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

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1answer
31 views

the partial derivatives computing

Given that: $u_{xx}^{''}=u_{yy}^{''} \;, \; u(x,2x)=x \;, \; u_{x}^{'}(x,2x)=x^{2}\;, \; $ How to find the following values? $u_{xx}^{''}(x,2x)=?\; u_{xy}^{''}(x,2x)=? \; u_{yy}^{''}(x,2x)=?\; $ ...
3
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3answers
87 views

How to solve this Ordinary Differential Equation (ODE)?

By using the method of undetermined coeficiens, find the particular solution $y_p$ for this inhomogenous differential equation. $$y''+y= \sin x + x\cos x$$ I have find the roots which are $\pm i$. ...
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2answers
89 views

Strange AP Calculus BC question help?

The question is $\int_0^3\frac{1}{(1-x)^2}$. I got an answer (from u-substitution) however the solution manual says that the integral does not converge. Someone told me that the integral is undefined ...
2
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1answer
64 views

dimension of graph

I have a really rudimentary question hope to clear out: If $f: V \rightarrow V$, where $f$ is a linear map and $V$ is a vector space. Then what is the dimension of $\operatorname{graph}(f)$? To my ...
2
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2answers
60 views

Simple Calculus question (Substitution)

My question is "To show $ \int_{1}^\alpha {1\over1+x^2} dx = \int_{1\over\alpha}^1 {1\over 1+u^2} du$ such that $u = \frac{1}{x}$ and $\alpha $ > 1, I got to $ \int_{1}^\alpha {1\over1+x^2} dx ...
3
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2answers
101 views

Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$

Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$ I thought about maybe breaking the polynomial in two different fractions in order to make the sum more manageable and reduce it to ...
10
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1answer
284 views

Does it exist a function for which the derivative changes sign more than countably many times?

Does there exist any function $f \in C^2[0,1]$; $f: [0,1] \mapsto [0,1]$, for which the derivative changes sign more than countably many times?
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1answer
500 views

Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .

That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...
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2answers
442 views

Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$

Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$? So I started by saying that by the geometric series test where $a=x^2$ and $|r| = |\frac{1}{e^x}| \leq 1$, the series ...
2
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2answers
68 views

Is $\sum_{n=1}^{\infty} {\frac{(-1)^n}{x+2^n}} $ in $(-2,\infty)$ uniformly convergent?

Is $\sum_{n=1}^{\infty} {\frac{(-1)^n}{x+2^n}} $ in $(-2,\infty)$ uniformly convergent? I started by checking if it is pointwise convergent, because if it wasn't then especially it is not ...
0
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1answer
159 views

Show the surface area of revolution of $e^{-x}$ is finite

I need to show that the surface area of revolution of $e^{-x}$ is finite when the region from $x=0$ to $x=\infty$ is rotated about the x-axis. I tried using the surface area formula, but got stuck on ...
1
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2answers
90 views

Convergence of infinite series 1/n!

I have this question: Do the series converge absolutely or conditionally? $$ \sum (-1)^{n+1}\frac{1}{n!} $$ I would say it does not converge absolutely, since I suggest, by using the ratio test, that ...
1
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1answer
600 views

Evaluate the given limit by recognizing it as a Riemann sum.

Please note that this is homework. Please excuse my lack of $\LaTeX{}$ knowledge. The Problem: Evaluate the given limit by first recognizing the sum (possibly after taking the logarithm to ...
3
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1answer
607 views

Integral proof of logarithm of a product property

In one of my textbooks, the expansion of a logarithm product is proved using integrals. $$\ln xy = \ln x + \ln y\iff \int_1^\left(xy\right)dt/t$$ $$\ = \int_1^xdt/t + \int_x^\left(xy\right)dt/t$$ ...
4
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3answers
374 views

Find point nearest to the origin

Find the points on the curve $5x^2 - 6xy + 5y^2 = 4$ that are nearest the origin. The first method I've tried is I've taken the derivative of the equation to optimize (Pythagorean Theorem) and also ...
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3answers
1k views

Derivative of $\sec^{-1}(x)$

I'm struggling with problem below which I eagerly want to solve. Let me know from where this problem is, if possible (the origin source of textbook). your answers might really helpful to get through ...
3
votes
4answers
442 views

Finding the area of this figure inside a circle?

Let's say I have a unit circle and I draw a triangular sector with an angle $\pi/2$. Next to it, I draw a sector with an angle $\pi/8$, next $\pi/18$, so that the $n^{\text{th}}$ sector has an angle ...
1
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4answers
53 views

From geometric sequence to function

I have this question: Find the functions which equal the sums: $$ x + x^3 + x^5 + .. $$ Now, I can see in my result list, that its supposed to give $$ \frac{x}{(1-x^2)} $$ I can see why the ...
4
votes
2answers
161 views

Integral of $\frac{{x^{1/2}}+3}{2+{x^{1/3}}}$

I want to solve this integral and think about doing the following steps: $1)\quad t=x^{1/3}$ $2)\quad x=t^3$ $3)\quad dx=2t^3\,dt$ How I can show $\sqrt{x}$ as $t$? ...
10
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2answers
328 views

How to show that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$

$\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$. I struggled on it, but i did't find any pattern to solve it.
2
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2answers
113 views

Need help solving $\int x \sqrt{\frac {a^2 - x^2} { a^2 + x^2 }} dx$

I have a complicated integral to solve. Can someone provide a better way to solve it than what i did - dividing by a inside the root, and then putting $ t = x / a $, and then putting $t^2 = \cos ...
2
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3answers
134 views

Not sure how to go about solving this integral

$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$ I've let $u=x^2+3$ but can't seem to get the right answer. Really not sure what to do.
2
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1answer
50 views

Relationship between Mean Value Theorem and the maximum norm

I am seeking assistance with the following application of the Mean Value Theorem: Let $x \in \Omega$ and construct an associated neighbourhood $N_x = (a, a+ \sqrt{\epsilon})$, such that $x \in N_x$ ...
1
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1answer
219 views

MCQ on a function

$f(x)=\left( \ln \left( \frac{\left( 7x-x^{2} \right)}{12} \right) \right)^{\frac{3}{2}}$ Choose correct options , more than one may be correct . (a) $f$ is defined on $R^+$ and is strictly ...
3
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2answers
295 views

Evaluate integral in terms of Gamma function

Need help evaluating the following integral in terms of the gamma function where gamma function is: $$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt.$$ The integral is the following: $$\large{\int_0^\infty ...
15
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1answer
225 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...
2
votes
2answers
144 views

If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$

If $a_1,a_2,\dotsc,a_n $ are positive real numbers, then prove that $$\lim_{x \to \infty} \Bigl[\frac {a_1^{1/x}+a_2^{1/x}+.....+a_n^{1/x}}{n}\Bigr]^{nx}=a_1 a_2 \dotsb a_n.$$ My Attempt: ...
19
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4answers
397 views

Finding the fraction $\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}$ when knowing the sums $a+b+c+d$ to $a^4+b^4+c^4+d^4$

How can I solve this question with out find a,b,c,d $$a+b+c+d=2$$ $$a^2+b^2+c^2+d^2=30$$ $$a^3+b^3+c^3+d^3=44$$ $$a^4+b^4+c^4+d^4=354$$ so :$$\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}=?$$ If the ...
4
votes
4answers
766 views

How to find inverse of the function $f(x)=\sin(x)\ln(x)$

My friend asked me to solve it, but I can't. If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$? I have no idea how to find the solution. I try to find ...
3
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1answer
94 views

When does $e^{f(x)}$ have an antiderivative?

today I tried to integrate $x^x$ by applying a reverse chain rule which turned out to be false. I was told $\int e^{f(x)}\,dx$ can be done when $f(x)$ is linear. This made me wonder what conditions we ...
1
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1answer
217 views

Need to deduce $f(x)$ from $f_x=e^{t(x)}$

I know that $$f_x=e^{t(x)}$$ (where the notation $f_x=\frac{df}{dx}$) (EDIT: $f=f(x)$ and $t$ parameterizes $x$, so $x=x(t) \Leftrightarrow t=t(x)$) and that therefore $$\frac{d^n ...
0
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3answers
122 views

How can I determine $\lim_{x\rightarrow 2} \frac{(x^3-5x^2+8x-4)}{x^4-5x-6}$?

This is the limit: $$\lim_{x\to2}\frac{x^3-5x^2+8x-4}{x^4-5x-6}$$ Thank you.
1
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1answer
846 views

Find Standard Error and fill in the blanks.

A survey organization draws a simple random sample of 1,000 registered voters in a certain town. In the sample, 32% approve of the Mayor. The organization estimates that 32% of all 50,000 registered ...
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2answers
388 views

A less known definition of the definite integral of a continuous function

The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110. (link to full book) (screenshots: page ...
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2answers
75 views

If $\lim_{x \to \infty} \frac{f'(x)}{x}=2$ does it follow that $\lim_{x \to \infty} \frac{f(x)}{x^2}=1$?

I need to show that the following statement is true or false. $$\displaystyle\lim_{x \to \infty} \frac{f'(x)}{x}=2 \Rightarrow \displaystyle\lim_{x \to \infty} \frac{f(x)}{x^2}=1$$ I considered ...
2
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3answers
116 views

Properties of limits when dealing with functions and parentheses .

My calculus instructor recently mentioned some odd properties of limits that I don't recall ever seeing, and seem alien to me. He says that the following statements are allowed: ...
3
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2answers
363 views

How to convert a geometric series so that exponent matches index of sum?

I need to convert the following series into a form that works for the equation $$\frac{a}{1-r}$$ so that I can calculate its sum. But the relevant laws of exponents are eluding me right now. ...
3
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1answer
66 views

Is my interpretation of integration correct?

I thought of this when I was thinking about the $dx$ under the integral sign. So we have a function $y=f(x)$. Therefore, $dy/dx=f'(x)$, so $dy=f'(x)dx$. Now the graph of $f$ is split into small ...
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7answers
486 views

Why isn't the derivative of $e^x$ equal to $xe^{(x-1)}$?

When we take a derivative of a function where the power rule applies, e.g. $x^3$, we multiply the function by the exponent and subtract the current exponent by one, receiving $3x^2$. Using this ...
4
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1answer
125 views

Integrate function $\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$

How can I integrate this function? It's originated by an exponential prior and a poisson likelihood. $\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$
2
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2answers
63 views

Finding radius of convergence for this power summation $\sum_{n=0}^\infty \left(\int_0^n \frac{\sin^2t}{\sqrt[3]{t^7+1}} dt\right) x^n$

I have been given this tough power summation that its' general $c_n$ has an integral. I am asked to find the radius of convergence $R$ $$\sum_{n=0}^\infty \left(\int_0^n ...
3
votes
1answer
198 views

Existence of a differentiable curve

Let $f: [a, b] \to \mathbb{R}^n$ where a and b are real numbers. Prove that the curve $f$ is differentiable if and only if there is an open interval $I$ that contains the closed interval $[a, b]$, ...
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3answers
1k views

Determinant of Vector

Is posible obtain the determinant of any vector?.How I will be able to obtain the determinant of any vector $v=[v_1,v_2,\cdots,v_n]\in \mathbb{R}^n$?
9
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1answer
112 views

prove 3rd derivative of a function in a point

Let $f$ be twice differentiable and $f'''$ exists in one point $x\in D$. I want to show for this one point $$f'''(x)=\lim_{h\rightarrow0}\frac{f(x+3h)-3f(x+2h)+3f(x+h)-f(x)}{h^3}$$ I did the ...
2
votes
1answer
32 views

Time of descent down a slope defined by a function

Let $\phi$ be a real, decreasing differentiable function defined on $[a, b]$. An object is dropped at the point $(a, \phi(a))$ and is accelerated only by gravity. How long does it take to reach $(b, ...
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0answers
169 views
1
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2answers
92 views

Show $x+\frac{\lambda}x \geq 2\sqrt{\lambda}$ all $x,\lambda>0$

For $\lambda>0 $ and $x > 0$, $$x+\frac{\lambda}x \geq 2\sqrt{\lambda}$$ I tried to let function $g(x) =$ the difference of them and then find $g'(x) = 0$. With the given $x$, I can get ...
1
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1answer
68 views

How to evaluate the following limit?

How to evaluate this limit? $$\underset{n\to \infty }{\mathop{\lim }}\,\left( {{n}^{-2}}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{n}^{2}}}{\frac{1}{\sqrt{{{n}^{2}}+ni+j}}}} \right)$$ Thanks!
2
votes
1answer
51 views

Derivative of order 16 - is there a method to do so?

I have the following exercise: Find the $16^{\text{th}}$ derivative of $y$, (i.e. $y^{(16)}$), for $y = \sin x$. Is there any method to do so, or I simply have to differentiate the function $16$ ...
3
votes
2answers
78 views

A Calculus Question on onto functions with a specified range.

The following question was from a mock test of a competitive exam. Suppose $f:\mathbb{R} \to [-8,8]$ is an onto function and $f(x) = \dfrac{bx}{(a-3)x^3 + x^2 + 4}$ where $a,b \in \mathbb{R}^+$. ...