Questions tagged [calculus]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
22 views

Theorem 5.13 in “Principles of Mathematical Analysis” by Walter Rudin L'Hospital's Rule L'Hopital's Rule

I am reading "Principles of Mathematical Analysis" by Walter Rudin. Thank you Saaqib Mahmood. I copied and pasted your text Theorem 5.13 on p.109: Suppose $f$ and $g$ are real and ...
3
votes
1answer
55 views

How to compare $\pi, e\cdot 2^{1/3}, \frac{1+\sqrt{2}}{\sqrt{3}-1}$

This is in the GRE exam where we are supposed to answer fast so I think there might be some trick behind this to allow us to do that. But so far the best I can do is to write $\frac{1+\sqrt{2}}{\sqrt{...
0
votes
1answer
18 views

For the function $g(x) = 8x^2 - x + 4$, at what tangent point is the instantaneous rate of change equal to $-1$?

It is a multiple-choice question with the options of: a) $(1,11)$ b) $(-1,13)$ c) $(0,4)$ d) $(2,34)$ Really struggling with how to approach this one. This style of question isn't covered at all ...
0
votes
1answer
23 views

Calculus for the Practical Man: Chapter 4, Problem 17 (Answer's differs from that of the book)

A tank is in the form of a cone with the point downward, and the height and diameter are each 10 feet. How fast is the water pouring in at the moment when it is 5 feet deep and the surface is rising ...
0
votes
2answers
31 views

Calculus for the Practical Man: Chapter 4, Problem 10 (Solve problem without trigonometry possible?)

My issue is that I am trying to avoid trigonometry as a means to solve this problem, because he goes into the calculus of trigonometric functions in the subsequent chapter, and, therefore, I think he ...
3
votes
4answers
51 views

(Pre-Calculus) How would you graph $𝑦 = (𝑥^2 − 1)(𝑥 − 2)^2$ by hand?

I've been doing some summer practice assignments for my upcoming calculus class, and I have been tasked with graphing $𝑦 = (𝑥^2 − 1)(𝑥 − 2)^2$ by hand. At first i started making a table of values, ...
0
votes
0answers
21 views

subdifferential of continuously differentiable function

Suppose $f(x)$ is continuously differentiable, $\partial f(x)$ is the subdifferential. Is the following true? See 8.8(c) on page 304 Variational Analysis Book $\partial f(x)=\{f'(x)\} \quad (1)$...
0
votes
0answers
38 views

which one is bigger $f(x)$ or $g(x)$? [on hold]

If we know $a+b=c+d$, and $1<c<a<b<d$, all of them are positive values. And $f(x)=\frac{1}{b-a} \frac {b^{x+1}-a^{x+1}} {x+1}$, $g(x)=\frac{1}{d-c} \frac {d^{x+1}-c^{x+1}} {x+1}$ for $x>...
0
votes
2answers
45 views

Find $a, b, c, d\in\mathbb{R}$ so that the $\lim_{x\to1} f(x)$ exists.

Find $a, b, c, d\in\mathbb{R}$ so that the limit: $$\lim_{x\to1} f(x)$$ exists, when: $$ f(x)=\begin{cases} \frac{ax^{3}-(b+c)x-(a+d)}{(x-1)^{2}} & x<1 \\ \frac{bx^{2}-ax+(b+d)}{(x-1)^{2}} &...
0
votes
3answers
39 views

Domain of $( \frac{x-1}{x-2\{x\}})^{1/2}$

Find the Domain of $\left(\frac{x-1}{x-2\{x\}}\right)^{1/2}$ where $\{\}$ denotes the fractional part function. $\left(\frac{x-1}{x-2\{x\}}\right)$ should be positive, $\left(\frac{x-1}{x-2\{x\}}\...
1
vote
1answer
32 views

Exponential of complex square root

Is there any way to simplify any further the exponential of a complex square root, as in the following expression: $$ e^{ a + \sqrt{x + i\cdot y}}, $$ where $a>0, x >0$ and $y<0$. If I were ...
2
votes
1answer
59 views

Find inflection points of $\frac{\cos x}{x}$

We have following function $$\frac{\cos x}{x}$$ I would like to have find inflection points of the function above First derivative $$f'(x) = \frac{-x\sin x - \cos x}{x^2}$$ Second derivative $$...
0
votes
1answer
25 views

Determining continuity in multivariable calculus

If f is a function defined throughout a disk centred at $(x_0, y_0)$, and if $f_x(x_0, y_0)$ and $f_y(x_0, y_0)$ both exist, then f is continuous at $(x_0,y_0)$. I know to be a function to be ...
3
votes
1answer
64 views

Why doesn't the graph of $x^2-\cos x$ look wiggly?

When I use Desmos to draw the graph of the function $f$ defined by $f(x):=x^2-\cos x$, the graph looks very similar to a quadratic function. Unlike the graph of, say, $x-\cos x$, it does not have any ...
0
votes
0answers
30 views

Not using Stokes theorem for line integral

I have a question for the integral of $f(x,y,z)=(y-1)dx+z^2dy+ydz$ over the curve of intersection between the surface $x^2+y^2=z^2/2$ and the plane $z=y+1$. I know that I can do this using Stokes ...
0
votes
2answers
26 views

Help with algebra in evaluating a limit

Good evening folks. I'm doing some self-study from the eighth edition of Stewart Calculus (metric version), and I ran into this problem on page 103: Evalute $$\lim_{x\to 1} \frac{x^{1/3} - 1}{x^{1/2}-...
0
votes
3answers
47 views

what did i do wrong when trying to prove the derivative of ln(x)

when trying to prove the derivative i ended up with $$\lim_{h \to 0}\frac{\ln(x+h)-\ln(x)}{h}=\lim_{h \to 0}\frac{\ln(\frac{x+h}{h})}{h}=\lim_{h \to 0}\ln((1+\frac{h}{x})^{(\frac{1}{h})}$$ and $$\...
2
votes
1answer
41 views

Calculate $\int\sqrt{(x^2+1)}dx$

Calculate $I$ =$\int\sqrt{(x^2+1)}dx$ I have tried calculating it using integration by parts: $$f'(x) = 1, f(x) = x$$ $$g(x) = \sqrt{x^2+1}, g'(x) = \frac{x}{\sqrt{x^2+1}}$$ $$\int\sqrt{x^2+1}dx = x\...
4
votes
2answers
68 views

How to prove $\sum_{n=-\infty}^ \infty {\rm sinc}\bigl( \pi(t-n)\bigr) = 1$?

Thank you by avance for your help. So, I found on this website, that $\sum_{n=-\infty}^{\infty} {\rm sinc}( \pi n)= 1$. But I could not find any way to prove it. I know it’s about fourrier, but I don’...
1
vote
2answers
30 views

How to evaluate $\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$, where t is constant

I need to evaluate the following one. Can't understand the method in my textbook. $$\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$$ My textbook is to let $\alpha=1-\sqrt{1-t^2}$, $\beta=1+\...
1
vote
1answer
33 views

Formation of differential equation of Circle two cases

Why is differential equation of circle at origin $x^{2}+y^{2}=c^{2}$ is $2x+2y \dfrac {dy}{dx}=0$ and the differential equation of circle at (h,k) , $\left( x-h\right) ^{2}+\left( y-k\right) ^{2}=c^{...
0
votes
1answer
29 views

Proving discontinuity of a function defined by a double integral

Show that $$f(\alpha)=\int_{\alpha}^1 dx \int_{\alpha}^1\dfrac{(x-y)}{(x+y)^3} dy$$ is not continuous at $\alpha=0$. I have found out the iterated integral at $\alpha=0$ which is $1/2$. But after ...
2
votes
4answers
90 views

Is my analysis of the series $\sum_{n=1}^\infty{\frac{\ln n}{n}}$ correct?

I came across the following series and I'm supposed to analyse whether it converges or not. $$\sum_{n=1}^\infty{\frac{\ln n}{n}}$$ My attempt: At first sight, the thought of using the integral test ...
9
votes
4answers
138 views

At what point of mathematical education can you start inventing new math?

I am a 2nd year student doing an honors program in math and statistics. Everything that I have been learning has been formulas, theorems, and mathematical concepts that other people have discovered/...
1
vote
1answer
36 views

Calculus for the Practical Man: Chapter 4, Problem 7

A man standing on a wharf is hauling in a rope attached to a boat at the rate of four feet a second. If his hands are nine feet above the point of attachment, how fast is the boast approaching the ...
0
votes
1answer
32 views

About double map function $D(x) = 2x\pmod 1$ iterations

Well, I have a basic question about double map function, let's go to it... The double map $D:[0,1) \to [0,1)$ is define by $$ D(x) = 2x\pmod 1 = \begin{cases} 2x & \quad \text{if } 0 \...
-3
votes
0answers
17 views

How do we calculate $\int_{i}^{f} (L_1 - L_2) dt$ where $L_1 = f(t)$ and $L_2 = f(t+dt)$? Thanks. [on hold]

I have two functions f(u) and g(u). I have to use these two functions to calculate a function $h(u) = \int_{i}^{f} (L_1 - L_2) du$ where $L_1 = func(f(u),g(u))$ and $L_2 = func(f(u+du),g(u+du))$.
1
vote
1answer
18 views

Proving limit of bounded convergent sequence is bounded

Question: Show that if $a \leq x_n \leq b$ for every $n$ and $x_n \rightarrow x$, then $a \leq x \leq b$. Proof: Let $\epsilon>0$. By assumption $a_n \leq x_n \leq b$ for all $n$. By definition ...
1
vote
0answers
73 views

Related rates problem: How rapidly is the area of an equilateral triangle increasing at the instant when each side is $69.28$ inches?

A metal plate in the shape of an equilateral triangle is being heated in such a way that each of the sides is increasing at the rate of ten inches per hour. How rapidly is the area increasing at the ...
-8
votes
0answers
52 views

Solve: $2y'' = e^y$ [on hold]

I wish to find the solution of the differential equation: $$ 2 y^{\prime \prime}=\mathrm{e}^{y} $$
0
votes
3answers
34 views

Find inflection points of $e^{\cos x}$

I got a function $$ f(x) = e^{\cos x}$$ I would like to find inflection points of the function above within range $[-2\pi, 2\pi]$ Find first derivative $$f'(x) = e^{\cos x}(-\sin x)$$ Find second ...
1
vote
0answers
29 views

What, if anything, is fundamentally special about plugging certain functions into certain ODEs?

While studying differential equations, I have noticed that several solution techniques are nearly identical. Reduction of order, variation of parameters (first-order version), and transforming ...
0
votes
1answer
38 views

“every regular functions is differentiable infinitely” [on hold]

Is there a theorem that states something like "every regular functions is differentiable infinitely"? I want to prove that $$\frac{1}{1+x^2}$$ (or similar "simple" functions) has a second derivative ...
2
votes
1answer
19 views

If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then…

If $f: R \to R$ is one one and differentiable function and graph of $y = f(x)$ is symmetrical about the point $(4,0)$ then, If $f'(-100) >0$ then roots of $x^{2} - f'(10)x - f'(10) = 0$ may be non ...
0
votes
1answer
28 views

Can the sum property of integrals be stated for indefinite integrals?

Some time ago I learned about the following property of integrals: If $ f $ and $ g $ are bounded, integrable functions on $\color{red}{[a, b]}$, then so is $f + g$ and $$ \displaystyle \int_\...
0
votes
4answers
89 views

Strange! Right solution of equation is partially wrong.

Let $0<a<1$ be arbitrary but fixed. The equation $$ \frac{x^2 (1+y)^2}{\left(a y + \sqrt{1+(1-a^2)y}\right)^2} = 1$$ in $y$ has according to straight-forward calculus and Mathematica two ...
2
votes
0answers
21 views

How to estimate $\sum_{2 \leq n \leq X} \frac{1}{(\log n)^A} $?

I was wondering how does one estimate $$ \sum_{2 \leq n \leq X} \frac{1}{(\log n)^A}? $$ where $A>0$. I feel like it should be $\ll \frac{X}{(\log X)^A}$... Any comments would be appreciated. ...
-2
votes
1answer
66 views

Calculating $\int_{\pi/4}^{\pi/2} \frac{x \cos x-\sin x}{x} dx$ [on hold]

How to calculate this trigonometric integral ? $$\int_{\pi/4}^{\pi/2} \frac{x \cos(x)-\sin(x)}{x} dx$$
0
votes
1answer
28 views

Taylor-Remainder divided by $(x-x_0)^n$ goes to zero as $x$ approaches $x_0$.

In a script about the Taylor-Theorem I'm reading I have stumbled across the property that for remainder $R_n(x,x_0): \lim_{x \to x_0} \frac{R_n(x,x_0)}{(x-x_0)^n} = 0$. Since there are no further ...
0
votes
0answers
24 views

Is the following integral setup to find the volume correct?

I need to find the volume of the cylinderical column standing on the area common to parabola $y= x^2$ and $x =y^2$ as base and cutoff by the surface $z= 1+y-x^2$. So I set up the integral as : ...
2
votes
0answers
47 views

Let $f(x)$ is a stricly decreasing, continuous function from $[0, +\infty)$ to $[0, +\infty)$ such as $\lim_{x\to \infty} f(x)=0$ [duplicate]

Prove that $$\int_0^\infty {f(x)-f(x+1)\over f(x)}dx=\infty$$ I already understood that it is enough to prove that exist some constant $C$, such as for all $f$, exist $a_f$, such that $$\int_0^\infty ...
1
vote
2answers
26 views

Values of θ for which r is minimized in a circle?

So, I've been stuck trying to figure out how to find the value(s) of θ where r is minimized in a circle? The circle's equation is r = 6 sin θ with an interval of 0≤θ≤2π. I believe the answer to be ...
0
votes
1answer
22 views

Which of given condition follows from mean value theorem?

Let $f:R \to R$ be differentiable which of the following follows from mean value theorem : (a) For all $a,b \in R$ if $c \in (a,b)$, then $\dfrac{f(b) - f(a)}{b-a} = f'(c)$ (b) For some $a,b \in R$ ...
2
votes
1answer
50 views

Find the range of $P= \frac{\sum_{1}^n x_i}{\prod _{1}^n (x_i^2+1)}$

When I solved the following problem. Find the range of $B\equiv\dfrac{x+y}{(x^2+1)(y^2+1)}$ where $x,y \in\mathbb{R}$. Solution. Setting $x=\tan u$ and $y=\tan v$, expression is $\frac{\sin 2u+\sin ...
2
votes
5answers
54 views

Proving $ \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^2+|y|}=0$

I'm unable to prove that $$ \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^2+|y|}=0$$ I tried with polar coordinates but I'm unable to reach a function that depends only on $\rho$ $$0\le\frac{\rho^3\cos^3(\...
1
vote
2answers
56 views

find largest $x$ based on $f(x,y)=c$

Consider the following equation: $$ \frac{p}{\left(a-x\right)^2+y^2}+\frac{1-p}{\left(b-x\right)^2+y^2}=1 $$ where $0<p<1$ and $a,b\in\mathbb{R}$. I want to look for the largest $x$ satisfying ...
0
votes
1answer
71 views

Difficult integrals with $\sin(\ln(1+x))$ [on hold]

Does anyone know how can I integrate these two functions : $\dfrac{x}{(1+x)^{1.5}} * \sin (\ln (x+1))$ and $\dfrac{(1-x)}{(1+x)^{1.5}} * \sin (\ln (x+1))$ Wolfram uses complex numbers and the ...
1
vote
1answer
57 views

Different results when calculating $\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x$ [duplicate]

1) Limit passing through $a^\cdot$ $$\begin{align*}\displaystyle\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x&=\displaystyle\lim_{x\to\infty}\left(\displaystyle\lim_{x\to\infty}\left(1+\frac{1}{x}\...
5
votes
2answers
119 views

How to find volume from Washer method?

So, I want to use Washer method to find $$\pi \int_{y=0}^2 (\sqrt{4y})^2-(\sqrt{4y}-0.1)^2 dy.$$ I've literally spent a day trying to do so, and when I treat each part separately and then subtract ...
1
vote
0answers
38 views

Behaviour of Laplace transform as $s \to \infty$

I have shown that if $f$ is a continuous function with scaling symmetry given by $f(mt)=m^kf(t)$ for any $m>0$, than $(\mathcal{Lf})(s) = m^{-k-1}(\mathcal{Lf})(\frac{s}{m})$. For this $f$, what ...