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

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0
votes
3answers
93 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...
3
votes
2answers
114 views

$\exists x_0$ such that $f(f(x_0))=x_0$ prove that $f$ has a fixed point

Let $f:\mathbb R\to \mathbb R$ be coninuous. Suppose there exists $x_0$ such that $f(f(x_0))=x_0$. Prove that $f$ has a fixed point or in other words: $\exists c\in\mathbb R: f(c)=c$ . Suppose ...
1
vote
4answers
486 views

Integrating powers of linear and quadratic functions

How can I integrate function such as $(x+9)^3$? I obviously know that I can expand the function and integrate it normally. However, that is possible and feasible only as it is of third degree. What if ...
0
votes
1answer
35 views

Prove the series converges

Prove $\sum_{n\ge0}\frac{\ln(n+1)}{n^2}$ Converges. I want to know what's wrong with my proof: By Cauchy condensation test: $\sum_{n\ge0}\frac{\ln(n+1)}{n^2}$ converges iff ...
2
votes
6answers
145 views

How to evaluate the following indefinite integral? $\int\frac{1}{x(x^2-1)}dx.$

I need the step by step solution of this integral please help me! I can't solve it! $$\int\frac{1}{x(x^2-1)}dx.$$
0
votes
1answer
53 views

Derivative and Taylor Series

In higher dimensions, is the derivative (jacobians,gradients etc.) defined using taylor series or taylor series formula proved through derivatives ?
0
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2answers
65 views

Laplace transform of $L({1-e^{-t}\over t})$

I have to find the Laplace transform of $$\mathcal{L}\left[\dfrac{1-e^{-t}}t\right],$$ then this is equivalent to $$\mathcal{L}\left[\dfrac{1}t\right]-\mathcal{L}\left[\dfrac{e^{-t}}t\right]$$ But ...
4
votes
4answers
233 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
2
votes
1answer
73 views

if $f$ is differentiable at $x_0$ then the limit exists

Let $f$ differentiable at $x_0$. Show that the following limit exists $$ \lim_{h\rightarrow0} \frac{f(x_0+h)-f(x_0-h)}{h}$$ If $f$ is differetiable at $x_0$ then it's one-sided derivative exists ...
2
votes
1answer
51 views

find all the indefinite integrals of a function in interval

Well, this is the function: $$\frac{\left(5x+3\right)}{x^3+2x^2-3x}$$, and i need to find all the indefinite integrals in the interval $\left(1,\infty \right)$. So far i used integration by partial ...
0
votes
1answer
37 views

convergence/divergence $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$

I am stuck with this series $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$. I used nth-root test, but the limit was $1$. Then I tried to think about it as $(-1)^n(\frac{n}{n-1})^n$ to use Leibniz, and I got ...
3
votes
2answers
127 views

Show that $\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$

I wish to show $$\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$$ I've tried substitution and integration by parts to get a recursive formula for ...
1
vote
0answers
55 views

Convergence of $\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$

I have to find for which $r\in\mathbb{R}$ a) series diverge b)converge absolutely c)converge not absolutely. $$\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$$ I don't know ...
0
votes
1answer
53 views

How do I solve this Integral using an infinite series

I'm supposed to use an infinite series to solve $$\int\frac{e^{2x}}{x}dx$$ How do I solve this? I know the answer already, I don't even have to use infinite series to solve this, however it was ...
0
votes
1answer
60 views

$F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
0
votes
4answers
54 views

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$ Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$. Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$ This was a question on my midterm a few days ago. I've been thinking about it for a while and still cannot ...
0
votes
3answers
57 views

Calculus question on application of the quotient rule

The question is using quotient rule, find and simplify $\large \frac{dE}{d\theta}$ for $$ E(\theta)= \frac{\theta-u\theta^2}{u+\theta} $$ My try: $$ ...
2
votes
2answers
139 views

A Sine-Cosine Integral

What is the value of the integral \begin{align} I(k) = \int_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) + \cos(kx)}{\sin(x) + \cos(x)} \ dx \end{align} where $k$ is an integer ? Is it possible to evaluate a ...
0
votes
2answers
41 views

Calculus quotient rule

$g(x)=\frac{x^{1/4}}{x^3+1}$, find $g'(x)$. Use the quotient rule. My attempt was: \begin{align}g'(x)&=\frac{\frac{1}{4}(x^{-3/4})(x^3+1)-(3x^2)(x^{1/4})}{(x^3+1)^2}\\ ...
2
votes
3answers
50 views

Calculus / tangent line

Find the exact value of $c$ in the figure shown below, where the line $l$ tangent to the graph of $y = 2^x$ at $(0, 1)$ intersects the $x$-axis. looking at the graph, find the exact slope of the ...
4
votes
2answers
399 views

Calculus / substituing in f(x)+g(x) to find h'(x)

$$\begin{align} f(x) &=7\\f'(x)&=2\\ g(x) &=2 \\ g'(x)&=-5 \\ h(x) &= f(x) + g(x)\end{align}$$ Find: $h'(2)$ My attempt was: $2+7=9$ but it seems to be wrong.
1
vote
2answers
73 views

Calculus / find the value of $x$ so that $f ''(x)=0$

Let $f(x)= 10xe^x$ $(a)$ Find the exact value of $x$ so that $f ''(x) = 0$. I tried: $$\begin{align}f'(x)& = 10e^x\\f''(x)&= e^x\end{align}$$ but at that point, the $f''(x)$ would never ...
-1
votes
1answer
98 views

Coin in City Problem [closed]

Please consider this problem. in one city common coin is 1dollar ,2dollar and 3dollar coin. how many way of paying the The price for an 20dollar candy which the seller has no money and number of ...
0
votes
2answers
345 views

Optimization problem, not sure how to proceed

So I'm a bit confused by this optimization word problem. I would be able to solve it I think given number values for the speeds but I'm uncertain how to get an exact answer when you don't know the ...
0
votes
2answers
43 views

Logorithms on a first level learning

Solve log$_{5x-1}$ $4$ $=$ $1/3$ $(5x-1)^{1/3}$=4 $((5x-1)^{1/3})^3$ = $4^3$ $5x-1=64$ $5x=65$ $13$ I am not sure where to go with this. I learned some things about logs before my class ended ...
2
votes
1answer
89 views

How to evaluate this definite integral? $\int_0^{0.5\ln3}\tfrac1{e^x+e^{-x}}dx.$

By using the substitution $u=e^x$, determine the value of: $$\int_0^{\tfrac12\ln3}\cfrac1{e^x+e^{-x}}\mathrm dx.$$ I've made the substitution and I'm already stuck... What's the next step?
1
vote
3answers
72 views

How to integrate $\frac{10}{4x^2-24x+61}$?

Show that $\displaystyle\int_3^{5.5}\dfrac{10}{4x^2-24x+61}\mathrm dx=\dfrac\pi4.$ I've completed the square, and now have: $10 \int \dfrac{1}{4(x-3)^2+25}dx$ Using common results, I know it ...
1
vote
2answers
445 views

Proving polynomial limit theorems

I am pretty confused on this math question. It is a two-parter but I'm not sure what part a is asking me, perhaps someone on StackExchange could help. The question reads as follows: (a) If p is a ...
1
vote
0answers
42 views

Change of Variables Polars

Let $r^2 = x^2 + y^2$ Find $\frac{dr}{dt}$? I think it is $$ \frac{dr}{dt}= \frac{1}{2}\left(x^2+y^2\right)^{-1/2}\left[2x \frac{dx}{dt} + 2y \frac{dy}{dt}\right], $$ would this be correct?
1
vote
2answers
241 views

Which is greater, $e^{\pi}$ or $\pi^e$? [duplicate]

I'm familiar with a simple method of demonstrating that $e^\pi$ is greater: $f(x) = \ln|x|/x$ $f'(x) = (1 - (\ln|x|))/(x^2)$ so f's max is at $(e, 1/e)$ so $1/e > \ln(\pi)/\pi$ and $e^{\pi} > ...
0
votes
1answer
54 views

Quotient of liminf

Given two sequences $\{a_n\}$ and $\{b_n\}$ with $b_n>0$ for any $n$. Does this hold? $$\lim\inf_{n\rightarrow \infty} \frac{a_n}{b_n}= \frac{\lim\inf_{n\rightarrow \infty} ...
0
votes
1answer
60 views

$f$ is differentiable twice, bounded and has a minimum on $x_0$, prove that there's a point $y\in\mathbb R$ such that $f''(y)=0$

Suppose $f:\mathbb R\to \mathbb R$ is differentiable and there's a constant $c>0$ such that $f'(x)>c$ for all $x\in(a,\infty)$. Prove that $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ ...
3
votes
2answers
90 views

very strange phenomenon $f(x,y)=x^4-6x^2y^2+y^4$ integral goes wild

I am going over my lecture's notes in preparation for exam and I saw something a bit strange I would like someone to explain how it is possible. Look at the function $f(x,y) = x^4-6x^2y^2+y^4$ if we ...
4
votes
1answer
147 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
1
vote
3answers
76 views

Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
1
vote
1answer
92 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
1
vote
1answer
56 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
2
votes
1answer
63 views

Integration of exponential functions: $\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$

How I am to solve this integral? I am not able to use any of the methods. $$\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$$
1
vote
1answer
75 views

Differentiability in a neighborhood of a strictly convex continuous function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a strictly convex continuous function, and let $f$ be differentiable at the point $x_0\in \mathbb{R}$. Can we say that $f$ is differentiable on some ...
3
votes
4answers
127 views

Definite Integral: $\int_0^1\frac{x^{2} -1}{\log x}\,dx$ [duplicate]

How to figure out the following integral? I have not been able to solve it from some time. $$\int_0^1\frac{x^{2} -1}{\log x}\,dx$$
4
votes
4answers
103 views

$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?

Question I'm trying to solve to prepare for an exam. I need to find out if $\displaystyle\iint_{\mathbb R^2} \frac{dx\,dy}{1+x^{10}y^{10}}$ diverges or converges. What I did: I switched to polar ...
5
votes
2answers
67 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
3
votes
1answer
77 views

Prove that a function is differentiable if…

I'm trying to prove that given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}^m$ in $p =(p_1, p_2) \in \mathbb{R}^2$, the function $$ g(x, y) = f(x, y) - \frac{\partial f}{\partial x}(p)(x ...
1
vote
2answers
152 views

Calculus Area Cubic curve

Why area bounded between the "line $AB$" and the "cubic curve" and area bounded between the "line $BC$" and the "cubic curve" is $16$ times?
2
votes
1answer
67 views

Does positivity of the integral implies positivity of a function

Let $f\colon [0,T]\mapsto \mathbb{R}$ be a continuous function with $\lim_{x\to 0}f(x)<\infty$ and let $\int\limits_0^tf(x)dx>0$ for every $t\in(0,T)$. Does it imply that there exists $t_0$ such ...
1
vote
4answers
50 views

Differentiate $\frac{\ln(x)}{\cos(x)}$

Please help me with this question. $$y= \frac{\ln(x)}{\cos(x)}$$ Just starting with calculus. Thank you.
1
vote
1answer
44 views

Rates of change question?

A boat is observed from top of a $100\ \text m$ high cliff. The boat is travelling towards the cliff at a speed of $50\ \text{m/min}$. How fast is angle of depression changing when angle of ...
0
votes
0answers
30 views

Difficult Integral in functional basis

Let $$g(x)=\int f\prime(x)\left[\frac{4}{3}x^2+4x^3+(2x^2+4x^3)f(x)+6x^2f^2(x)+xf^3(x)\right]dx$$ express $g(x)$ in terms of $\{1,x,x^2,x^3,....\}$ and $\{f(x),f^2(x),f^3(x),...\}$. Is there a clever ...
3
votes
2answers
75 views

Proof of the derivative of $a^x$ [duplicate]

I've tried for a while myself from first principles and applying various rules, but always end up going in circles. I've gotten as far as $$ y = a^x $$ $$ \frac{dy}{dx} = a^x \left( \lim_{x ...
1
vote
3answers
105 views

Question releating to the $\int^x_1\frac{\ln(t)}{t+1}$

If $f(x)=\int^x_1\frac{\ln(t)}{t+1}dt$ if $x > 0$. Compute $f(x) + f(1/x)$. As a check, you should obtain $f(2)+f(1/2)=(\ln2)^2$ I have tried evaluating the integral ...