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

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1answer
44 views

Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $$\cdots=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=\cdots$$ I know the ...
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2answers
81 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
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1answer
71 views

L'Hopital's Rule with $\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$

(a) Show that $$\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$$ is a standard indeterminate form, but that L'Hopital's Rule does not give you any information about the limit. (b) Show ...
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3answers
80 views

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$. I need to show that for each $\epsilon$ there exists an $n_0 \in \mathbb{N}$ such that $ \forall n \geq n_0: |2^{\frac{-1}{\sqrt{n}}}-1|\lt ...
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1answer
85 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
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1answer
29 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
3
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0answers
71 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
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2answers
63 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
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1answer
67 views

Determining the infinite limit of a Riemann' sum

I need to evaluate the following $f(x)=x^2 + 2x - 5$, on $[1,4]$, by using the riemans sum and limiting it to infinity. I have set up everything $\Delta x=\frac{3}{n}$. $x_i=1+\frac{3i}{n}$, I would ...
17
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3answers
471 views

How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?

Question: Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$ I think this ...
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1answer
54 views

Given that the graph of $f$ passes through the point $(1, 6)$ and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$, find $f(2)$.

As in the title - we assume that the graph of $f$ passes through $(1,6)$ (i.e. $f(1) = 6$) and that the slope of its tangent line at $(x, f(x))$ is $2x + 1$ and we are asked to find $f(2)$. How does ...
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2answers
72 views

Find $\lim_{h\to0^+}\frac{\sqrt{h^2+4h+5}-\sqrt{5}}{h}$

I am trying to find the limit $$\lim_{h\to0^+}\frac{\sqrt{h^2+4h+5}-\sqrt{5}}{h}$$ My approach is to divide both the numerator and denominator by $h=\sqrt{h^2}$ since $h>0$. Then we get ...
3
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1answer
131 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
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3answers
42 views

Isosceles Triangle With Height Limiting To Zero

The figure shows an isosceles triangle $ABC$ with $\angle B = \angle C$ . The bisector of angle $B$ intersects the side $AC$ at the point $P$. Suppose that the base $BC$ remains fixed but the altitude ...
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1answer
96 views

Evaluate $\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$ [duplicate]

How can we compute $$I=\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$$ Mathematica gives $I=0$.
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1answer
118 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
0
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1answer
169 views

Partial derivative of a function of a matrix

I have this function of a matrix and two vectors: $$f(x, V, y) = x^T V y$$ where $$V \in R^{d \times d}, x \in R^{d}, y \in R^{d}$$ Basically I need to optimize (find the minimum) of this function ...
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2answers
58 views

Prove that $f$ Contains A Fixed Point Using The Intermediate Value Theorem

Use the Intermediate Value Theorem to prove that any continuous function with domain $[0, 1]$ and range a subset of $[0, 1]$ must have a fixed point. My approach: I will call this continuous ...
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2answers
47 views

Upper bound on $ \binom{a}{m+1}\sum ^m_{j=0} \binom{a-m-1}{j}/\binom{b}{j+m+1}$

Given $a,b,m$ such that $0<2m<a<b$. I would like to find out upper bound of $$S = \binom{a}{m+1}\sum ^m_{j=0} \frac{\binom{a-m-1}{j}}{\binom{b}{j+m+1}}$$ Anyone can help me please? Thank you ...
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0answers
38 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
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3answers
320 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
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3answers
103 views

simplify the expression $\arctan\frac{x\sin t}{1-x\cos t}$

Same as above, how to simplify it. I am to calculate its $n$th derivative w.r.t x where t is const, but I can't simplify it. Any help would be appreciated. Thank you.
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0answers
16 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
1
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1answer
169 views

How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is accurate to within 0.0001?

I'm very lost on the following problem and will appreciate your help very much. How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is ...
3
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3answers
382 views

Finding a limit to negative infinity with square roots: $\lim\limits_{x\to -\infty}x+\sqrt{x^2+2x}$

Find the limit of the equation $$\lim_{x\to-\infty} x+\sqrt{x^2 + 2x}$$ I start by multiplying with the conjugate: $$\lim_{x\to-\infty} x+\sqrt{x^2 + 2x}\left({x - \sqrt{x^2 + 2x}\over x - ...
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2answers
136 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
2
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3answers
129 views

Separable differentiable equations

Which of the following is a solution to the separable differentiable equation: $$\frac{dy}{dx}=\frac{xy}{\ln y }$$ $A.\ \displaystyle e^{|x|}$ $B.\ \displaystyle e^{\sqrt{\frac{x^2}2}}$ $C.\ ...
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1answer
33 views

How to calculate the partial derivatives of the composition $F(u(s,t),v(s,t))$?

Could someone help me to understand how to do this problem? I believe Partial Derivatives are used. Thanks!
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1answer
78 views

Typical material covered in Calculus 1 course?

I have a copy of Larson's Calculus: early transcendental functions, 2nd edition. I was wondering what material I would need to cover to have the equivalent of a Calculus 1 course at a University. I ...
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4answers
65 views

Min and Max of $ f(x,y)=\frac{x-y}{a-x-y}$

I'd like to find max and min of $$ f(x,y)=\frac{x-y}{a-x-y}$$ where $0\le x<y\le a/2$. Any one can suggest? Thank you
2
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2answers
129 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
3
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1answer
24 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
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2answers
59 views

Limit Different if I Factor and Simplify an Equation First.

So I have $$\lim_{t\to -1} \frac {t^2 - t - 2}{t^2 - 1} = \frac 32$$ The solution given was found by factoring and simplifying the equation and then taking the limit of numerator and denominator and ...
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0answers
105 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
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1answer
29 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
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2answers
64 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
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4answers
53 views

Surjective function - proving

$f: \mathbb{R}\to \mathbb{R}$ $f(x) = x^3 -2x^4$ In order to prove that $f$ is not surjective, my teacher told me to find that in most the $f$ is negative. And indeed, only for $0<x<0.5$ it's ...
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0answers
49 views

Minimize distance between the ships

I'm studying calculus from a the book "Calculus with Analytic Geometry by Georfe F Simmons", and I have a certain difficulty to solve the following problem: Ah noon the ship A is at a distance at ...
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0answers
43 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
1
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1answer
40 views

Surjective functions and cal'

$f,g: \mathbb{R}\to \mathbb{R}$ Both are also surjective functions. My question is if $f+g$ will be also surjective. I need to dis/prove it if it's true or false. Now, my friend told me it's false ...
4
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1answer
140 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
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2answers
59 views

Finding the differential of $y=(u+1)/(u-1)$

I'm having trouble with differentials. I've been trying to learn about them online using great resources like PatrickJMT but I'm having trouble finding examples for this kind or problem. I hate asking ...
4
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4answers
165 views

Finding $\lim_{n\to\infty}\sqrt[n]{3^n + 4^n}$ [closed]

Evaluate the limit $$\lim_{n\to\infty}\sqrt[\large n]{3^n + 4^n}.$$ Sorry, I'm not really sure how to find this limit. Can someone share a clever trick for this?
2
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1answer
106 views

Calculus - The proof of arithmetic infinity limits

I having trouble to understand the proof of arithmetic infinity limits. (I'm quoting from my learning book) f,g are functions and lets assume that : $$\lim_{x \to x_0}f(x)=L \mbox{ (final)}$$ $$ ...
3
votes
1answer
42 views

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$ I would like for you guys to have a look at my approach and give my advice regarding the solution and whether there's a ...
2
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0answers
66 views

Derivative changes sign for continuous and differentiable function

Give $f$ is continuous and differentiable, if $f'(a) < 0 < f'(b)$, can we say there exists a $c\in (a,b)$ such that $f'(c) = 0$ ? My gut feeling is yes, using Rolle's theorem. If $f(a) = ...
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2answers
69 views

Maclaurin series of the function $\frac{x^2}{2+3x^2}$

I got this question: Find the Maclaurin series of the function $\frac{x^2}{2+3x^2}$ and find its domain of convergence. I tried using the binomial series $(1+x)^m = 1 + \sum_{k=1}^{\infty}{m \choose ...
3
votes
3answers
145 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
3
votes
2answers
108 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
2
votes
3answers
217 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$