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

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1answer
48 views

How to evaluate the integral $\int_0^{\ln3} e^{x-e^x}\,\mathrm dx$?

How to evaluate the following definite integral? $$\int_0^{\ln3} e^{x-e^x}\,\mathrm dx.$$ Should I use some sort of U Substitution?
2
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3answers
134 views

Using L'Hospital's Rule to evaluate limit to infinity

I'm given this problem and I'm not sure how to solve it. I was only ever given one example in class on using L'Hospital's rule like this, but it is very different from this particular problem. Can ...
0
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4answers
48 views

Infinite Limits at Infinity

Prove that $\lim_{x\to\infty}\frac{3x^3-4x+1}{2x^2+2}=\infty$. I am having trouble proving this question. You must use the limit definition for infinity. That is given any N>0 , there exist an M>0 ...
0
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1answer
59 views

Evaluate the integral $(1/(1+x^2)) (\arctan x)^2$

I got $\arctan x(1/(1+x^2))^2$. Because wouldn't the antiderivative of $1/1+x^2$ be $\arctan x$ and then $(\arctan x)^2$ just be $(1/(1+x^2))^2$ ?
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1answer
36 views

Functions Defined by Integrals of Non-Elementary Functions

In calculus I, I encountered the function $e^{-x^2}$. The teacher told us that it was impossible to integrate. Curious, I graphed the function on my calculator, together with its integral. The ...
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3answers
344 views

Is this definite integral impossible?

From my understanding when you integrate $f(x)$ you get $F(x)+C$, and when finding a definite integral the $C's$ cancels out due to subtraction. However, I came across an example where the $C$ doesn't ...
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3answers
148 views

Integration by parts order question + integration question

In integration by parts, does it matter which of your terms is v (or, rather $f(x)$) and which term is du (or, rather $ f'(x) $? Also, I'm having trouble finding $\int e^{2x} \cos(3x)dx $. I keep ...
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4answers
173 views

A small airplane is flying due north at $150\,\rm km/h$ when it encounters a wind of $80\,\rm km/h$ from the east.

A small airplane is flying due north at $150\,\rm km/h$ when it encounters a wind of $80\,\rm km/h$ from the east. what is the resultant ground velocity of the airplane?
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0answers
46 views

What is wrong with this proof in topology?

Let $X$ be a $T_4$-space and $M \subset X$, then $M$ is a subspace that is also $T_4$. Proof: If $A,B$ are closed in $M$, then $A=W_A \cap M$ and $B = W_B \cap M$ for some closed sets $W_A,W_B$. ...
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1answer
88 views

Second derivative identity proof?

I have been told that if $y=g(x)$ then $$\dfrac{d^2y}{dx^2}=\dfrac{dy}{dx}\cdot \left( \dfrac{d}{dy}\dfrac{dy}{dx} \right) $$ if this is true please can some one tell me how we get this result? Here ...
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1answer
53 views

Polar coordinates: $ \iint_D (\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2})\:\mathrm{d}x\:\mathrm{d}y$

I need to calculate the following integral $$\iint_D \left(\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2}\right)\:\mathrm{d}x\:\mathrm{d}y$$ where $D$ is the disk $x^2 + y^2 \leq a^2$ Using the ...
0
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3answers
45 views

How to prove that $\frac{d}{dx}\epsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant

How to prove that $\frac{d}{dx}\varepsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant? Thanks in advance for your immense help.
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1answer
33 views

Line integrals and vector fields

We had this example in class the other day, and the professor didn't not walk through how he obtained it. Compute $\int_C \vec f \cdot d\vec r = \langle 4x^3y^2 - 2xy^3, 2x^4y - 3x^2y^2 + ...
1
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1answer
22 views

minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
3
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4answers
176 views

Series: prove/disprove a statement

Let $a_n$, a sequence suh that $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$ and the series: $\sum a_n$, $\sum (a_n + a_{n+1})$ Prove/Disprove: The series converge/diverge together. I'll be ...
0
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1answer
35 views

Show by means of an example that a limit exist?

Show by means of an example that $\lim_{x\to 4} $[$f(x)$ + $g(x)$] may exist even though neither $\lim_{x\to 4} $$f(x)$ nor $\lim_{x\to 4} $$g(x)$ exists. I'm having some problems with this question. ...
2
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2answers
59 views

A problem with the domain of function in the defintion of limits

My Stewart's Calculus gives the following definition of limit: $f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for ...
2
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3answers
145 views

1 to the infinty indeterminate limit

If $\lim_{x\to a}f(x)=1$ and $\lim_{x\to a}g(x)=\infty$ then show that $$\lim_{x\to a}\{f(x)\}^{g(x)}=e^{\lim\limits_{x\to a}{g(x)\{f(x)-1\}}}.$$ I started off as: $$\lim_{x\to a}\{f(x)\}^{g(x)} = ...
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2answers
341 views

minimising the value of an integral

Given the integral $$ I \equiv \int_{0}^{\pi/2}\left\vert\,\cos\left(x\right) - kx^{2}\,\right\vert \,{\rm d}x $$ Find the value of $k$ so that $I$ is minimum. Help would be ...
2
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1answer
142 views

question about Laguerre polynomials

how to prove that $$L_{n+1}(x)=\frac{1}{n+1}((2n+1-x)L_n(x)-nL_{n-1}(x))$$ I see it on wikipedia but I dont know how they prove it
2
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1answer
36 views

Numerical sum problem.

I just started the series chapter and I come across some series that I don't know how should I resolve them. All of them have the same structure : $$ \sum_{n=0}^{\infty} (-1)^n ... $$ For example: $$ ...
0
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3answers
42 views

What is the domain of $z=\arcsin\dfrac{x}{y}$?

I get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well, ...
2
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2answers
113 views

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Intuition?

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Is there an intuitive explanation to what this means as well as an algebraic proof? Also I understand that $\operatorname{Curl} ...
2
votes
4answers
135 views

Evaluate $\lim_{x \rightarrow 0} x^x$

I know when evaluated it gives one but I can't figure out how to prove it. Can anyone help? I believe it requires L'Hopitals rule and taking a natural log but i cannot figure out the exact math.
0
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0answers
57 views

Riemman sum to get surface area of cone, and cone height

I have this itch to think about math ideas in my free time. Like when I play a game or something, I kind of "work in parallel" on math ideas. Back in one of my calculus classes we worked on Riemann ...
1
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2answers
86 views

Show there is a unique solution to the equation F(x)=c if c>0

Define $F(x)=\int_1^x \frac{1}{2\sqrt{t} -1} dt$ for $x\ge1$. If c>0, prove there is a unique solution to the equation F(x)=c, for x>1. I know I need to use the intermediate value theorem, but how do ...
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2answers
119 views

Show that each of the series converges.Help Please

Show that each of the series converges on their respective domains. a) $$\sum_{n=1}^\infty \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$ b)$$\sum_{n=1}^\infty e^{-nx}, x \in(0,\infty)$$ For the first ...
0
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1answer
64 views

Integration Question

If we know the integral $$\int \frac{\mathrm{d}x}{f(x)+1}$$ can we find the integral of $$\int\frac{\mathrm{d}x}{f(x)+c}$$ for arbitrary $c\in\mathbb{R}$ (where defined)? Does it make a difference if ...
4
votes
1answer
107 views

What is the general limit theorem?

There are simple limit theorems like http://archives.math.utk.edu/visual.calculus/1/limits.18/ But they are just special cases. I am quite sure there is an established general result for them. In ...
3
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1answer
108 views

Spivak Ch 11 Theorem 7

Can someone please explain how to supply a rigorous $\epsilon,\delta$ argument for this theorem as Spivak says ? My argument is: $f'(a)=\lim_{h\to 0} f'(\alpha_h)$ equivalent to ...
2
votes
3answers
138 views

Compute the integral $\int_0^{\frac\pi2}\frac{\mathrm d \theta}{\sqrt{\sin \theta}}$

Compute the Riemann integral $$\int_0^{\frac\pi2}\frac{\mathrm d \theta}{\sqrt{\sin \theta}}$$ It seems very difficult, I don't know how to go ahead. Thank you very much for your help!
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1answer
34 views

Dense set in $T_2$-space.

Given a $T_2$-space $X$ and a dense set $M \subset X$ with cardinality $m$ we shall conclude from this that $\operatorname{card}(X) \le 2^{2^m}$. I don't get this: $M$ is always closed as it consists ...
0
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1answer
34 views

Calculate the Derivative of a univariable integral at a point $4$

Considering the function below: the objective is to calculate $F'(4)$ (the derivative of $F(x)$ in the point $4$)? we know that: and that: so if I try to replace $x$ by $4$ in $F'(x)$ I get ...
1
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1answer
110 views

Resolve $\lim_{x \to 0} \frac {\tan 3x}{\tan 5x}$ [duplicate]

I started resolving this limit: $$\lim_{x \to 0} \dfrac{\tan 3x}{\tan 5x}$$ so: $$\begin{align}\lim_{x \to 0} \dfrac{\tan 3x}{ \tan 5x} & = \lim_{x \to 0} \dfrac{\frac{\sin 3x}{\cos 3x ...
0
votes
2answers
74 views

Integral Inequality with Exponential

Prove that $$ \int_{0}^{1} e^{x^2} dx <\frac{e+2}{3} $$ I know a solution (using power series) that gives an even lower bound, but I would really like to see what is the solution intended by the ...
5
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2answers
3k views

What is the formula for nth derivative of arcsin x, arctan x, sec x and tan x?

Are there formulae for the nth derivatives of the following functions? 1) arcsin x 2) arctan x 3) sec x, and 4) tan x Thanks.
2
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1answer
61 views

Integral $\int_{0}^{\infty }x^2\cos\left(\mathrm{e}^x\right)~\mathrm{d}x $ convergence proof? [closed]

Tried a few things, none of them worked. I am really out of ideas at the moment. $$\int_0^{\infty}\! x^2\cos(\mathrm{e}^x)~\mathrm{d}x $$ How do I prove that this integral converges?
0
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1answer
26 views

Finding radius of convergence

I have gotten the problem almost solved, but I'm hung up on how to solve this inequality: $$|x|/|2x+1|<1 $$ I could move the denominator to the right side of the equation: $$|x|<|2x+1|$$ But ...
1
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2answers
69 views

Use Logarithmic Differentiation to find $\frac{d}{dx} (x^{{x}^{x}})$ at $x=1$

How do do this? Use Logarithmic Differentiation to find $\frac{d}{dx} (x^{{x}^{x}})$ at $x=1$.
2
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2answers
80 views

why $\int \frac{1}{1+\sin(x)+\cos(x)}dx = \ln\left | \tan(\frac{x}{2})+1 \right |+const.$?

This is how I solve $$\int \frac{1}{1+\sin(x)+\cos(x)}dx$$, but I got the wrong answer, and the correct answer is $$\ln\left | \tan(\frac{x}{2})+1 \right |+\text{const}.$$ How to solve this?
2
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4answers
2k views

Finding the limit of a recursive sequence $x_{n+1}=\frac{1}{2}(x_{n}+\frac{M}{x_{n}})$

Define the sequence $\{x_n\}_{n\in\mathbb{N}}$ by $$x_0=\frac{M}{2}, \qquad x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{M}{x_{n}}\right)$$ where $M\in\mathbb{R}$, $M\geq 0$. Find ...
3
votes
3answers
317 views

Is a sequence of all the same numbers monotonic?

I'm wondering based on the definition of monotonicity: A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. So given that the sequence $a_n = 3$ is all the same numbers and ...
1
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2answers
128 views

Recurrence relation: $x_{n+1} = \frac 12 x_n + \frac 1 {x_n},$ [duplicate]

$$x_{n+1} = \frac 12 x_n + \frac 1 {x_n}, x_0 \neq 0$$ $$ a = \frac a2 + \frac 1a \Rightarrow a = \frac {a^2 + 2} {2a} \Rightarrow 2a^2 = a^2 + 2 \Rightarrow a^2 = 2 \Rightarrow a = \pm \sqrt 2$$ If ...
0
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1answer
15 views

Consider a function $g(x)$ with derivative of $g(x)$ prime = $x^3(x-2)^2(x+8)^9$. For what value(s) of $x$ does $g(x)$ have a local maximum

I know that the answer is -8 but I keep getting 0 when I do $g(0), g(2)$, and $g(-8)$. I know that the critical points are 0, -2, and -8. I'm about to pull my hair out with this problem. Can someone ...
0
votes
0answers
16 views

Integrals of logarithm singularity over a quadrilateral

When applying boundary element method, one have to evaluate the integral over a quadrilateral: $$I=\iint \log(r+|z+z_1|)dxdy,$$ where $$r=\sqrt{(x-x_1)^2+(y-y_1)^2+(z+z_1)^2}.$$ $P(x_1,y_1,z_1)$ is ...
3
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2answers
56 views

How Do I Integrate? $\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}$

How do I integrate this one? $$\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}\,dx$$ Is my answer correct: $$-3\ln\left \| x+2 \right \|+\ln\left \| x \right \|+\frac{4}{x}+C$$
0
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1answer
22 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
1
vote
1answer
78 views

Confused with Leibniz notation of a derivative

If $f$ is a function and $x$ is function of $t$, how do you find the derivative of $f(x)$ in terms of the derivative of $f(t)$? With Leibniz' notation this is shown as (using the chain rule) ...
0
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2answers
84 views

Painful? Moment Generating Function

Part 1 Let $X$ be a random variable with the p.d.f. $f(x)=\frac{1}{4\pi}e^{\frac{-x^2}{4}}$, compute the MGF of $X$. So I know I want ...
1
vote
1answer
52 views

Sigmoid derivative using quotient rule

Sigmoid function defined by $f(x)=\frac{1}{1+e^{-x}}$ can be derived easily with derivative of a composed function like here: Derivative of sigmoid function. However I was asking myself how this could ...