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

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1answer
16 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 $\frac{dr}{dx}$ ?
0
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0answers
6 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 ...
11
votes
2answers
53 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?
1
vote
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.
0
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0answers
25 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} = ...
12
votes
4answers
74 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 ...
0
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1answer
9 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
12 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
13 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 ...
0
votes
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
votes
1answer
40 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} ...
0
votes
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 ...
0
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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?
0
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1answer
25 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 ...
0
votes
3answers
20 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.
0
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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 ...
0
votes
2answers
31 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
20 views
0
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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 ...
1
vote
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 ...
4
votes
1answer
68 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 ...
0
votes
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 } ...
0
votes
2answers
38 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 ...
1
vote
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 ...
1
vote
1answer
20 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} = - ...
0
votes
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 $(1,0)$. ...
1
vote
2answers
30 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 ...
0
votes
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 ...
1
vote
2answers
39 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 ...
0
votes
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 ...
0
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2answers
26 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.
0
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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
votes
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 ...
2
votes
2answers
28 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
votes
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
votes
3answers
91 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
votes
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 ...
0
votes
2answers
19 views

Find the asymptote of $(9x^3 + 5)/(2x^3+\sqrt{x^6+2})$ , if any

Find the asymtode of $(9x^3 + 5)/(2x^3+\sqrt{x^6+2})$ i know how to find the horizontal and oblique asymptotes but i am stuck with this question. how to evaluate the square root part as that too will ...
0
votes
1answer
14 views

Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$.

Statement of the problem: Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$. I've gone through Calc. I, II in the ...
0
votes
3answers
50 views

how to integrate $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $? [on hold]

what to find $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $ ? Could you give me a hint? Thanks!
-4
votes
1answer
33 views

integrate by parts: $\int \cosh^2(x)dx$ please show solution step by step [on hold]

Integrate by parts: $$\int \cosh^2(x)dx$$ Please show the solution step by step. I actually somehow found my self in a loop solving the integral: = cosh(x) sinh(x) - int (sinh(x) (-sinh(x)) (x) = ...
3
votes
2answers
29 views

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation

If $4x^2+5x+xy=4$ and $y(4)=-20$, find $y'(4)$ by implicit differentiation. I implicitly differentiated the equation, but I don't see how I can use $y(4)=-20$ to my advantage.
0
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0answers
14 views

The vector form for the tangent line to the graph of y=4x^2−5x+1 at x=3 is

Attempt: y = 4x^2-5x+1, x = 3, y = 22 => (3,22) y' = 8x-5, y'(3) = 19 y-y_1=m(x-x_1), y-22 = 19(x-3), y = 19x-57+22, y= 19x-35 Finding the y-intercept and x-intercept To save time I got ...
2
votes
3answers
36 views

What is the derivative of $y = x^{x+1}$

Can someone walk me through how to solve this derivative? We went over it as such: $y = x^{x+1}$ Then take the natural log (ln) of both sides $\ln(y) = (x+1)\ln(x)$ I get lost as to why that ...
1
vote
0answers
16 views

Differences between directional derivatives

In our Calc 3 class, we have started doing directional derivative and their applications. So, for a function $f(x,y)$, the value of $f_{xx}f_{yy}-f_{xy}f_{yx}$ is used to determine what type of ...
1
vote
2answers
34 views

Integration with square root in denominator

I am honestly embarrassed to ask this because i feel like i should know how to do this but: $ \int \frac{x}{\sqrt{2x-1}}dx $ Try to use u-substitution please
0
votes
1answer
32 views

Calculus sequences And series

Find the values of $x$ for which the series $\sum_o^\infty \frac {(x+3)^n}{2^n}$ converges. I took it as $(\frac {x+3}2)^n$ then used the rule of summation of $r^n= \frac 1{1-r}$ then found ...
0
votes
3answers
42 views

find the derivative of an integral

Find $f'(x)$ where $f(x)$ is the integral from ${\sqrt{x}}$ to $x$ of $e^x-e^{t^2} dt$ Is there an easy way to do this using the fundamental theorem of calculus because if I try to ingretate w.r.t ...
1
vote
2answers
83 views

How to integrate $\int \frac{\sqrt{x}}{x+1}dx$?

How to integrate $$\int \frac{\sqrt{x}}{x+1}dx$$ Can I substitute $x+1$ with $u$?