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Questions tagged [calculus]

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

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1answer
2k views

Difference between function, derivative and second derivative?

If you are looking at three graphs: one is the original function, one is the derivative and the other is the second derivative, what is the accepted way of determining which is which? For example ...
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5answers
18k views

Difference between maximum and minimum?

If I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...
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1answer
143 views

A Newton-like identity

Let $\xi_1, \xi_2, \cdots, \xi_n$ be indeterminates. Define the following indeterminates: $$s_k := \sum\limits_{i=1}^n\xi_i^k, 1\le k <\infty ,$$ $$\sigma_k := \sum\limits_{1\le i_1<i_2<\...
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3answers
404 views

Basic integrals

I have been trying to study for a test on monday but I can't do any of the basic problems. I know what to do but I am just not good enough at math to get the proper answer. I am supposed to use part ...
6
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1answer
299 views

Continuous $f: \mathbb R \to \mathbb R$ with infinitely many zeroes in every interval

Is there a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for any two real numbers $a>b$, $f(x)=0$ has exactly a countable infinite many solutions with $a>x>b$?
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3answers
521 views

A deceiving Taylor series

When we try to expand $$ \begin{align} f:&\mathbb R \to \mathbb R\\ &x \mapsto \begin{cases} \mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\ 0 &\Leftarrow x=0 \end{...
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1answer
91 views

Effect of $k$ on turning point?

In the function $$y=(k-x)e^x ,$$ What is the effect of $k$ on the turning point of the function? I can't see any clear pattern when I change the variable. What are some real-life scenarios to which ...
2
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2answers
10k views

Is $f'$ continuous at $0$ if $f(x)=x^2\sin(1/x)$

Let $f(x)=x^2\sin(1/x)$ for $x≠ 0$ and $f(0)=0$ for $x=0$. Is $f'$ continuous at $0$? My attempt: $f'(x)=2x\sin(1/x)-\cos(1/x)$. Since when $x$ goes to $0$, the limit of $\cos(1/x)$ does not ...
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2answers
856 views

integrate square of $\arctan x$. Tricky

$$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$ I ran across an integral I am having a time solving. The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
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2answers
1k views

$\cos x = kx$, finding $k$ that gives two solutions

$\cos x = 0.3x$ has three solutions. $\cos x = 0.4x$ has one solution. How to find $k$ so that $\cos x = kx$ has two solutions?
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1answer
182 views

How do I remove the polynomial from a fraction?

TLDR: I want to solve function $(-4x^2-6x+4)/(x^2+1)^2$ for $0$. How can I get the polynomial out of the denominator so I can apply the quadratic formula? Long form: I'm trying to find the ...
0
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3answers
86 views

From given equality find that $p$ for which equality have at least one positive root

Having $px^3+(p-3)x^2+(2-p)x=0$ how to find p that this equality have at least 1 positive root ? How can we solve this and similiar things? Because i'm stuck... i did it for quaratic, but i can't ...
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6answers
3k views

Challenging problems in calculus

I'm looking for a textbook (or website etc.) which contains challenging problems in calculus. Problem in Real Analysis by Titu Andreescu is good, but slightly too advanced for me. Spivak's Calculus is ...
2
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3answers
471 views

Limits of sets in relation to their infimum and supremum plus monotonic sequences concept

I have tried to solve the following question but I haven't gotten an answer. Show that: $\lim\ [0,1-1/n] = [0,1)$. $\lim\ [0,1-1/n) = [0,1)$. all the limits having $n\to\infty$.
3
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1answer
921 views

Proving that integration is continuous

Define $C([a,b], \mathbb R)$ to be the space of continuous functions $f : [a,b] \to \mathbb R$ with the norm $\| \cdot \|_{\infty}$. Let $H : C([a, b], \mathbb{R}) \rightarrow \mathbb{R}$ be the map ...
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3answers
8k views

Example of a function that is not twice differentiable

Give an example of a function f that is defined in a neighborhood of a s.t. $\lim_{h\to 0}(f(a+h)+f(a-h)-2f(a))/h^2$ exists, but is not twice differentiable. Note: this follows a problem where I ...
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1answer
1k views

Prove $\lim\limits_{x→∞} f''(x) = 0$

If $\lim\limits_{x→∞} f(x)$ and $\lim\limits_{x→∞} f''(x)$ both exist, then $\lim\limits_{x→∞} f''(x) = 0.$ You may use the fact that $\lim\limits_{x→∞} f(x)$and $\lim\limits_{x→∞} f'(x)$ both exist, ...
3
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2answers
128 views

Evaluating integrals

I am having trouble figuring out an algebraic trick to make this work Evaluate the integral $$\int_1^9\frac{x-1}{\sqrt{x}}dx$$ I know I can turn the integrand into $(x-1) (1/\sqrt{x})$ but I ...
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1answer
621 views

Test review: intervals of increasing/decrease and L'Hôpital's rule

I am not sure what I am doing wrong, but I just got this whole test wrong except one question. Anyways I was having trouble with 1) Find the intervals on which $f(x)=x-2\cos x$, $0\leq x \leq 2\...
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2answers
2k views

Integrating a product of exponentials and error functions

I have the following integral $$ \int\limits_0^\infty x^2\exp(-\delta x^2)\operatorname{erf}(\gamma x)\,dx. $$ Ideally, I would like a closed-form in terms of common functions, but a series answer ...
150
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6answers
168k views

Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$

In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate ...
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2answers
152 views

Stuck on question with asymptotes

Ive been stuck on this quest from textbook and I can't find an answer for it.. Question is, what may the equation of a rational function with below conditions look like? Explain how you got the ...
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1answer
92 views

Inverse and Derivative of $g'(x)=(1+x^{3})^{-1/2}$ Question

I am stuck on the following question: Suppose that $g$ is differentiable with derivative $g'(x)=(1+x^{3})^{-1/2}$. Show that the inverse function $h=g^{-1}$ satisfies $h''(x)=\frac{...
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1answer
250 views

Polynomial Function and $n$-times differentiable

If a function $f$ is $n$-times differentiable on $\mathbb R$ and $f^{(n)}=0$, prove $f$ is a polynomial of degree $\leq n-1$. A hint would suffice.
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1answer
1k views

Tangent Lines and Implicit Function

I am asked to find the equations of the tangent lines at three different points for the following function: $$y^{5}-y-x^{2}=-1$$ I am provided with a graph/sketch of the function and asked to find ...
5
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4answers
3k views

Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?

why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just ...
4
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3answers
84 views

Nice way of showing the following equality?

Is there a good way of showing that ${d^n\over dx^n}(x^2-1)^n|_{x=1}=2^nn!$? I have tried  binomial expanding the thing then differntiate term-by-term, which seems a bit clumsy. Perhaps there's a ...
7
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1answer
679 views

Does the Wronskian have anything to do with the product rule in calculus

Does the Wronskian have anything to do with the product rule in calculus. I ask this because i noticed the form looking similar to the product rule. $$W=g(x)f'(x)-g'(x)f(x)$$ where as the ...
3
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1answer
283 views

Sine and cosine series

I'm trying to prove that $|\sin(x)| \le 1$, $|\cos(x)| \le 1$ and $|\sin(x)| \le |x|$ for all $x \in \mathbb{R}$ using the power series of sine and cosine : $$\begin{align*} \sin(x) &= \sum_{k=...
5
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3answers
325 views

Sum of the series $\sum_{n=1}^\infty \frac{(-1)^n}{n2^{n+1}}$

How do I calculate the sum of this series (studying for a test, not homework)? $$\sum_{n=1}^\infty \frac{(-1)^n}{n2^{n+1}}$$
2
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1answer
85 views

monotonicity of a function which is right differentiable

If $f$ is continuous on $[a,b]$ and $f\;'> 0$ on $(a,b)$, then $f$ is monotonically increasing on $[a,b]$ and proof follows from mean value theorem. Now suppose we have $f \in C[a,b]$ and $f\;&#...
2
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1answer
799 views

The quotient of two functions

If $f$ and $g$ are two smooth functions in $\mathbb R^n$ such that if ${\partial ^\alpha }f(x)=0$ for arbitrary index $\alpha$ and $x \in \mathbb R^n$, then ${\partial ^\alpha }g(x) = 0$. Then is ...
0
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2answers
374 views

Supremum and infimum for a sequence

I came across this statement while I was reading some topic: we have a sequence $\{ a_n \}$ of real numbers, such that $|f(x)|\geq \frac{1}{a_n}$ for all $n\in \mathbb Z$, then " if $\inf a_{n}=a>...
6
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3answers
2k views

Forced wave equation

If a system satisfies the equation $$\nu^2 {\partial^2 \psi\over \partial x^2}={\partial^2 \psi\over \partial t^2}+a{\partial \psi\over \partial t}-b\sin\left({\pi x \over L}\right)\cos\left({\pi \nu ...
1
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2answers
112 views

Slope at $-\infty$

Given a twice differentiable function $f(x)$ on $\mathbb R$ with the following properties: $f$ is an increasing function in $\mathbb R$ There is a sequence of real numbers $\{x_{n}\}_{n=-\infty}^{n=...
2
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0answers
124 views

How to solve this equation

How do I solve the equation given below for values for $\epsilon$? $$ \sum_{j=1}^{n-1} (2{\alpha_{j}}+1) ~~ \prod_{k=1,~k\neq j}^{n-1} \bigg(({\alpha_{k}}-1){\Big((2{\alpha_{k}}+3) + {\epsilon}(2{\...
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2answers
5k views

Using Intermediate Value Theorem to prove # of polynomial roots

I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots. I'm not in school ...
5
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3answers
82 views

quotient rule difficulties

I'm trying to use the quotient rule to differentiate $\frac{r}{\sqrt{r^2+1}}$ but I'm getting the wrong answer. So far I have $$\begin{align*} \frac {d}{dr} \frac{r}{\sqrt{r^2+1}} &= \frac {\...
12
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2answers
7k views

Sufficient and necessary condition for strictly monotone differentiable functions

I read from a book without proof the following theorem: Let $f(x)$ be a differentiable function, then $f(x)$ is strictly increasing if and only if $f'(x) \geq 0$, and $f'(x) \gt 0$ almost ...
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1answer
2k views

A question about the interval of convergence for alternating series

Say we are given the simple power series $$\sum_{i=0}^{\infty}(-1)^k\frac{(x-4)^k}{2^k}$$ The interval of convergence can easily be shown to be $x\in(2,6)$ using the Root Test, and, since absolute ...
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1answer
3k views

Midpoints and Riemann sums

I am trying to figure out these questions and I think I understand how this complicated formula works. I do not have it memorized but I do have it written down for reference. If $f(x)=e^x-2$, $0\le ...
0
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1answer
4k views

Speed of a runner

I am trying to do this homework problem for calculus. It is an intro to integrals and I have no idea what I am doing wrong. The speed of a runner is increase steadily during the first $3$ seconds ...
0
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2answers
277 views

Behavior of $\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/n^{0.5-\epsilon})}}{2a+b/n^{0.5-\epsilon}}\right)^{\frac{n}{2}}$

I am having trouble expressing the behavior of the following limit: $$\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/n^{0.5-\epsilon})}}{2a+b/n^{0.5-\epsilon}}\right)^{\frac{n}{2}}$$ After some ...
6
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1answer
309 views

How to prove that $ f(x) = \sum_{k=1}^\infty \frac{\sin((k + 1)!\;x )}{k!}$ is nowhere differentiable

This function is continuous, it follows by M-Weierstrass Test. But proving non-differentiability, I think it's too hard. Does someone know how can I prove this? Or at least have a paper with the proof?...
2
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1answer
162 views

Are there times when convergence tests contradict each other?

In my Calc II class, we're just starting convergence tests and all the examples are very convinient and they work perfectly (obviously, since they are examples), but my professor couldn't really ...
3
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1answer
4k views

Explanation: Volume of a trapezium

A trough is filled with water at a rate of 1 cubic meter per second. The trough has a trapezoidal cross section with the lower base of length half a meter and one meter sides opening outwards at an ...
2
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3answers
784 views

Plotting $\frac{1}{\ln x}$

I need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?
1
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2answers
185 views

If $\int_V f \;dV = 0$ when can we say that $f=0$ everywhere

If $\int\limits_V f \; \mathrm dV = 0$ can we say that $f=0$ everywhere? Or what conditions are there on concluding this. In particular I want to solve the PDE $\nabla^2 f=f^3$ on the region $$D=\{(x,...
3
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2answers
6k views

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
11
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2answers
307 views

How to prove that $\lim\limits_{h \to 0} \frac{a^h - 1}{h} = \ln a$

In order to find the derivative of a exponential function, on its general form $a^x$ by the definition, I used limits. $\begin{align*} \frac{d}{dx} a^x & = \lim_{h \to 0} \left [ \frac{a^{x+h}-a^...