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

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1
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1answer
52 views

Approximating an integral with taylor series

I am working on the following homework problem: "Assume that $\sin(x)$ equals its Maclaurin series for all $x$. Use the Maclaurin series for $\sin(5x^2)$ to evaluate the integral $$\int_0^{0....
0
votes
3answers
242 views

How to evaluate $\lim\limits_{n\to\infty} {\sin({b\over n})+\sin({2b\over n}) + \ldots + \sin({nb\over n})\over n}$ by relating it to a Riemann sum? [closed]

Evaluate the following limit by relating it to a Riemann sum: $$\lim_{n\to\infty} \frac{\sin\left(\frac{b}{n}\right)+\sin\left(\frac{2b}{n}\right) + \ldots + \sin\left(\frac{nb}{n}\right)}{n}$$
6
votes
2answers
223 views

What's a good primer from linear algebra to spherical harmonics?

I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials ...
1
vote
0answers
86 views

A question on continuity of a piecewise function with 4 constants

I have this function, and I need to find the values of $a, b, c$ and $d$ so that $f(x)$ will be differentiable everywhere. $$f(x)=\begin{cases} ax+b, & x<-2 \\ x^2+c, & -2\le x\le2\\ {d\...
20
votes
5answers
1k views

Is there an epsilon-delta definition of the second derivative?

Is there an epsilon-delta definition for the second derivative? I know that there is such a definition for the first derivate $f'(x)$ which can be derived from the limit $f'(x) = \lim_{y\rightarrow x}...
1
vote
1answer
133 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
5
votes
2answers
252 views

Why does the Hessian work?

I am working through Susskind's 'The Theoretical Minimum' (on physics) – it also includes some maths. In particular, there is an interlude for which he discusses partial differentiation. He discusses ...
4
votes
1answer
105 views

Prove that for every positive integer $n, \exists c_n$ such that $f(c_n) = f(c_n+1/n)$

Let $f:[0,1] \to [0,1]$ be a continuous function with $f(0) = f(1)$. Prove that for every positive integer, $n, \exists c_n \in [0,1]$ such that $f(c_n) = f(c_n+1/n), c_n \in [0,1-\frac{1}{n}].$ With ...
1
vote
3answers
276 views

Solving the differential equation $dr=(r\cos\theta +r\sin\theta)d\theta$

$dr=(r\cos\theta +r\sin\theta)d\theta$ In my book this is under separation of variables then i tried to factor out r and divide both sides then integrate both sides but where can i find my $C_1$? I ...
2
votes
1answer
62 views

Distance between two 3D lines

What is the distance between the 3D lines $x = \begin{pmatrix} 1 \\ 2 \\ -4 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix} t$ and $y = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix} + \begin{...
-2
votes
5answers
72 views

Understanding rates of change. [closed]

I have just started my unit on understanding rates of change and began by doing a warmup exercise. After reading through it I was asked to answer this question to make sure I understand but alas I do ...
3
votes
3answers
109 views

Is this inequality true? If yes, for what functions?

Let $B=B(0,1)\subset \mathbb R^2$. Let $u$ be a radially symmetric differentiable function on $B$ and $v=Ax+b$ be a linear function where $A$ is a $2\times 2$ matrix satisfies $A=-A^T$, and $b=(b_1,...
1
vote
1answer
59 views

Prove that $\{(x,y)\mid x\in\mathbb Q\}\cup\{(x,y)\mid y\in\mathbb Q\}$ is a connected subset of $\mathbb R ^2.$ [duplicate]

In my notebook, something is very briefly, not in detail whatsoever, path connectedness mentioned, and two assumptions are made about $x_1, x_2 \in \mathbb Q.$ If anyone can prove this I would greatly ...
2
votes
6answers
110 views

Differentiate the Function: $y=x^x$

$y=x^x$ Use $\frac{d}{dx}(a^x)=a^x \ln a$ My answer is: $x^x \ln x$ The book has the answer as $x^x\ (1+ \ln\ x)$ Am I missing a step?
10
votes
3answers
154 views

Logarithmic Integral II

While reviewing an old calculus book the following integral was assigned: \begin{align} \int_{0}^{1} \left( x^{a-1} - x^{n-a-1} \right) \, \frac{\ln^{2}x \, dx}{1-x^{n}} = \frac{2 \, \pi^{3} \, \cos\...
0
votes
0answers
11 views

minimization of adaptive basis functions

i am self-studying the topic of boosting, and trying to understand the following argument.i am failing to see the connection between 16.39 and 16.40 - why is this function of $\phi_m$ the optimal ...
2
votes
2answers
110 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
0
votes
2answers
59 views

Left and Right limits of $\lfloor\lfloor x\rfloor\rfloor$ at $x = 0$

I'm self-studying calculus from Larson's Calculus 8E and on page 102 and I don't understand why $$\lim_{x\to 0^-} \frac{f(x) - f(0)}{x-0} = \frac{\lfloor\lfloor x\rfloor\rfloor - 0}{x} = \infty $$ ...
3
votes
5answers
70 views

$U_n=\int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ .

$U_n= \int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ where Find $\lim_{n\to \infty} U_n$ without finding the integration I don't know how to start
4
votes
3answers
95 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
3
votes
2answers
2k views

Volume of hyperellipsoid

How can I compute the volume of the hyperellipsoid corresponding to a Mahalanobis distance $r^2 = (x-\mu)^{T}\Sigma^{-1}(x-\mu)$? I'm a bit confused because the answer involves $r$: $$V = V_{d} |\...
2
votes
10answers
127 views

Point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and the $x$-axis

How can I algebraically (without looking at the graph) find the point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and $x$-axis, in the interval $[0, \pi]$?
1
vote
2answers
63 views

What is the difference between these two limits, one with $\lim\limits_{x\to0^{+}}$, the other with $\lim\limits_{x\to 0}$?

I don't need an exact answer, I just need to know how these two limits would affect the answer and if there is a huge difference on how they are worked out, if they have a different step-by-step ...
1
vote
0answers
34 views

Sum of Bell Polynomials of the Second Kind

A problem of interest that has come up for me recently is solving the following $$\frac{d^{n}}{dt^{n}}e^{g(t)}$$ There is a formula for a general $n$-th order derivative of a composition as shown ...
2
votes
4answers
56 views

Roots of $f(x)=x-2+\frac{a-3}{x}$

I wanted to find the values of (a) for which the function $f(x)=x-2+\frac{a-3}{x}$ has more than one root. I know that the equation needs to be set equal to zero, from that step onward I have no idea ...
0
votes
3answers
73 views

How to intuitively arrive at the total derivative limit and the jacobian matrix?

I'm following this PDF and I need to understand how to arrive at the definition of total derivative geometrically. For now, what I understand is that, from the original definition of derivative: $$\...
2
votes
1answer
53 views

The Euler-Lagrange equation yields a constant function?

My functional is $J[f] = \int_{-\infty}^{\infty} f(x) \log f(x)\,dx$. I want to maximize it using the calculus of variations. In order to use the Euler-Lagrange equation, I define $L(t, y, y')$ such ...
0
votes
2answers
79 views

Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the ...
0
votes
1answer
25 views

Derivative as a rate of change

Could someone please help explain this answer to me? The question is: The equations for free fall at surfaces of Mars and Jupiter ($s$ in meters, $t$ in seconds) are $s$ = $1.86t^2$ on Mars and $s$ ...
0
votes
5answers
134 views

What is the mistake in doing integration by this method?

Integration Of a given function can be found out in many ways, For a specific function ∫1/xlogx, if we do integration by parts (∫f(x) g(x)= f(x) ∫ g(x)- ∫ [d/dx (f(x)) ∫g(x)] dx ) we get this way ...
0
votes
3answers
49 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
0
votes
3answers
214 views

Differentiate the Function: $y=\log_2(e^{-x} \cos(\pi x))$

Differentiate the Function : $y=\log_2(e^{-x} \cos(\pi x))$ Here is my work. What I have I done wrong?
0
votes
1answer
71 views

Find the x-coordinates of two other points of inflection of $f(x)= \int \frac{x+1}{x^2+1}$, given there is an inflection point at $(1,1) $

$$f(x)= \int {\frac{x+1}{x^2+1}}$$ I have to find the x-coordinates of two other points of inflection, given there is an inflection point at (1,1). My approach is to differentiate the equation, and ...
2
votes
4answers
415 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
2
votes
1answer
81 views

Question about the derivative of distance vs displacment.

Displacement and Distance are not exactly the same things. I have seen everywhere on the Internet that the derivative of a distance function is it's velocity function, however to my understanding this ...
1
vote
2answers
25 views

what is the value of $\theta$ used in calculate volume bounded by $z=x^2+y^2$ and $x^2+y^2=2x$

This is an example from my textbook, it explain everything well except the reason why $\frac{\pi}{2} < \theta < \frac{-\pi}{2}$ but not $2\pi < \theta < 0$. It's not explained and I can't ...
1
vote
1answer
50 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi \...
1
vote
2answers
71 views

Maxima and Minima of Functions of Two Variables $ f(x,y) = e^{x+y^2}\cdot y $ and $ f(x,y) = e^{x^2-y^2}\cdot y $

I'm having trouble finding the local minimum and maximum of the next functions: $$1. f(x,y) = e^{x+y^2} \cdot y $$ $ f_x'= (e^{x+y^2}\cdot y) ; $ $ f_y'= (e^{x+y^2}(1+2y^2)) $ $$ 2. f(x,y) = e^{...
0
votes
3answers
74 views

How do I solve this deceleration problem?

Question: A car is traveling at 100km/hr, when the driver sees an accident 80 meters ahead. What constant deceleration is required to stop the car in time to avoid a pileup? So far I have approached ...
0
votes
2answers
218 views

Is $\int\left(\sin^2x + \cos^2x\right)\;dx = \int 1 \; dx$ ?

I have just begun my 2nd calculus course and so far have just been applying the substitution method for solving anti derivatives and other basic rules. I have a question that is probably very easy to ...
12
votes
4answers
518 views

Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $

What is the asymptotic behavior of the sequence: \begin{equation} s_n=\sum_{k=1}^{n}k^{1/4} \end{equation} when $n\to \infty$?
9
votes
5answers
789 views

Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$

I'd love your help proving the following claim: If $f$ is continuous and $\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$. Here the counter example of all these ...
2
votes
2answers
27 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
5
votes
3answers
12k views

Integrating $e^{f(x)}$

can someone tell me a way of integrating functions like $e^{f(x)}$ I have a specific case: $\int e^{-3x}\,\mathrm{d}x$ PS: I'm not looking for the answer of this, but the way of doing it. Thanks ...
0
votes
1answer
65 views

Asymptotics of function of $n^a$, $2^n$ and $\sqrt{n}$, when $n\to\infty$

I am having trouble with estimation of the following$$\frac{n^a}{2^{n-\frac{\sqrt n+1}{2}}(1-\frac{1}{2 \sqrt n})^{n-\frac{\sqrt n-1}{2}}} $$ Where $n \in N$ and $a$ is a real number greater or equal ...
-2
votes
3answers
131 views

How to bound $ax^2+2bxy+cy^2$ by multiples of $x^2+y^2$ [closed]

Let $(x,y)\in[0,1)\times[0,1)$ cum $x^2+y^2<1$. Are there any $\mu\geq\lambda>0$ such that $$\lambda\xi_1^2+\lambda\xi_2^2\leq(1-x^2)\xi_1^2+2xy\xi_1\xi_2+(1-y^2)\xi_2^2\leq\mu\xi_1^2+\mu\xi_2^...
1
vote
2answers
355 views

Limit of $x^2\cos(1/x^2)$ when $x\to0$ by squeeze theorem

How can I argue that $$\lim_{x \to 0} x^2 \cos\left(\frac{1}{x^2}\right) = 0$$ I understand I have to use a squeeze theorem and that one piece goes to zero but I'm not sure how to tackle this ...
2
votes
1answer
64 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
0
votes
1answer
42 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
0
votes
1answer
399 views

On the pointwise limit of $\sqrt[n]{p_n(x)}$ when $n\to\infty$, for some polynomials $(p_n)$

For every $n$, I have a polynomial $p_n(x)=a^{(n)}_{n-1}x^{n-1}+a^{(n)}_{n-2}x^{n-2}+\dots+a^{(n)}_0$ (the $n$ in the exponent of the coefficients is merely an index). I can show that $\lim_{n\to\...