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Questions tagged [calculus]

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

0
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1answer
11 views

If $f>0$ and $(\ln f)'=f'/f$ is Lipschitz continuous, are we able to conclude that $(\ln f)'''$ is bounded?

Let $f\in C^2(\mathbb R)$ be positive and $g:=\ln f$. Assume $$g'=\frac{f'}f$$ is Lipschitz continuous and hence $g''$ is bounded. Is $g'''$ bounded too?
2
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4answers
34 views

How to check if the sum of infinite series is convergent?

I have this exercise where I need to find if the sum of infinite series is convergent: $\sum_{n=1}^ \inf \frac{(\sin^2(x) - \sin (x) +1)^n}{\ln(1+n)} $ for x $ \in (\pi/2,\pi) $ Now I decided to do ...
-1
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1answer
20 views

What is the technique where you perform a “u-substitution” on a limit called?

I say u-substitution for lack of a better term, it's somewhat similar to u-substitution in integrals but a bit different. For example, if you want to evaluate: $ \lim_{x \to 0^+}x^{x^2}$ What you ...
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1answer
24 views

Prove that image of a continuously differential path is a null set

I got by the following exercise: prove that the image of a continuously differential path $\gamma:[0,1] \rightarrow R^2$ is a null set. I know that I need to find some cover of intervals $I_j=[a_j,...
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2answers
21 views

Implicit Differentiation with a Tangent Line

I was looking to implicitly differentiate $$-22x^6+4x^{33}y+y^7=-17$$ and found it to be $$\dfrac{dy}{dx}=\dfrac{132x^5-132x^{32}y}{4x^{33}+7y^6}$$Now, I am trying to find the equation of the tangent ...
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0answers
11 views

If $u(x)$ is harmonic and equal to $\phi(|x|)$, is $\phi$ continuously differentiable?

I was trying to show that radial harmonic functions on the unit ball (in $\mathbb{R}^n$) are constant. To this end, I suppose that $u$ is a radial harmonic function on the unit ball and write $$ u(x) =...
1
vote
1answer
21 views

extrema of function

In my class we learned the following theorem: If $f$ is an $n$-times differentiable function at a point $x$ and all the derivatives up until $m$ are equal to $0$ with $m$ being different than $0$ ...
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0answers
11 views

How to calculate type 1 and type 2 error for a proportion?

I'm having trouble calculating the $\alpha$(type 1) and $\beta$(type II) errors. It is well known that 40% of the cancer patients get better due to a new treatment. To test this claim, the treatment ...
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0answers
39 views

If $f'$ is continuous at the point $x_0$, does this imply $f$ is Lipschitz around $x_0$

Let $f : R^n \to R^m $ which is differentiable around the point $x_0$, whose derivative is continuous at the point $x_0$. Does this imply $f$ is Lipschitz around $x_0$ My intuition: Since $f'$ is ...
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1answer
19 views

Simulation: Generate random numbers that cluster around an average?

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
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2answers
23 views

Show that $\sin(x+a)/\sin(x+b)$ possess neither a maximum nor a minimum. [on hold]

Show that $\sin(x+a)/\sin(x+b)$ possess neither a maximum nor a minimum. How to solve this?
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0answers
10 views

Drawing a saddle on a contour graph

When drawing a contour graph of say, $f(x,y) = \frac12x^2 + \frac13y^3 - xy - x + y$ , there is a saddle point at point $(1,0)$. How can I determine the slopes of the two lines that meet at this ...
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0answers
18 views

Calculus-probability

I have been working on a problem that needs to be proved. Then, in the middle of the proof I was blocked because I could not upper bound this probability given bellow. Actually, the double sum of the ...
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0answers
17 views

Help with the proof that a sequence is convergent iff it is bounded and has a single subsequential limit.

Let $\{x_n\}$ denote a sequence for $n \in \Bbb N$. Prove that $\{x_n\}$ is convergent if and only if it is bounded and has a single subsequential limit. Let $P$ denote "a sequence is convergent" and ...
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2answers
36 views

Indefinite integration by parts

I was given an excersice to solve wich asks you to prove: $$\int_0^1 f(r) r dr = 0 $$ knowing that: $$\int_0^1 f(t) dt = 0 $$ After doing integration by parts I ended up with: $$\int f(r) r dr = ...
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2answers
24 views

Integration of $\int_0^\frac{\pi}{6} \cos^{-3}2x \sin2x \,\ dx $?

I tried substituting $x=\frac{\cos t}{2}$ but I didn't got anywhere. Thanks!
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0answers
10 views

Tangent line to a concave down function is greater than the function proof?

Suppose $f(3)=2, f'(3)=0.5$, also $f'(x)>0$ and $f''(x)<0$ on all real number x. It makes sense intuitively and graphically, but how do I prove rigorously that the tangent line $L$ of $f(x)$ at ...
4
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1answer
43 views

Find function $f(x)$ that satisfying differential relation

Suppose the functions $F(x)$ and $G(x)$ satisfying $$F(x)=f(x)-\frac{1}{f(x)}$$ $$G(x)=f(x)+\frac{1}{f(x)}$$ such that $F'(x)=(G\circ G)(x)$, with initial condition $f(\frac{\pi}{4})=1$ is given....
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0answers
20 views

Asymptote of the given curve

What is Asymptote of $y=x+\frac{1}{x}$ Please explain...in my book a expansion is written but i get asymptotes easily just equation the coefficients of x and y to zero.
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1answer
30 views

Question regarding gradient of a normal in Calculus [on hold]

I have this questions regarding gradient of a normal, but I'm struggling with it $H(x)=(f\circ g)(x) $ Given that $g(3)=7$ , $g'(3)=4$ , $f'(7)= -5$ Find the gradient of the normal to the curve of $...
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2answers
46 views

An application of mean value theorem

Let $f:[0,1] \to \Bbb{R}$ a differentiable function and $M>0$ such that $f'(x) \geq M,\forall x \in [0,1].$ Prove that exists an interval $I$ with length $\frac{1}{4}$ such that $|f(x)| \geq \...
3
votes
3answers
42 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
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2answers
39 views

How to calculate area of an ellipse based on its formula?

How can I determine the area of a half-ellipse if all that is given is $y = \sqrt{1-n^2x^2}$? I have tried both geometry and calculus, but without convincing results… Thank you
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0answers
14 views

How this term is generalised?

Q. Find the assymptotes of $r=\frac{2a}{1+2cos\theta}$ Sol. Putting $u=\frac{1}{u}$, the equation of the curve becomes $2au=1+2cos\theta$ When $\lim_{u \to 0}$, then $cos\theta_1=\frac{-1}{2}=...
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votes
1answer
20 views

Derivative of a logarithmic function with $\frac{C}{x}$

The function is $$G(x)=4^{\frac{C}{x}}$$ I have $u=\frac{C}{x}$, then I calculate $$\frac{d}{du}4^u=u(4)^{u-1}$$ But does $$\frac{d}{dx}\frac{C}{x}=C$$ because I don't know if $C$ is a constant or ...
1
vote
1answer
49 views

How to calculate the following integrals?

How the calculate the following integrals? Therein $D$ is a constant. $$(1)\;\;\int_{0}^{2\pi}\frac{1}{1-D\cdot\cos\theta} d\theta$$ and $$(2)\;\;\int_{0}^{2\pi}\frac{1}{1+D\cdot\cos\theta} d\theta$...
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0answers
16 views

Integrating 3D function with time

I have an unknown function $\phi (x,y,z,t)$ and i want to try and compute $\int_{-h}^{\eta(x,y,t)} \phi_t \phi_{zz} dz$, is there a simple method to do this? or any method at all? Thank you :) edit: ...
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0answers
18 views

Darboux sums elementary question - am I correct

I'm very new to this material and I would like someone more experienced to give input if possible. $Q \subset \mathbb R^n$ is a box and $f: Q \to \mathbb R$ is a function. Let $\Xi_Q$ be the set of ...
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1answer
9 views

Partial derivative of 3D function with time

Having a little bit of trouble wrapping my head round this one. I have an undefined function of $\phi (x,y,z,t)$ and its partial derivatives $\phi_x$ and $\phi_t$, and i want to find $\frac{\partial}{\...
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0answers
32 views

i need some help with this integral [on hold]

$$\int _2^{\infty}\frac{1}{x(x^2-1)} dx$$ I worked out the integral to be $$-\ln (x)+\frac{1}{2}\ln (x+1)+\frac{1}{2}\ln (x-1)$$
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4answers
70 views

Integrating $\int \frac{1}{x(x^2-1)} dx$ [on hold]

$$\int \frac{1}{x(x^2-1)} dx$$ I tried a $u=x^2$ substitution, then partial fractions method, but the Wolfram answer had a different sign to what I got in one of the logarithms.
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0answers
37 views

I don't know how to solve the problem [on hold]

that is parallel to the line x-2y+5 squaltion of the tangent line to the curve y O. Make a sketci
3
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3answers
63 views

Finding the extrema of $f(x,y) = 2x^{4} - 3x^{2}y + y^{2}$

$$f(x,y) = 2x^{4} - 3x^{2}y + y^{2}$$ I found the stationary points of this function using the equations - $$\frac{\partial f}{\partial x} = 0 \qquad \frac{\partial f}{\partial y} = 0$$ I got $(0,0)...
1
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5answers
84 views

Evaluate the limit $\lim_{x\to + \infty}\left(\sqrt{(x+a)(x+b)} -x\right)$ for $a, b \in \Bbb R$ [on hold]

Determine the limit of the following or prove it doesn't exist: $$ \lim_{x\to + \infty}\left(\sqrt{(x+a)(x+b)} - x\right) \space \space \text{where}\ \space a,b \in \mathbb R$$ I know that the ...
0
votes
1answer
32 views

The parametric equation of a cone $z = \sqrt{x^{2} + y^{2}}$.

The given equation is - $z = \sqrt{x^{2} + y^{2}} , 0 \le z \le 1$ Let $x = r \cos t$, $y = r \sin t$ and $z = r$; where $0 \le r \le 1$ and $0 \le t \le 2 \pi$. Since $z$ is taken from $0$ to $...
2
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1answer
44 views

Why do most theories about derivatives use closed/open intervals?

How come most theories like the Mean Value Theorem, Intermediate Value Theorem, Bounded Derivative Theorem, all start with if $f$ is continuous on $[a, b]$, and differentiable on $(a, b)$?
2
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2answers
54 views

Is this a legitimate way to write the multivariable Taylor series formula?

While researching multivariable Taylor series I found this expression for a Taylor polynomial in two variables: $$T_n(x,y)=\sum_{i=1}^n\left((x-x_0)\frac{\partial}{\partial x}+(y-y_0)\frac{\partial}{\...
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votes
1answer
59 views

How do I solve the equation: $\log_2(x-1)+\log_2(3)=\log_{16}(2x)$ [on hold]

How do I solve the equation: $\log_2(x-1)+\log_2(3)=\log_{16}(2x)$ I know the basics of logarithm. I solved up to the point: $(x-1)^4 \cdot 81=2x$. I don't know if I did the right steps to get to ...
21
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1answer
785 views

Why does trying to compute $\lim_{x\to-\infty} {2x-1\over \sqrt{3x^2+x+1}}$ results in the negative of the answer given?

My textbook asks me to evaluate the limit $$\lim_{x\to-\infty} {2x-1\over \sqrt{3x^2+x+1}}$$ which evaluates to $-2\over\sqrt{3}$. The method in the book is to factor out $x^2$ from the root in the ...
0
votes
0answers
12 views

Solution by Fourier Metod to Elastic Wave Equation

I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media. \begin{equation} \begin{cases} \rho u_{...
1
vote
1answer
21 views

Find an equation of the tangent line of the exponential function at the point (0,1)

So I differentiated the expression $$ y=e^{2x}\cos(\pi x) $$ And I got $$ y'=2e^{2x}\cos(\pi x)-\pi e^{2x}\sin(\pi x)$$ But when from the given point $(0,1)$ when I plug in zero I get 2 I looked the ...
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votes
2answers
36 views

Do I have this idea of antiderivatives correct?

So is all it's saying that if there are two functions that have the same derivatives for every single $x$ in the interval, then $f(x) = g(x) + \alpha$, means that the second function is just the exact ...
0
votes
2answers
45 views

How to find the taylor series for $f(x) = \frac{1}{16-x^2}$ centered at 9 in summation notation

Having trouble finding the taylor series for the following function: $$f(x) = \frac{1}{16-x^2} \textrm{ centered at c=9}$$ I was trying to look at it in a way such that I could modify $\frac{1}{1-x}...
1
vote
3answers
38 views

Minimizing distance from a point to a parabola

Problem: The point on the curve $x^2 + 2y = 0$ that is nearest the point $\left(0, -\frac{1}{2}\right)$ occurs at what value of y? Using the distance formula, I get my primary equation: $L^2 = (x-0)^...
0
votes
0answers
8 views

Verifying the correct value of 3D Flux

I am attempting to calculate the 3D flux of the vector field $\vec{F} \ = \ 3x\vec{i} \ + \ 3y\vec{j} \ + \ 2z\vec{k}$ through the triangular surface bounded by the $(x, \ y, \ z)$ coordinates $(3, \ ...
0
votes
2answers
43 views

Differentiate the exponential function $f(x)= \frac{x^2e^x}{x^2+e^x}$

$$f(x)= \frac{x^2e^x}{x^2+e^x}$$ Using product rule and quotient rule I computed $$f'(x)=\frac{(x^2+e^x)e^x(x^2 + 2 x ) - x^2e^x(2x+e^x)}{(x^2+e^x)^2}$$ Is my computation correct so far?
0
votes
1answer
42 views

How to get derivative of integral with $\ln(x)$?

Find $f'(x)$ if $f(x) = \int_1^{ln(x)} e^{t^2} \,dt$ The correct way to solve it: $$f'(x) = e^{(\ln{x})^2} \frac{1}{x}$$ $$f'(x) = \frac{1}{x}e^{(\ln{x})^2}$$ I haven't seen an example like this ...
2
votes
0answers
39 views

The value of $a$ besides $1$ for which $\gamma =\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k^a}-\int_{1}^{n}\frac{1}{x^a}dx\right)$

I know that Euler-Mascheroni constant is given by $$\gamma =\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\int_{1}^{n}\frac{1}{x}dx\right)$$ I am not sure whether there exists a value $a$ ...
0
votes
0answers
27 views

Can someone explain how the error “M” was found in this question?

I'm doing error in linear approximation rn, and the equation is $M \over2 $ $(x-a)^2$. To find M, I have to assume $f$ is such that $f''(x) \leq M$, for each a. I'm confused on what M is and how to ...
0
votes
1answer
38 views

Solving integral with absolute value

With a given $x > 0$ (I think we could restrict it to $x \in [0, 3]$), I'm trying to find the following integral: $$ \int_{z = 0}^{\min(x,1)} \int_{y = 0}^{\min(x - z, 1)} |x -z -y -1| dydz $$ ...