've completed part a and b. B ...
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### Another limit task, I've multiplied by the conjugate, now what?

The task: $\lim_{x\to\infty} \sqrt{x^2+1} -x$ I've multiplied with the conjugate expression ($\sqrt{x^2+1} +x$), then I get this $\lim_{x\to\infty} \frac{1}{\sqrt{x^2+1} +x}$ Is this correct ...
6answers
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### Calculus, find the limit, Exp vs Power?

$\lim_{x\to\infty} \frac{e^x}{x^n}$ n is any natural number. Using L'hopital doesn't make much sense to me. I did find this in the book: "In a struggle between a power and an exp, the exp wins." ...
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### Integral with Tanh: $\int_{0}^{b} \tanh(x)/x \mathrm{d} x$ . What would be the solution when 'b' does not tends to infinity though a large one?

two integrals that got my attention because I really don't know how to solve them. They are a solution to the CDW equation below critical temperature of a 1D strongly correlated electron-phonon system....
3answers
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### When do the Freshman's dream product and quotient rules for differentiation hold?

This is motivated by looking at the calculus exams of some of my undergraduate students. A recurring mistake is assuming that the derivative of the product of functions is a product of derivatives and ...
1answer
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### Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
2answers
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### Easy; understanding question about implicit diff?

If you have a function, implicitly defined, how would understand the following; Assume $y=y(x)$, find $y'(0)$ and $y''(0)$. After I implicitly derivate it the first time, I have an ...
2answers
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### Help seeing algebra in calculus

I'm going through an examples section (on improper integrals) but I got lost at this bit: $$\lim_{t\to-\infty} \frac{t}{e^{-t}} = \lim_{t\to-\infty}\frac{1}{-e^{-t}}.$$ I think it's a simple ...
1answer
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### Does the series $\sum\limits_{n=1}^{\infty}\frac{\sin(n-\sqrt{n^2+n})}{n}$ converge?

I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it'...
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### Tractrix-like curves

Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line? What is a parametric equation of such ...
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### $\prod_{k=3}^{\infty} 1 - \tan( \pi/2^k)^4$

So I found this $$\prod_{k=3}^{\infty} 1 - \tan( \pi/2^k)^4$$ here. I have only ever done tests for convergence of infinite sums. At this link it shows a way to convert but in this case an is less ...
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### Intuition behind the ILATE rule

Often I have wondered about this question, but today I had a chance to recollect it and hence I am posting it here. During high-school days one generally learns Integration and I still loving doing ...
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### if $r\in (0,1), 1-r$ are the only sub limits of $a_{n}$ then $f(a_{n})$ converges when

This is actually a generalized version I wrote of a homework question that intrigued me: Let $f$ be continuous in $[0,1]$ and $\forall x\in [0,1], \ f(x)=f(1-x)$. If $r\in [0,1]$ and $1-r$ are ...
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### Interesting calculus problems of medium difficulty?

I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could ...
4answers
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### $(1-\frac{x}{n})^n\lt \exp(-x)$

Why is the following inequality true: if $x \geq 0$ then $(1-\frac{x}{n})^{n} \leq e^{-x}$ ? here $n$ is a positive integer. Is there a quick way to see this?
2answers
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### How to find a closed form solution to $\iint_{S}[[x+y]] dA$

I'm trying to find a general solution for $\displaystyle\iint_{S}[[x+y]] dA$ where $S=[a,b] \times [c,d]$ and $[[x]]$ is the greatest integer function. I know that if $a,b,c,d \in \mathbb{Z}$, then ...
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### Reference for matrix calculus

Could someone provide a good reference for learning matrix calculus? I've recently moved to a more engineering-oriented field where it's commonly used and don't have much experience with it.
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### Convergence/Divergence of $\sum_{n=1}^{\infty} \sin(1/n)$

it is a question Convergence/Divergence of calculus II! Please give me a hand! Determine convergence or divergence using any method covered so far. $$\sum_{n = 1}^{\infty} \sin (1/n)$$
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### Parametric Equation Question

Ok this is a really silly question and I should know this, but I can't seem to figure something out: for the last step, how do they know that $0 \leq x \leq 4$? If we use the minimum value of theta, ...
7answers
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### Applications of the Mean Value Theorem

What are some interesting applications of the Mean Value Theorem for derivatives (both the 'extended' or 'non-extended' versions as seen here are of interest)? So far I've seen some trivial ...