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

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1answer
366 views

Show that the directional derivative is linear by definition

If $f$ is differentiable at $x$, the map $h\mapsto f(x+h)-f(x)$ should be approximately linear. The scalar multiplicativity can be seen by noting that $$\lim_{h\to 0}\frac{f(x+ch)-f(x)}{h} = ...
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116 views

Calculate the limit $\displaystyle\lim_{x\to0}\frac {(\tan(x)-x)^2} {(\ln(x+1)-x)^3} $

Calculate: $$\displaystyle\lim_{x\to0}\frac {(\tan(x)-x)^2} {(\ln(x+1)-x)^3} $$ So if we expand Taylor polynomials we get: $$\frac {(x+{x^3\over3}+o(x^3)-x)^2}{(x-{x^2\over2}+o(x^2)-x)^3}=\frac ...
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2answers
100 views

Showing $f(x)\le 2$ for all $x\in [0,1]$

Let $F:[0,1]\to \mathbb R$ be continous such that for all $x\in \mathbb Q \cap [0,1]$ we have $f(x)\le 2$. Show that for all $x\in [0,1]$ we have $f(x)\le 2$. I'm going to write the answer I got ...
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2answers
39 views

Show $a_n$ is monotone

let $a_n$ such that $a_1=1$ and $a_{n+1} = 3 - {1 \over a_{n}}$. Well, We're asking if $a_{n+1} > a_n$, then $3-{1\over a_n} > a_n$. Hence, $a_n^2 - 3a_n +1 < 0$ We got a quadratic ...
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2answers
56 views

Solve $y'(t) = \dfrac{1}{1+ty}$

Does the following reasoning make sense? \begin{gather} \dfrac{dy}{dt} = \dfrac{1}{1+ty},\\ 1+ty \; dy = dt, \\ \int1+ty \;dy = \int dt,\\ y+t\dfrac{y^2}{2} = t+C. \end{gather}
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4answers
81 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
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1answer
183 views

An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
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4answers
149 views

If something holds for n+1 distinct values then it holds for all values.Proving a property of polynomial

Ok I am stuck on proving a property of polynomial.It basically goes like this. If $$f(x) = \sum_{k=0}^{n} c_kx^k $$ is equal to zero for n+1 distinct real values x,then f(x) is equal to 0 for all ...
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1answer
124 views

How would we show by comparison that $\sum\limits_{i=1}^\infty \sin(\frac{1}{n})$ diverges?

By using the integral test, I know that $\sum\limits_{i=1}^\infty \sin\left(\frac{1}{n}\right)$ diverges. However, how would I show that the series diverges using the limit comparison test? Would I ...
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4answers
269 views

Open or closed intervals?

Say you have a graph for v[t] Question asks “when is speed constant” The flat part of v[t] is from 2 to 3 seconds. Is the answer 2 < t < 3 or is it $2 \le t \le 3$ ? $$\lim_{t\to 2} ...
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1answer
278 views

Calculation of a curvilinear integral

Please help to calculate the following integral. Calculate $$\int_\gamma \frac{x\,dx + y\,dy+z\,dz}{x^2+y^2+z^2}$$ where $\gamma$ is the way of class $\mathcal C^1$ which unites point on the ...
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4answers
96 views

Prove that $\sum_{k=0}^{\infty} (k-1)/2^k = 0$

How to prove that this series converges, and that the limit is 0 ?
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2answers
133 views

Showing summation is bounded

I'm currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this ...
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3answers
135 views

Manipulation of differentials

In solving some differential equations in physics, one has to do some peculiar manipulation of differentials to wind up with a desired function. My professor has warned up that doing so, "Will make ...
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3answers
73 views

Explanation for limits equality.

$$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{1}{x}}} = \exp \left( {\mathop {\lim }\limits_{x \to 0} \frac{{\frac{{{a^x} + {b^x}}}{2} - 1}}{x}} \right)$$ I am ...
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75 views

Taylor expansion - what order would be preferred?

Let say you want to calculate the following limit: $$\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{1 - \cos x}}\ln \left( {\frac{{\sin x}}{x}} \right)} \right)$$ Obviously, Taylor Expansion ...
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1answer
71 views

Area between two functions?

Find the area between the functions $x+y = 2$ and $x + 4 = y^2$. The question is relatively simple: The area between the functions is: $$\int^{2}_{-3} 2-y-y^2+4 \text{ }dy$$ But can the above ...
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3answers
313 views

Find the derivative of this integral: $h(x)=\int_5^{1/x}10\arctan(t)\,dt$

$$h(x)=\int_5^{1/x}10\arctan(t)\,dt$$ Find $h'(x) $. I know how to calculate the derivative of basic integrals,but this one I've been trying to solve for quite a long time,and have not yet ...
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2answers
346 views

Area under the line tangent to a curve at a point $(a, b)$

I have this question and It can simply be solved by using the formula of triangle, but the problem here we want to method of integration, is there is away to find the area of shaded area (Triangle) ...
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2answers
63 views

Lagrange method for inequality

i have one question and any hint would be very helpful for me,we know how Lagrange multiplier works ,for example consider following problem Example 2 Find the maximum and minimum of subject to ...
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2answers
71 views

prove property of the definite integral of a function of sine.

Show that $$\int_0^{\pi}x\;f(\sin x)dx=\frac{\pi}{2}\int_0^{\pi}f(\sin x)dx$$ I tried to prove this using integration by parts and u-subs but I can't work it out.
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1answer
84 views

Can this polynomial have two distinct roots in $[-1,1]$? [closed]

how can I prove that $f_m(x) = x^3 + 3x +m$ can not have two distinct roots in $[-1,1]$? I tried Rolle's theorem but this hasn't worked for me. Help, please.
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2answers
70 views

How to calculate a complicated geometrical series?

I have a geometrical series (I don't know if its geometrical series or not): $$ \sum_{n=1}^{\infty }n\rho ^{n}(1-\rho) $$ how can I simplify it ? ( assume that $ 0 \le \rho \le 1$ ) The last answer ...
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3answers
77 views

Limit of two sequences

$ \lim_{n\to \infty} \sqrt[n]{3^n+4^n} $ . I think the limit is $4$. I did : $ \sqrt[n]{3^n+4^n} = 4 \sqrt[n]{(\frac{3}{4}) ^n+1}$ .Am I right? $ \lim_{n\to \infty} \frac{1}{1\cdot 4 } + ...
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2answers
73 views

Derivative of trigonometric function help

I am having trouble computing derivatives at $x=2k \pi$. Let $f(x)=x \sqrt{1-\cos(x)}$. Then $\displaystyle f'(x)=\frac{2(1-\cos x)+x\sin x}{2\sqrt{1-\cos x}}$. Then $\displaystyle ...
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4answers
106 views

Find $\int\frac{x^2}{\sqrt{1-x^2}}$

I need to compute the following integral: $$\int\frac{x^2}{\sqrt{1-x^2}}~dx$$ I tried everything I know, including defining $u=x^2$ and $u=\sqrt{1-x^2}$, but it failed.
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4answers
82 views

How many ways can you show that $ \displaystyle \int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2}\ \mathrm{d}x\ $ is divergent?

How many ways can you show that this integral is divergent? $\displaystyle\int_0^1 \dfrac{\sin(\pi x)}{(1-x)^2} \,dx$ The only way I was able to show this was using a hint which was given to me that ...
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2answers
79 views

Find the equation of the normal to the curve $y = 8/(4 + x^2)$ , at $x = 1$.

When you first differentiate the above, you get $-8/25$, right? Then you derive the gradient for a normal and proceed so on and so forth. The textbook I'm using says when you differentiate, you get ...
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3answers
96 views

How find this integral

$\int_{- \cos x}^{\sin x} \frac{1}{\sqrt{ 1-t^{2}}} dt$ my solution is put $t=sin {\theta}$ then $$\int_{- \cos x}^{\sin x} \frac{1}{\sqrt{ 1-t^{2}}} dt = \int_{- \cos x}^{\sin x} ...
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1answer
73 views

Does convergence of one series imply convergence of another series?

Let's assume that $$ \sum_{n=1}^{\infty} a_{n}$$ is convergent. Does it imply that $$ \sum_{n=1}^{\infty} \frac {a_{n}}{n^{1/2}} $$ is a)convergent b)absolutely convergent ??? Please, ...
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2answers
211 views

Limits involving trigonometric functions $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$

I was asked to find for what values of x the function is differentiable and write down the derivative. $f(x)=\lfloor{x^2 \rfloor} \sin^2(\pi x)$ for $x \geq 0$. There are two steps: When $x \in ...
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2answers
55 views

Prove $0 \le e^{-\theta x^2} \le 1$ for $0 \le \theta \le 1$

Why is it that $$0 \le e^{-\theta x^2} \le 1$$ for $0 \le \theta \le 1$? My textbook told me this in the context of langrange remainder for taylor series, and I can't figure it out. (Also, I don't ...
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1answer
49 views

Exact expression for series

Can the exact expression for the following series be found, given $|x|<1$? Just curious. $f(x) = \frac{x^2}{17}+\frac{x^3}{3}+\frac{x^4}{3}+\frac{x^5}{3}+ \ldots$
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3answers
103 views

Derivative of $x^2-\frac{1}{x^2}$ not matching WolframAlpha result

I was calculating a very simple derivative of $$ f(x) = x^2-\frac{1}{X^2} $$ and my result is $$ f^{\prime}(x) = 2x + \frac{2}{x^3} $$ But I can't explain why WolframAlpha says the result is $2x$. ...
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196 views

Find lower bound of function $\frac{x}{x^{1/x}}$

Can someone help me finding a lower bound to the function $$f(x)=\frac{x}{x^{1/x}},$$ where $x\in[3,+\infty[$? I suppose that a lower bound function can be $y=x$ but I don't really know how to start, ...
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3answers
105 views

Help understand chain rule derivative

I was verifying a larger function derivative on wolfram alpha and came across this derivative: $\frac{d}{dx} (1-x)^2 = 2(x -1)$ Using the chain rule, I was expecting to get: $2(1 - x)$ Instead. I ...
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1answer
224 views

Showing there does not exist a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f'(x)=h(x)$

If $h(x)=0$ if $x<0$ and $h(x)=1$ if $x\geq 0$, prove there exists does there does not exist a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f'(x)=h(x)$. Proof: We will show that $h$ is ...
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3answers
67 views

integrate the following equation (what am I doing wrong here 2)

Here is the equation: $$\int 3x \sqrt{1-2x^2}dt$$ Here is my answer: $$ \dfrac14 \int (1-2x^2)^{1/2} . 3x = -\dfrac14 \dfrac{(1-2x^2)^{3/2}}{3/2} = -\dfrac14 \cdot \dfrac23 (1-2x^2)^{3/2} + c$$ ...
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1answer
45 views

Finding the derivative of $\;\operatorname{arccoth}(\sin x)$

I have tried to solve it but I don't why it's wrong. I need to take the derivative of $\;\operatorname{arccoth}(\sin x)$: By using chain rule, I get: $$\dfrac 1{1 - \sin^2 x}\cdot \cos x = \dfrac ...
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2answers
58 views

How to find the derivative of $\operatorname{arcsinh}(3x)$?

I know The derivative of $\operatorname{arcsinh}(x) = 1/(x^2+1)^{1/2}$ But if I derivative $\operatorname{arcsinh}(3x)$ Why it doesn't equal to $(\operatorname{arcsinh}(3x))^{-1} (1/(9x^2+1)) (3)$ ...
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2answers
101 views

How do I compute this improper integral?

$$\int_{0}^{1}\dfrac{1}{2x^2-x}dx$$ This is a Type II improper integral because the function is undefined at $x=\frac{1}{2}$. If I were to do this problem on my own I would split this integral in ...
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2answers
134 views

does this series converge? $\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $

show the the following series converge\diverge $\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $ all the test i tried failed (root test, ratio test,direct comparison) ...
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131 views

Definite Integral questions

Evaluate the definite integral of the function. $$ \int_{-\pi/4}^{\pi/2} \; |\sin x| \; dx $$ My solution was : $$ \cos \left(\frac{\pi}{2}\right) - \cos\left(\frac{-\pi}{4}\right)$$ $$ ...
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3answers
231 views

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
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3answers
74 views

Finding $\displaystyle \lim_{x\to 0} \frac{\ln(3^{x}+1)-\ln(2)}{x}$

I have to find the limit of $\displaystyle \lim_{x\to 0} \frac{\ln(3^{x}+1)-\ln(2)}{x}$ using only notable limits avoiding l'Hopital's method and derivatives. I tried to use logarithm's properties ...
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3answers
111 views

How to show the following cool equality:

I am looking for a proof of the following relationship: $\newcommand{\ds}[1]{\displaystyle{#1}}$ $$ \frac{\ds{\int_{0}^{\pi}\sin^{n-2}\left(t\right)\,{\rm d}t}} ...
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2answers
39 views

$\lim_{n\to \infty} {a^n-b^n\over a^n+b^n}.$

Find the limit $$\lim_{n\to \infty} {a^n-b^n\over a^n+b^n}.$$ How do I find this limit? Thank you.
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3answers
75 views

Find the limit of $\frac{\ln(a^n+b^n+c^n)}{\sqrt{n^2+n+1}}$ as $n\to \infty$

What is $$\lim_{n\to \infty}\frac{\ln(a^n+b^n+c^n)}{\sqrt{n^2+n+1}}$$ where $0<a<b<c\ $? I tried like this: ...
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2answers
60 views

Why does the following series diverges? [closed]

Why does the following series diverges? $$\sum_{i=1}^\infty \tan(\frac{\pi}{i+2})$$
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2answers
588 views

Tangent line to the curve $x^3+xy^2+x^3y^5=3$

Does the tangent line to the curve $x^3+xy^2+x^3y^5=3$ at the point $(1,1)$ pass through the point $(-2,3)$? (using implicit differentiation) I got the implicit differentiation as ...