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

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1answer
91 views

Find the limit using L'Hopital's rule?

Find $$\lim _{x\to 0}\frac{1-\cos x}{x+x^2}$$ Give your answer as an exact number. So I did this and got .5 as my answer. What does it mean by "exact number"? .5 was not a correct answer so I am ...
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4answers
137 views

How does one work backwards from an infinite series to a function?

How does one go about working backwards from an infinite series to a function? For example, how would one determine what function this infinite series represents: $$C + \frac{C}{x} + \frac{C}{x^2} + ...
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1answer
4k views

Using triple integral to find the volume of a sphere with cylindrical coordinates

I'm reviewing for my Calculus 3 midterm, and one of the practice problems I'm going over asks to find the volume of the below solid 1. by using a triple integral with spherical coordinates, and 2. by ...
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2answers
59 views

Differentiation of logarithmic functions

I'm going over my homework from calc, and am having some trouble with a few questions. It seems as if I'm just not understanding how to solve the problem: $$f(x) = \sqrt[5]{\ln x}$$ $$f(x) = ...
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2answers
321 views

Finding the critical point and local extreme value?

How would I find the critical point of the following function. $f(x)=x^{2/3}+2x^{-1/3}$ This is what I did. $\frac{2}{3}x^{-1/3}-\frac{2}{3}x{^\frac{-4}{3}}$ ...
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2answers
122 views

Partial Fraction Decomposition $\frac{2x^2+3x+3}{(x+1)(x^2+1)}$

How would one go about decomposing this fraction? $$ 2x^2+3x+3\over (x+1)(x^2+1) $$ Here is what I have so far: $$ {2x^2+3x+3\over (x+1)(x^2+1) } = { A\over x+1 } + { Bx+C\over x^2+1} $$ $$ ...
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4answers
95 views

Simple analytic proof.

If asked to prove that $$e^x>1+x: x>0$$ Can I argue that $$\lim_{x\rightarrow0}\frac{e^x-1}{x}=1$$ and this limit is approached from right side. However, am not confident how I justify it ...
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3answers
285 views

Derivative of $x^2\sqrt{1+x}$

Given that $f(x)=x^2\sqrt{1+x}$, show that $f'(x)=\dfrac{x(ax+b)}{2\sqrt{1+x}}$ where $a$ and $b$ are constants to be found. I first tried using the product rule: ...
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3answers
329 views

Formula to limit a number within a minimum and maximum value

I'm looking for a formula that can limit a specific input within a specific range. I want to limit a number within 0 and 100. Example: Input: -100 / Output: 0 Input: -1 / Output: 0 Input: 0 / ...
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1answer
111 views

Solving for x with radicals and negative exponents

How do I go about solving for $x$ in this equation? $$\displaystyle -x^{-\large\frac{3}{4}} + \frac{15^{\large\frac{1}{4}}}{15} = 0$$
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2answers
141 views

How to evaluate this double integral $\int_0^1 \int_0^1 \cos (n\pi (x+y))\text{d}x\text{d}y$?

Evaluate : $$\int_0^1 \int_0^1 \cos (n\pi (x+y))\text{d}x\text{d}y$$ Can someone please point me in the correct direction? Thanks
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2answers
108 views

How to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$

I'm having troubles to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$ I've tried a lot using so many techniques without success. I found $\int_{-1}^1 \frac{x^{2012}}{1+e^x}dx=1/2013$, ...
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5answers
95 views

Need help evaluating the definite integral.

$$ \int_0^6 \frac{dx}{x^2+36} $$ This question is killing me. Ive done it 5 different times and not one answer was right.
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3answers
130 views

Solving $f=(f^2)'$

Find all differentiable mappings $f:\mathbb{R}\to \mathbb{R}$ so that $f=(f^2)'=2ff'$. My problem is that $f$ may very well be $0$ at some points ($f=0$ is for example a solution and so is $\frac12x$) ...
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2answers
174 views

Differentiation question?

How would I solve the following problem? Where would the function $|2x-1|$ not be differentiable? I am thinking it would not be differentiable at $x=1/2$ because there it would be zero.
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3answers
85 views

limit of $\frac{e^{2x}-x^2+x}{\cos(x)-1}$ as $x\to 0$.

The title said it, what is: $$ \lim_{x\to 0} \frac{e^{2x}-x^2+x}{\cos(x)-1} = ~? $$ If I evaluate the term I get $1/0$, by looking at a graph I see that it goes to $-\infty$, but I don't now how ...
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2answers
527 views

Trig limit question involving cot

How would I solve the following trig limit $$\lim_{x\rightarrow 0} \frac{4x}{\cot 3x}$$ This is what I did $$\frac{4}{3} \frac{\sin3x}{\cos3x}$$ $$\frac{4}{3} (\frac{\sin3x}{3x}) ...
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1answer
101 views

Is $f(x)=|x\sin x|$ positive and not bounded function

Can I say that the function $f(x)=|x\sin x|$ is positive not bounded and doesn't have a point that converges to infinity? Thank you!
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2answers
148 views

Trigonometry limits

How would I solve the following limits? As $$\lim_{x\to 0}x^2(1+ \cot^2{3x})$$ I know $\cot^2(3x)$ is $\displaystyle\frac{\cos^2{3x}}{\sin^2{3x}}$ but I am not sure how to solve it. The second ...
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2answers
236 views

Some integral with sine [closed]

$$\begin{align} & \int_{0}^{+\infty }{\frac{\sin px}{1+{{\text{e}}^{qx}}}}\text{d}x ,\ \ p,\ q>0\\ \\ \\ & \int_{0}^{+\infty }{{{\left( \frac{\sin x}{x} \right)}^{n}}\text{d}x} \\ ...
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3answers
2k views

Area of the intersection of a cone and a sphere

Calculate the area of the surface $x^2+y^2 = z^2 $ with $z \geq 0$, limited by $x^2 +y^2+z^2=2ax$. I think the method to solve it is to calculate the parametric equation of the curve and then ...
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2answers
172 views

What is the sum of this power series?

This is the series: $$ \sum_{n=1}^\infty \frac{x^{2n}}{(2+\sqrt{2})^n} $$ My problem is that I don't know how to rid of that $ (2+\sqrt{2})^{-n} $.
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2answers
2k views

Integral $ \int \frac{\operatorname d\!x}{\sin^3 x} $ [duplicate]

Possible Duplicate: Integral $\int \frac {1}{\sin^3(x)} dx$ Can someone help me compute $$\int \frac {1}{\sin^3 x } dx $$ Thanks !
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2answers
395 views

Distinct natural numbers such that $ab=cd=a+b+c+d-3$

Find the distinct natural numbers $a,b​​,c,d$ who satisfying $ab=cd=a+b+c+d-3$.
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2answers
94 views

finding a limit of a function by definition

calculate the limit of the following functions and prove directly from definition (using $\epsilon$ and $\delta$) a) $$\lim _{x\to 2}(x^3+3x)$$ We just learned the definition of limits in ...
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2answers
192 views

Show that $g$ is continuous at $(0,0)$.

Let $g:\mathbb{R^2}\rightarrow \mathbb{R} $ so that, in $M=[0,1]\times[0,1]$, $$g(x,y)=\begin{cases}\frac{x^2+y^2}{x+y} &\text{ if }x+y \neq 0,\\\\ 0&\text{ if }x+y=0\end{cases}$$ Show that ...
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3answers
81 views

Induction prove, how to come to $n\cdot(n+1)$

I am trying to solve an induction problem. Here are the steps for the example. Prove this equation $$ 1\cdot2 + 2\cdot3 + 3\cdot 4 + 4\cdot 5+\dots + \cdots +(n-1)\cdot n ...
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2answers
117 views

Evaluate $\int\frac{1}{r \ln(r)} \ dr$

What is the antiderivative of $$\int\frac{1}{r \ln(r)} \ dr$$ I'm trying to use substitution, but substituting $u=r$ doesn't help as that just changes the variable.
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2answers
60 views

Graph of $x^y$ at zero

I decided to review the properties of expontents after posting a question and getting downvoted. The answer was easy; however, got a bit confused by $ n^{x^m} $ with $n,m \in \mathbb{Z}$ So I ...
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2answers
1k views

Definition of a derivative of sine squared x.

Question: Let $f(x) = \sin^2x$. Use the definition of a derivative to find the derivative. Definition: $f'(x) = \lim_{h\to 0} \frac{f(x-h)-f(x)}{h}$ I have forgotten the trick to solving this one.
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2answers
90 views

How to find these integrals?

How to find the following two integrals? $$\int_{0}^{1}{\sqrt{{{x}^{3}}-{{x}^{4}}}dx}$$ and $$\int_{0}^{1}{x\sqrt{{{x}^{3}}-{{x}^{4}}}dx}$$
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46 views

Proof Help involving continuity and the FTC

Prove that if $f$ is a continuous function, then $$\frac{d}{dx}\left( \int_x^b f(t)dt\right)= -f(x).$$ Any help would be appreciated. Thanks!
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3answers
147 views

Infinite series with addition and subtraction of fractions

The question is to find whether it converges or diverges. If it converges, find what it converges to? $$\sum_{n = 1}^{\infty} \frac{2}{n^{1/3}} - \frac{2}{(n+1)^{1/3}}$$ I tried splitting up into ...
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1answer
718 views

How to show the normal density integrates to 1?

How could you show that the normal density integrates to 1? $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1 $$
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2answers
1k views

Limits of functions with square root and sin

Can someone help me calculate the following limits(without L'Hopital!) : 1) $\lim_{x\to 1 } \frac { \sqrt{x}-1}{\sqrt[3]{x}-1} $ . I have tried taking the logarithm of this limit, but without any ...
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2answers
92 views

Limit of sequences

This is part of some exercises we got to prepare ourselves for an exam in two weeks. It isn't homework. What are counter-examples to parts 1 and 3? How can we prove part 2? Let $a_n,b_n$ be ...
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2answers
60 views

Evaluating $\int \tfrac{e^{-x}}{1+e^{-x}}dx$

answer from book $x - \ln(e^x + 1)$ $~~~$ *I get * $- \ln \vert 1 + e^{-x} \vert + C $ $$\int \frac{e^{-x}}{1+e^{-x}}\, \mathrm{d}x $$ $$\begin{array}{l c l} u & = & 1 + e^{-x} ...
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129 views
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2answers
2k views

How to calculate $\lim\limits_{x\to \pi/4}{{\tan(x)}^{\tan(2x)}}$?

I guess I should apply natural logarithm here or something like that, but I can't unterstand what to do. I shouldn't apply L'Hôpital's rule as I haven't studied it yet.
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2answers
95 views

Coefficients in series expansion of $\left(\frac{x}{1-x}\right)^3$

How can I find the coefficient of $x^m$ in $$ \left(\sum_{n=1}^{\infty}x^n\right)^3 $$ for some $m\in{\Bbb N}$? I attempted it using the Cauchy products, but things got messy. I think one might want ...
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3answers
149 views

Calculate the limit of a sum

Calculate $$ \lim_{n\to\infty}S_n $$ when $$ S_n= \sum _{i=-n}^{n-1}\frac{2n+i}{2n^2} $$ by treating the sum as an area. The problem here is that I have no idea how I would start. Can someone ...
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1answer
440 views

Principles of Mathematical Analysis

I just got through reading the first chapter of principles of mathematical analysis by Walter Rudin, the first chapter goes on and on about Dedekind cuts and then starts defining properties of them, ...
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4answers
108 views

The integrals from $1$ to $\infty$ for $\dfrac{1}{x}$ and $\dfrac{1}{x^2}$

I have two integrals: $$ A=\int\limits_1^\infty \dfrac{1}{x}dx\,, B=\int\limits_1^\infty \dfrac{1}{x^2}dx $$ Calculus says that A is an improper integral as it diverges, but the B converges and is ...
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2answers
603 views

Inverse function of $f(x)=e^{x/2}$

How would you find the inverse function of $f(x)=e^{x/2}$?
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3answers
89 views

Find $\lim_{x \to 0} (f(3+1/x)/f(3))^x$

Let $f(3)=5$ and $f'(3)=2$. Find $$\lim_{x\rightarrow0} \left(\frac{f(3+\frac{1}{x})}{f(3)} \right)^{x}$$ My question is : Is the problem correct? Is the answer $e^{\frac{2}{5}}$ correct? If the ...
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5answers
314 views

Use product rule to differentiate $x^3 e^{-2x}$

I've got a question asking me to "differentiate $x^3 e^{-2x}$ using the product rule. So I differentiate using it $(u v)'=u'v+uv'$ and get $u: x^3$ $u':3x^2$ $v: e^{-2x}$ $v':-2e^{-2x}$ ...
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1answer
66 views

Differentiability at 0

I am having a problem with this exercise. Please help. Let $\alpha >1$. Show that if $|f(x)| \leq |x|^\alpha$, then $f$ is differentiable at $0$.
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3answers
365 views

How to find the inverse laplace transform of an arbitrary function

How to find $$\mathcal{L^{-1}} \left[ \frac{F(s)}{s+a} \right]$$where $F(s)$ is the Laplace transform of $f(t).$
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3answers
609 views

how do you find a derivative of $y=1-\cos(x)\sin(x)$

The question asks to find the derivative of the function $1-\cos(x)\sin(x)$, and I thought maybe using some derivative rules I could, but I don't know where to start.
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4answers
211 views

Indefinite integral

So I'm currently trying to solve $$\int \sqrt{ 1+\frac{1}{3x} } \, \, dx$$ I know that this can also be represented as ((x+1/3)/x)^1/2 but I dont like that form. I also know that this can be done ...