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

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0
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3answers
362 views

How to show that a limit of a function exists with a variable

$$f(x) = \begin{cases} \frac{x}{\ln(5x+1)} , x>0 \\ \frac{2x+\alpha}{x+3} , x\le 0 \\ \end{cases}$$ For which $\alpha$ there exists a limit for $f$ in $x_0=0 $ ? I tried to go from the ...
2
votes
2answers
83 views

On the $\max\int_0^1 fg$ where $f$ is fixed

Let $f$ be a continuous function. What is the maximum of $\int_0^1 fg$ among all continuous functions $g$ with $\int_0^1 |g| = 1$?
1
vote
3answers
189 views

Find the total area between the curve and the x-axis.

Let $$y= \frac{2}{x^2},\;\quad 1 \le x \le 2$$ I'm asked to find the area between the curve and the x-axis. I think we have to use integrals to solve this? I'm not sure.
1
vote
1answer
89 views

question about an inequality in calculus [duplicate]

Please, carefully show that $$ e^{\pi} > \pi^e $$ You are not allowed to use a calculator! thanks
14
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2answers
306 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
0
votes
2answers
37 views

Please help me understand how to differentiate with logarithm?

$$\dfrac{\partial\ln E(p_x,p_y,V)}{\partial\ln p_x}=\dfrac1{E(p_x,p_y,V)}\cdot\frac{\partial E(p_x,p_y,V)}{\partial p_x}\cdot\dfrac{\partial p_x}{\partial\ln p_x}$$ How did we get this ...
0
votes
1answer
39 views

limit of the triple integral

I found $\iiint_E$ $1\over (x^2+y^2+z^2)^{n/2}$ dV, where E is the region bounded by the spheres with radiuses r and R (both positive), is $4\pi\left(\frac{R^{3-n}}{3-n}-\frac{r^{3-n}}{3-n}\right)$. ...
3
votes
1answer
64 views

If $I_{n} = \int_{0}^{1}x^n\cdot e^xdx,$ Then $\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) =$

(1) If $\displaystyle I_{n} = \int_{0}^{1}x^n\cdot e^xdx$. Then $\displaystyle \lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) = $ $(2)$ Value of $\displaystyle ...
2
votes
3answers
875 views

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
1
vote
2answers
168 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, ...
1
vote
1answer
33 views

Comparison of $S_{n}$ and $T_{n}$, where $S_{n} = \sum_{k=1}^{n}\frac{n}{n^2+nk+k^2}$ and $T_{n} = \sum_{k=1}^{n-1}\frac{n}{n^2+kn+k^2}$

Let $\displaystyle S_{n} = \sum_{k=1}^{n}\frac{n}{n^2+nk+k^2}$ and $\displaystyle T_{n} = \sum_{k=1}^{n-1}\frac{n}{n^2+kn+k^2}$ for $n=1,2,3,\dots$ Then which of the following options are Right. ...
1
vote
3answers
76 views

Let $\,f \colon [0,1] \to [0,\infty)\,$ be continuous . Suppose ..

I am stuck on the following problem that says: Let $\,f \colon [0,1] \to [0,\infty)\,$ be continuous . Suppose $$\int_o^xf(t)\mathrm{d}t \ge f(x) \,\,\quad\forall x \in [0,1] \tag{1}$$ Then ...
3
votes
4answers
116 views

Evaluating $\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{1-\sqrt{2}\sin x}$

How can I evaluate $$\lim_{x\to\frac{\pi}{4}}\frac{1-\tan x}{1-\sqrt{2}\sin x}$$ without L'Hopital rule. Using L'Hopital rule, it evaluates to 2. Is there a way to do it without using L'Hopital?
1
vote
1answer
32 views

Visual Optimization Question

I have the following problem: You want to put up a billboard and the ad costs $10,000 per day. Your market research team has produced the graph below. In it, R(t) predicts the extra sales, in ...
4
votes
3answers
280 views

How to evaluate this definite integral $\int_0^2(1-x^2)^\frac{1}{3}~dx$

A student asked me to help him calculate this definite integral $$\int_0^2(1-x^2)^\frac{1}{3}~dx$$ Although I have tried almost all the methods I have learned, I can not still do with it. I have tried ...
0
votes
1answer
74 views

Implicit Differentiation (Textbook says ignore dy/dx)

Find, $$x^2y^2-2xy+x=2$$ find the slope and the equation of the tangent line at $(2,0)$ by implicit differentiation. I have gotten this question wrong on a quiz and exam. I'm trying to understand ...
0
votes
1answer
24 views

Find the inflection points.

Find the local extreme points and inflection points of $y=2x^3 + 3x^2 - 12x.$ I know how to find the extreme points, but am confused on the inflection points. We use the second derivative to found ...
1
vote
1answer
38 views

Question about the conditions that satisfy the Alternating Series Test.

The series $\sum_{n=1}^{\infty}(-1)^{n+1}u_n$ converges if all three of the following conditions are satisfied: The $u_n$'s are all positive. The positive $u_n$'s are (eventually) nonincreasing: ...
2
votes
2answers
276 views

Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent

$p>1$ is a integer, Show a convergent series $\sum\limits_{n=1}^\infty a_n$, $a_n\in\Bbb R$, such that the series $$\sum_{n=1}^\infty a_n^p$$ is divergent p.s. If $p>1$ is not an integer ...
0
votes
1answer
58 views

Using negative powers in a derivative

I am misunderstanding how to use negative powers when taking the derivative of a function that was originally a quotient. For example: $$f(t)=\frac{t^2+t^3-1}{t^4}$$ $$f(t)=t^2+t^3-1+t^{-4}$$ ...
0
votes
1answer
31 views

Find $f'$ and state the domains of $f$ and $f'$ for the function $f(x) = x^2sec^23x$.

Find $f'$ and state the domains of $f$ and $f'$. $$f(x) = x^2sec^23x$$ $$f' = (2x)(sec^23x) + (x^2)(2sec(3x) * sec3xtan3x * 3)$$ $$f' = (2x)(sec^23x) + (x^2)(6sec(3x) * sec3xtan3x)$$ The problem ...
1
vote
1answer
63 views

Simplfiying a trigonometric polynomial

I was doing a derivative problem for calculus. The problem reads: $y=(\sec{x}+\tan{x})^5$ find $y'$. I have found a derivative, I believe is almost certainly correct as I have checked it with a ...
0
votes
2answers
41 views

Trigonometric Derivative

I have the derivative: $$\frac{\sin(5-\theta)}{\theta^2}$$ When I go to solve it I am using the quotient rule; However when I went to check it on wolframalpha it states that I should be using the ...
0
votes
3answers
229 views

Differentiate the function: $y = \tan\theta(\sin\theta + \cos\theta)$

Differentiate the function: $$y = \tan\theta(\sin\theta + \cos\theta)$$ How I approached the question: $$y = \tan\theta(\sin\theta + \cos\theta)$$ $$y = [\tan\theta\sin\theta] + ...
1
vote
1answer
63 views

Differentiate the function: $v = x\sqrt{x} + \frac1{x^2\sqrt{x}}$

Differentiate the function: $$v = x\sqrt{x} + \frac1{x^2\sqrt{x}}$$ My attempt: $$v = (x)({x^\frac12}) + {(x^{-2})(x^\frac{-1}2)}$$ $$v = ({x^\frac32}) + {(x^\frac{-5}2)}$$ $$v' = ...
2
votes
1answer
71 views

Regarding Cauchy Integral and Cauchy - Goursat Theorem on $g(z)=\int_C \frac{2s^2-s-2}{s-z} dz$

If $C$ is the circle $|z|=3$ $$g(z)=\int_C \frac{2s^2-s-2}{s-z} ds$$ then using Cauchy Integral $$g(2) =\int_C \frac{2s^2-s-2}{s-2} dz = 2\pi i (2(2^2)-2-2) = 8\pi i$$ But what can we say about ...
4
votes
1answer
92 views

Coloring $[0,1]$.

Here is an interesting coloring problem that I am unable to prove. Any help is appreciated. Can we partition the closed interval $[0,1]$ into finitely many intervals, such that each sub-interval is ...
3
votes
3answers
138 views

Finding $\displaystyle \lim_{x \to 0^+} x^{\sin x}$

Find $\displaystyle \lim_{x \to 0^+} x^{\sin x}$ This is how I started but I get to a dead end fast: $\displaystyle\lim_{x \to 0^+} e^{\ln x^{\sin x}}=\lim_{x \to 0^+} e^{\sin x \ln x}$ I ...
0
votes
2answers
362 views

how to calculate derivatives of integer valued functions?

Definitions: $[x]$ is the integer value of $x$. For example: $[4.3]=4$. $\{x\}=x-[x]$. Can someone help me calculate the derivative of the following functions, and determine where the derivative ...
0
votes
1answer
87 views

Help with solving for a flow curve:

So I'm preparing for a final exam in multivariable and our textbook posed the following question: find the flow lines of F(x,y) = (-y, x) Which I can't seem to solve correctly. We are told that a ...
2
votes
3answers
91 views

if f'(x)<g'(x) is f(x)<g(x)?

show that : (x+1)ln(x+1)-1$<$$x^2$/2 okay so i want to show that f(x) $<$ g(x) when x$>$0 f(x)=(x+1)ln(x+1)-1 and g(x)= $x^2$/2 (x+1)ln(x+1)-1<$x^2$/2 deriving the functions give ...
4
votes
1answer
59 views

Get from point A to point B efficiently.

This is a question I thought about while crossing the street. Suppose you're standing at the bottom-left corner of a rectangle. Your goal is moving to the the top-right corner, efficiently, ...
2
votes
1answer
74 views

Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
1
vote
2answers
76 views

cyclic functions

Can anyone help me with this exercise: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable function: Prove the following: If $f$ is cyclic with cycle $T$, then also $f'$ is cyclic with ...
1
vote
2answers
91 views

question about taylor series

Can someone explain why 1 and 2 use different Taylor series? Why i cant use $1/(1+r)$ = $\sum_{n=0}^{inf}(-1)^n r^n$ on 2,vice versa?
5
votes
2answers
702 views

Is there such a thing as partial integration?

Recently in my mathematics courses I was taught partial derivatives, and I wondered if the reverse exists for integrals. This may sound like a stupid question, and it probably is, but let me explain: ...
4
votes
3answers
431 views

Matrix derivative $(Ax-b)^T(Ax-b)$

I am trying to find the minimum of $(Ax-b)^T(Ax-b)$ but I am not sure whether I am taking the derivative of this expression properly. What I did is the following: \begin{align*} \frac{\delta}{\delta ...
1
vote
5answers
358 views

Domain of function $f(x) = \frac{1 - \sqrt{1 - x^2}}{x}$

If I had this function: $$f(x) = \frac{1 - \sqrt{1 - x^2}}{x}$$ How do I find the domain of $f(\cos x)$? I've tried but I haven't gotten anywhere. I set $\cos x = 0$ but it's $0$ in infinitely many ...
1
vote
2answers
24 views

Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution

How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?
2
votes
4answers
5k views

Use implicit differentiation to find dy/dx

$xy+x=2$ I know the answer is $-(1+y)\over x$, but I don't know how to solve to get the answer. Thank you!
0
votes
2answers
211 views

Proving a limit by Cauchy definition

for $a>1$: $$\mathop {\lim }\limits_{x \to \infty } \frac{{{a^x}}}{x} = \infty $$ So, by the definition of Cauchy for limits, for any $M>0$ I need to find a $D>0$ such that: $x>D$ ...
0
votes
1answer
42 views

Ordinary Differential Equations - Step by step

hi i'm in first year calculus 2 Differential Equations, I'm having trouble with the steps of integrate the trig identity. find the general solution of the equation (1/x)y'= 4 cos (2x)
0
votes
2answers
57 views

$|f(x)| \le |x|^\frac53$ for all $x$, then $f$ is differentiable at $x = 0$

Prove that if $|f(x)| \le |x|^\frac53$ for all $x$, then $f$ is differentiable at $x = 0$. I have no idea where to begin with this question, can someone help me out with this?
2
votes
3answers
160 views

Integral with a limit; integral and inequality

I am trying to solve the following problem. $$ \lim_{h \to 0} \int_0^h\frac{\sqrt{t^2+9}}{h}\mathrm{d}t $$ My presumption is that I should just evaluate the function at $0$, but I can't justify why ...
1
vote
1answer
90 views

Review of calculus course over the break

I am deeply sorry if this thread or discussion topic does not belong to this forum, but I have no idea on where to post this issue of mine. Essentially I have finished a Calculus 1 course, but kind ...
1
vote
2answers
285 views

Epsilon delta proof min

http://www.milefoot.com/math/calculus/limits/DeltaEpsilonProofs03.htm I've been studying these épsilon delta proofs. In the non-linear case, he gets: ...
0
votes
1answer
53 views

basic calculus inequality

Can anyone give me a hint on how to prove this inequality? I've tried various algebraic manipulations with exponentiation, but I haven't gotten anywhere. Thanks! $\frac{1}{b-a}\int_a^b ...
2
votes
1answer
95 views

Can all conservative vector fields from $\mathbb{R}^2 \to \mathbb{R}^2$ be represented as complex functions?

Considering that such a vector field $(M,N)$ is conservative iff for $M,N$ differentiable, $\frac{\partial{N}}{\partial{x}} = \frac{\partial{M}}{\partial{y}}$, we only have one of the two ...
0
votes
2answers
308 views

What is the area outside of $r=1$ and inside $r=2 \cos(3 \theta)$?

What is the area inside the polar curve $r = 2 \cos(3 \theta)$ but outside of the circle given by the polar equation $r = 1$? A picture of the polar curve is at ...
1
vote
2answers
224 views

Mean Value Theorem to find inequality

How do I use the mean value theorem to find $$ 1+x < e^x < 1+xe^x $$ for all x>0 I'm not really sure what function I can use or if I can use any function to show with MVT. I tried using ...