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

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Bounding surface area from below

We're being introduced to the surface area formula, the integral from a to b of 2πf(x)sqrt(1+(dy/dx)^2). We are using a test function, 1/x from x=1 to x=5, and the surface area is the integral from 1 ...
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0answers
11 views

Justification of using logarithm to simplify limits

Sometimes one needs to find limits like $$L=\lim_{x\to a}f(x)^{g(x)},$$ which usually are simplified to $$L=\exp\left(\lim_{x\to a}g(x)\ln f(x)\right),$$ where the new limit is found e.g. via ...
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0answers
11 views

Construction of a function

Give an example of a function that is partial differentiable and differentiable but not continuous partial differentiable . One example I thought (but is wrong) is the function: $$f(x, ...
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1answer
17 views

What is wrong with my general solution and how to find $f(x)$ such that $\lim_{x\to \infty} (\frac{y}{ f(x)})=1$

Given that $$y=\frac{1}{w}$$ Here is my working: $$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Auxillary Equation: $$a^2+2a+5=0$$ $$a=-1+2i,-1-2i$$ C.F $$w=e^{-x}(C \cos 2x+ E ...
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1answer
32 views

Convergence to zero

I need to prove that if $n \rightarrow \infty$ then this sum converges to zero. $$ \sum_{k = [\frac{n}{2}]}^n {n \choose k} Q^k \cdot \left( 1-Q \right)^{n-k} $$ In this case $Q$ is constant ...
4
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1answer
34 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
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0answers
13 views

Real world situation/model using (factored) higher degree polynomial function?

(Form A) $$y=3x^5+23x^4-7x^3+x^2-4x+9$$ (Form B) something in factored form $$y=(x-1)^2(x+2)^5(x-5)^4(x+7)$$ 2 part question: 1) Anyone know of examples where these types of functions arise? 2) Is ...
4
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1answer
30 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
2
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1answer
24 views

How to get rid of $(\frac{dw}{dx})^2$ term in a differential equation

My try: $$y=w^{-1}$$ $$y'=-w^{-2} \frac{dw}{dx}$$ $$y''=\frac{2}{w^3} \frac{dw}{dx} - \frac{1}{w^2} \frac {d^2w}{dx^2}$$ Substituting these to the first expression : ...
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1answer
8 views

Are the following vectors in the range of A

Let $A$ be the following matrix: $$ \left( \begin{array}{cccccc} 1 & 2 & 1 & 3 & 2 & 1\\ 2 & 0 & 3 & 2 & 3 & 0 \\ 4 & 2 & 1 & 1 & 2 & 1 \\ ...
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1answer
5 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
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4answers
52 views

If $r=\sqrt{x^2+y^2}$, what is $\frac{dx}{dr}$ and $\frac{dr}{dx}$?

If $r=\sqrt{x^2+y^2}$, what is $\frac{dx}{dr}$ and $\frac{dr}{dx}$ ?
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0answers
8 views

Uniform bound of the integral $ \int_{r}^{\infty}{(\frac{1}{\sinh s}\frac{\partial}{\partial s})^2 K_{2+i\sigma}(s) ds} $

Denote $K_{z}(s)=(\frac{s}{2})^{-z-\frac{1}{2}}J_{z+\frac{1}{2}}(s)$, Where $J_z$ is the standard Bessel function of order $z$. Now Set $$ g(\sigma)=\int_{r}^{\infty}{(\frac{1}{\sinh ...
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2answers
90 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
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5answers
53 views

Does $\int_0^{\infty}\frac{x\hspace{1mm}dx}{x^3+1}$ converge?

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge? Can some explain how to approach this problem? All ideas are appreciated
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1answer
29 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$?

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
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0answers
27 views

Trouble with derivative

I will like assistance in differentiating the following function: $$ f(X,\alpha) = \frac{A_{i-1}}{\alpha_i} \exp\left(- \frac{X-t_{i-1}}{\alpha_i}\right)$$ with respect to $X$ where $$ A_{i} = ...
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4answers
92 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
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1answer
11 views

Domain of dericative of a function [on hold]

Give an example of a function the domain of whose derivative is a PROPER subset of its own domain.
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0answers
13 views

Differentiability find a and b

If I have the function $f(x,y)=|x|^a|y|^b$. How can I find the values of $a$ and $b$ that make the function differentiable on $\mathbb{R^2}$? How would I approach this problem I am quite confused?
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0answers
15 views

Asymptotic limit of a definite integral

I would like to solve the following integral: $$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$ in the limit where $b$ is small and $a$ is large ($a$ and $b$ are positive ...
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1answer
18 views

Average velocity of a ball thrown up in the air.

A ball is thrown straight up in the air with a velocity of $65 \,m/s$. After $t$ seconds the height is given as $y=65t-16t^2$. Give the average velocity for the time period beginning when $t=1$ and ...
3
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1answer
43 views

Solution to trigonometric derivative

Version 2 For \begin{align} &x(t)\text{:=}\cos (t)+\cos (2 t)+1&\\ &y(t)\text{:=}\sin (t)+\sin (2 t)&\\ \end{align} how would I go about proving that the solutions to \begin{align} ...
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2answers
36 views

If $f(x) = x^3-3x+1.$, then no. of distinct real roots of $f(f(x)) = 0$

If $f(x) = x^3-3x+1.\;,$ Then no. of different real solution of the equation $f(f(x)) = 0$ $\bf{My\; Try::}$ Given $f(x) = x^3-3x+1\;,$ Then $f'(x) = 3x^2-3 = 3(x-1)(x+1)$ Now for max. and ...
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0answers
15 views

When a function is non differentiable in every point does it mean that it has no instantaneous rate at every point? [on hold]

For example Weierstrass function,does it has instantaneous rate at every point?
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1answer
26 views

find the centroid of shown plane figure

Here is what I have so far equation of top curve : $f(x) = -\frac{4}{4.5^2}x^2 + 4$ equation of circle : $g(x) = \sqrt{1.8^2 - x^2}$ By symmetry, $\overline{x} = 0$ How do I go about finding ...
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3answers
21 views

Tangents and Exponential Curves

How do you find the gradient of a line given that it is a tangent to a curve? For example, if $y = mx$ is a tangent to the curve $y = e^{2x}$, how do I find $m$?
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2answers
18 views

Is it possible that a continuous function in every point has a discontinuous instantaneous rate at every point?

when x changes continuously,so does y.However as x changes continuously, Dy/Dx changes abruptly it goes nuts.
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1answer
14 views

Show that if a is a positive constant, then x = 0 is the only critical point of f(x) = x + a √ x.

Okay. So I've plugged in a positive constant (2) to see what happens when you take the derivative. Always gets me some variant of root x in the denominator, and giving me no critical points, rather ...
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2answers
33 views

When $a\ll b$, how to approximate $f = \int_0^a \sqrt{b^2+x^2}/\sqrt{a^2-x^2} \, \, dx$?

Suppose $a\ll b$. How do I then approximate $$\int_0^a \frac{\sqrt{b^2+x^2}}{\sqrt{a^2-x^2}}dx$$ ? I think that maybe Taylor approximation may help, but I am not sure how to proceed. My physics ...
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0answers
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1answer
32 views

Is it true that $\lim_{ x\to a}(f(x)/g(x)) = f(a)/g(a)$ without the assumption $g(a)\ne 0$?

Test prep problem: If f and g are continuous functions, then $\lim_{ x\to a}(f(x)/g(x)) = f(a)/g(a)$. State whether this is True or False. Provide proof or a counterexample. While this is the ...
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1answer
24 views

Does the definition of derivative exclude the possibility for discontinuous rate of change?

Is it possible to have a function whose instantaneous rate at every X are different to each other such that there are no pattern of gradual change between them but the definition of derivative fails ...
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1answer
70 views

If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$, [duplicate]

If $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function such that $f(x)=0$ for only finitely many values of $x$, then prove/disprove the following : $(a)$ Either $f(x) \geq 0 ~~\forall ...
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2answers
39 views

How can I prove that no derivative exist withing this function?

Our teacher challenge us a question and it goes like this: The derivative of a function is define such as $$\begin{cases} 1 & \text{if } x>0 \\ 2 & \text{if } x=0 \\ -1 &\text{if } ...
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2answers
39 views

How to properly state as to why $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges.

So I know that $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges, because the highest power in the numerator is $n^\frac{3}{2}$ and the highest power in the numerator is $n^4$, so I have ...
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1answer
23 views

Is the line through $(-4, -6, 1)$ and $(-2, 0, -3)$ parallel to the line through $(10, 18, 4)$ and $(5, 3, 14)$?

Problem statement: Is the line through $(-4, -6, 1)$ and $(-2, 0, -3)$ parallel to the line through $(10, 18, 4)$ and $(5, 3, 14)$? My attempt: For the first line, we know the vector equation ...
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1answer
21 views

Multivariate Calculus - Partial Derivatives - Implicit Differentiation - Chain Rule

Let $z = z(x,y)$ be defined implicitly by $F(x, y, z(x,y)) = 0$, where $F$ is a given function of three variables. Prove that if $z(x,y)$ and $F$ are differentiable, then $$\frac{dz}{dx} = - ...
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2answers
23 views

Find the equation of the parabola given the tangent to a point and another point.

I have a problem with derivatives, I've been trying to solve but I was not able to do it. A parabola is tangent to the line $3x-y+6 = 0$ in the point $(0,6)$ and goes through the point $(1,0)$. ...
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2answers
34 views

Integral of pdf

I need to find the integral for this pdf but I don't know if I need to, or can, take the integral of two variables at the same time. $$ f(x;\theta)=\frac{x}{\theta^2} e^{-x^2/(2\theta^2)} ,\quad ...
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1answer
13 views

Gauss Curvature…Product of Minimum and maximum values

The function g(ϑ ) = cos2 (ϑ ) fxx (x0 , y0 ) + 2 cos(ϑ )sin(ϑ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in ...
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2answers
41 views

How to determine whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges.

I am trying to find whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges. I used the limit test, and it comes out as inconclusive since ...
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2answers
16 views

Normal vectors and tangent planes

Could you check my work please? Let me know if it's right or wrong. We have the level surface $$f(x, y, z) = xyz -6$$ The normal vector is equal to the gradient, so at the point $(a, b, c)$ $$\nabla ...
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2answers
27 views

Explain why it is necessary to restrict the range of inverse trig functions?

This is very confusing. Please help and use lower level vocabulary that is easy to understand.
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1answer
20 views

Differentials word problem

The Questions Use differentials to find the approximate amount of copper in the four sides and bottom of a rectangular tank that is 6 feet long, 4 feet wide, and 3 feet deep inside, if the copper is ...
4
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1answer
37 views

Area between curves $y=x^3$ and $y=x$

I've tried to done one of my homework problems for several times, but the answer doesn't make sense to me. The question asks to find the area between $y=x^3$ and $y=x$. Those are odd functions, and ...
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2answers
29 views

Fourier series, instantly determining $b_n$ once $a_n$ is found.

Find the Fourier series of the following function: $f(x) = \left\{\begin{align} 1+x,\quad -1\lt x \lt 0 \\ 1-x,\;\;\;\quad 0\lt x \lt 1\end{align} \right.$ $f(x+2) = f(x),\quad\quad -\infty \lt x ...
2
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0answers
11 views

Explanation of solving intersection of two planes

I understand that in order to solve for intersection line of two planes, you must find the cross product of the normal vectors of each plane which will be parallel to the line of intersection. That ...
3
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3answers
93 views

Evaluate the limit of $\ \tan \frac{\pi \sqrt{3x-11}}{x-5}$ as $x\to 5$

$$f(x)= \lim_{x \to 5} \left[5 \tan\left( \frac{\pi \sqrt{3x-11}}{x-5} \right)\right]$$ I'm not sure on how to approach this. Am I allowed to plug $x=5$ in the numerator and end up with ...
0
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1answer
29 views

find $\lim_{x\to2}\frac{|x-2|}{(x^2)-4}$

$$\lim_{x\to2} \frac{|x-2|}{x^2-4}$$ in this question when i replace $x$ with $h$, such that $h\to0$ and check for RHL and LHL.I get the same values for both RHL and LHL. what i do is ...