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

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3
votes
2answers
65 views

Power rule derivative in complex

Problem: Prove that if $f(z)= z^n$, then $f' (z)$ = $n z^{n-1} $ using the definition of the derivative.
0
votes
1answer
110 views

Calculus question with optimization homework

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area ...
2
votes
1answer
193 views

Showing that a set of points equidistant to two other points form a plane.

Question: if p and q are two distinct points in space, show that the set of points equidistant from p and q form a plane. Work Done: Note: I'm pretty sure this can be done with vectors and cross ...
0
votes
2answers
69 views

Tangent line Optimization Homework

Find the point on the line $6x + y = 9$ that is closest to the point $(2,7)$ Find $x$ and $y$. This is what I have attempted so far, but my $6x$'s cancel out, which leaves me with $0$. I do not ...
3
votes
1answer
651 views

Calculus of variations: Lagrange multipliers

Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the ...
2
votes
1answer
97 views

Optimization Homework

I need help with this math question: A farmer with 720 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is ...
1
vote
1answer
101 views

Using the Test for Divergence

So I have the problem $$\sum_{n = 1}^\infty (-1)^n \frac{n}{n+5}$$ and I have to figure out if the series converges or diverges. I start out by testing the conditions of the Alternating Serie Test, ...
1
vote
1answer
67 views

Equation $\sin(\pi x)=|\sin(\pi ax)|$?

When $a$ is an integer and $x\in[0, 0.5]$, is there any closed-form solution to the equation $\sin(\pi x)=|\sin(\pi ax)|$? I just want to find the largest solution $x\in[0, 0.5]$ (there are more ...
1
vote
1answer
33 views

Prove $u_n$ has the limit given

Let $u_n = \sqrt{u_n+1}~ and ~u_1=1$. Prove that $\displaystyle\lim_{n\to\infty} u_n = \frac{1}{2}(1+\sqrt5)$ Here is what I did. First, to show the sequence converges: $u_1 = 1, \space u_2 = \sqrt ...
1
vote
1answer
298 views

How to prove $\frac{\ln x}{\sqrt x}$ is decreasing.

I need to prove this before I can use the Integral Test to determine if the series is is converging or diverging. My series is from [1,infinity). I tried plugging in number and the function seems to ...
0
votes
2answers
66 views

Double integration problem

I have a problem regarding the integration below: $$\Omega : x^{2}+y^{2}\le1$$ $$\iint_{\Omega } (1-x^{2}-y^{2}) dxdy=?$$ How to integrate this? Could anyone give me a hint? Thank you Thanks for ...
2
votes
2answers
51 views

Partial Fractions and power of a factor with $x^2$

I just started working with partial fractions and hit a wall with splitting this one: $$ \frac{3x^2 + 2x + 1}{(x + 2)(x^2 + x + 1)^2} $$ I get here: $$ \frac{Ax + B}{(x^2 + x + 1)^2} + \frac{Cx + ...
1
vote
1answer
67 views

show that speed of particle remains constant

This is the question: Im not sure what to do with this problem.
2
votes
2answers
52 views

Finding $\int \frac{x^2}{(a^2-x^2)^{\frac{3}{2}}}dx$ using trigonometric substitution. Where did I go wrong?

Evaluate the following integral using trigonometric substitution $$\int \frac{x^2}{(a^2-x^2)^{\frac{3}{2}}}dx$$ I used the substitution $x=a \sin(u)$, then $dx = a \cos(u) du$. The integral then ...
1
vote
3answers
78 views

What function $f(x)$satisfies $\int{f(x)}dx=f(x)$ and how to prove it?

The answer may be $e^x$, but also $ne^x$. There must be a proof to know all those possible answers. EDIT: I think the question might be wrong because you don't understand me. Let's change the ...
6
votes
3answers
236 views

Differentiating both sides of a non-differential equation

I'm working on solving for $t$ in the expression $$\ln t=3\left(1-\frac{1}{t}\right)$$ and although I can easily tell by inspection and by graphing that $t=1$, I'd like to prove it more rigorously. I ...
0
votes
2answers
181 views

Difficult Integral Question

I'm trying to evaluate the following integral; $$\int e^{(x^2 - z^2)} (2x \cos(2xz) - 2z \sin(2xz)) dz$$ I've tried splitting it up, and using integration by parts, but it just isn't coming out in a ...
1
vote
3answers
105 views

Evaluating integral of $\int e^{-ax} \,dP$ where $P$ is the normal distribution $N(\mu,\sigma^2)$.

I realize questions regarding integrating the normal distribution are numerous, but I wasn't able to find an already answered question that helped me with this. The integral is: \begin{align*} ...
0
votes
7answers
152 views

Finding the limit of $\frac{\sqrt{1+x^2}}{x^2}$

I am kind of confused when it comes to finding this limit: $\displaystyle\lim_{x\rightarrow\infty}\frac{\sqrt{1+x^2}}{x^2}$ I did $\dfrac{\dfrac{1}{2}\dfrac{1}{\sqrt{\arctan(x)}}}{2x}$ then I am ...
1
vote
3answers
68 views

Double partial derivative at maximum or minimum points

Is this true? Let $U\subset \mathbb R^n$ be open set, and $u:U \to \mathbb R$ be differentiable every order . If $x_0\in U$ is a maximum point of $u$, then $$u_{x_i}(x)=0$$ and ...
3
votes
1answer
34 views

At how many points is this function continuous?

Question: Let $f$ be a function with domain $[-1, 1]$ such that the coordinates of each point $(x,y)$ satisfy $x^2 + y^2 = 1$. What is the total number of points at which f is necessarily continuous? ...
4
votes
2answers
181 views

Finding the limit of $x \sin\frac{\pi}{x}$

How can I find the limit of the following $x\rightarrow\infty$ $x \sin\frac{\pi}{x}$ I did $\dfrac{\sin\frac{\pi}{x}}{\frac{1}{x}}$ I took the derivative using l hospital and got. ...
0
votes
3answers
45 views

How to calculate this limit

I need find the follow limit: $\lim_{n\to\infty}\sqrt[n]{1^{\pi}+2^{\pi}+\cdots+n^{\pi}}$. Please help me. Thanks for your attention.
8
votes
5answers
581 views

Finding the limit of $\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$

How would one find the limit of $\displaystyle\lim_{x\to 0}\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$ I know I have to use the L'Hospital rule. $\displaystyle\lim_{x\to ...
0
votes
1answer
91 views

How find this integral $\int\frac{x^2+2x+1+(3x+1)\sqrt{x+\ln{x}}}{x\sqrt{x+\ln{x}}(x+\sqrt{x+\ln{x}})}dx$

Today,when I found some problem on wiki,and find the integral,and I suddenly saw one of these nice problem. $$\int\dfrac{x^2+2x+1+(3x+1)\sqrt{x+\ln{x}}}{x\sqrt{x+\ln{x}}(x+\sqrt{x+\ln{x}})}dx$$ ...
1
vote
2answers
94 views

What *really* are the local maxima and local minima

In math is the local max and local min just any peak ... point where slope of the function changes from positive to negative or vice-versa... Or are the LOCAL max and min just the highest point of the ...
1
vote
1answer
72 views

diffeomorphisme-exercice on differential calcul

Let $f:\mathbb{R^2}\to \mathbb{R^2}$ an application defined by $f(x,y)=(x+a\sin y,y+b\sin x)$ with $a$ et $b$ are two positives reel such as $ab<1$. 1- Prouve that $f:\mathbb{R^2}\to ...
5
votes
1answer
69 views

suspended cable problem with slack

Suspending a cable produces a hyperbolic cosine shape, but what happens if we orientate the problem in such a way where the rope begins at $(0,y)$ and ends at $(x,0)$ and we specify a rope of length ...
2
votes
3answers
230 views

Clarification of L'hospital's rule

I have a question regarding L'hospital's rule. Why can I apply L'hospital's rule to $$\lim_{x\to 0}\frac{\sin 2x}{ x}$$ and not to $$\lim_{x\to 0} \frac{\sin x}{x}~~?$$
2
votes
2answers
66 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
1
vote
2answers
29 views

Function Continuity on an Interval.

h is a continuous function on interval [a,b] and h(x) belongs in Q for all x. Which statement is true? (a) h is constant on the interval
2
votes
0answers
56 views

Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
2
votes
0answers
43 views

Proof verification - If g is integrable and $\frac 1g$ is bounded then $\frac 1g $is integrable.

Question: Given that if $g$ is integrable, $\frac 1g$ is bounded in $[a,b]$, then $\frac 1g$ is integrable as well. I would like to verify correctness of this proof. Thanks. $\omega_i(g) = ...
0
votes
1answer
35 views

Prove that there is at least one value c ∈ R such that the tangent line to f at (c,f(c)) is parallel to the tangent line to g at (c,g(c))

Let f and g be differentiable function on R such that f(a)=g(a) and f(b)=g(b) for some a < b. Prove that there is at least one value c ∈ R such that the tangent line to f at (c,f(c)) is parallel to ...
0
votes
3answers
2k views

What's the the integral of tan(4x) dx?

How is $\int \tan(4x) dx = \cos^2 (4x)$? Shouldn't it be $\ln(\sec(4x))$? I don't understand... please help Thank you
1
vote
2answers
224 views

Prove f(x)sinx+f'(x)cosx=1

Let f be a twice differentiable function on $\mathbb{R}$ such that $f''(x)=-f(x)$ for all $x$ are real. Suppose that $f(0)=0$ and $f'(0)=1$ i) Prove that $f(x)\sin x + f'(x)\cos x = 1$ for all $x$ ...
0
votes
1answer
20 views

Some question about even function

Given that $f$ is an even function, $f''(x)>0$. Which of the following are true? I. $f(0)<f(1)$ II. $f(4)-f(3)<f(6)-f(5)$ III. $f(-2)<\frac{f(-3)+f(-1)}{2}$ I understand I is true ...
1
vote
1answer
96 views

Maximal unique solution to an IVP.

In class we learned the existence and uniqueness theorems for differential equations. The weaker Picard-Lindelof states that for any IVP, $$ \begin{cases} x'(t) = f(t, x(t))\\ x(t_0) = x_0 \end{cases} ...
0
votes
2answers
67 views

Where are all the constants of integration?

I am an mechanical engineering student so I'm kind of ashamed to ask this question but I have a weak math background and am digging into some of my knowledge gaps. So my question is where are all of ...
1
vote
2answers
29 views

Finding $H'(1)$

Given $H(x)=F(x)G(x)$ Find $H'(1)$ Suppose: $F(1)=2$ $F'(1)=3$ $G(1)=5$ $G'(1)=-2$ Then using the product rule I assumed $H(1)=11$ because: $H(1)=((2)(-2))+((3)(5))=11$ using the product rule. ...
1
vote
1answer
23 views

Limits involving indeterminate forms

I am trying to find $\lim_{x \to \infty} (\ln x)^{1/x}$ and I have gotten it to $\lim\limits_{x \to \infty} \frac{1}{x \ln x}$. The answer is $1$ and I just can't figure out why.
2
votes
2answers
212 views

How can I calculate this integral by hand?

In solving this problem, I come up with the following integral: $$\int_{-1/(4\pi)}^0\frac{(s\log{(-4\pi s)})^{(2+n)/2}}{s^2}ds$$ where $n=1,2,3...$ By using Mathematica, I could get that the ...
1
vote
2answers
39 views

Derivative where x=a

If $f(x)=x^3+3x+2$ Find the number(s) a such that the tangent lines to the graphs of $f(x)$ and $f'(x)$ at $x=a$ are the same. So far I have come up with: $f'(x)=3(x^2+1)$ And when I graph both ...
1
vote
0answers
461 views

function that is continuous everywhere but not differentiable at x=1

So I easily came up with an example of a function that is continuous on $(-\infty, \infty)$, but not differentiable at $x=1$. I used $f(x)=|x-1|$. However, a friend in my calculus course asked me a ...
1
vote
0answers
25 views

If $f$ decreases and $\sum n^{a-1}{f(n)}^a<\infty$ then $\sum 2^n (2^n)^{a-1}{f(2^n)}^a$ converges.

Let $a\in(0,1]$ and $f:\mathbb{Z}_+\to\mathbb{R}_+$ be a decrasing function such that $$\sum_{n=1}^\infty n^{a-1}{f(n)}^a<\infty.$$ Prove that $$\sum_{n=0}^\infty 2^n ...
2
votes
4answers
107 views

Calculating $\int e^{-|x|} \, dx$.

I have been trying to calculate $\int e^{|x|} \, dx$, but merely splitting up in two cases for $x<0$ and $x>0$ does not give the desired result, which I got by calculating it on a CAS. ...
1
vote
2answers
114 views

What is the definite integral $\int_{-1}^1 \frac{1}{x} dx$ equal to?

What is the definite integral $\displaystyle\int_{-1}^1 \frac{1}{x} dx$ equal to? Is it $0$ like it would be if you just integrated and plugged in the bounds, or does the discontinuity at $x=0$ ...
0
votes
1answer
61 views

Calculate $\lim_{n\to\infty}\frac{1+a+a^2+\dots+a^n}{1+b+b^2+\dots+b^n}$

I'm fairly confident I got the right idea, but I'm not quite sure how to state the answer... \begin{align} ...
2
votes
3answers
196 views

How do I find $\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan 2x^2}$?

Can't understand how to solve limit like this: $$\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan2x^2}$$ My attempt is: $$\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan2x^2}=\lim_{x \to 0} \frac{\cos3x}{\tan2x^2}- ...
0
votes
1answer
56 views

How do I solve $y'''-5y''+11y'-15y=0$?

How do I solve the following linear ordinary differential equation with constant coefficients? $$y'''-5y''+11y'-15y=0.$$ Please help. Thank you.