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

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0
votes
2answers
102 views

1st and 2nd derivatives of $x\sqrt{9-x}$

Do I start with the power rule? I know to rewrite it as $x(9-x)^{1/2}$ and if I use the power rule I get $\sqrt{9-x}/2(9-x)^{3/2}$ and I have no idea if that's right, and then I need to find the ...
0
votes
1answer
51 views

Prove: $1/2(a+b) \times (a - b) = b \times a$ - in three space

Let $\times$ = cross product and $a$ and $b$ have vector signs on them. $a = [a_1, a_2, a_3]$ $b = [b_1, b_2, b_3]$ How would I prove this? Do I expand out cross product? Any help is appreciated!
0
votes
2answers
157 views

function composition - n times

Please consider this function: $$f(x) = \frac{x}{{\sqrt[6]{{1 + {x^6}}}}} $$ What would be the value of the composition (n times): $$f \circ f... \circ f = ? $$ I tried doing it manually, maybe ...
1
vote
1answer
62 views

Convergence of a sequence with repeated sines [duplicate]

Let $x\in(0,\pi/2)$ and $\{a_n\}_{n\in\mathbb N}$ defined recursively as follows: $$ a_0=x, \quad \text{and} \quad a_{n+1}=\sin(a_n). $$ Show that $$ \lim_{n\to\infty}{n\,a_n^2}=3. $$ Note. There is ...
0
votes
1answer
88 views

What direction does the n vector (normal to the surface) have to be when doing Stokes' theorem?

The author uses $g=y+z-2$ instead of $g=2-y-z$ to ensure that n has a positive k component so that it points outward. But why was it necessary that n points outward? Is it because C is in the ...
1
vote
3answers
101 views

Help understand chain rule derivative

I was verifying a larger function derivative on wolfram alpha and came across this derivative: $\frac{d}{dx} (1-x)^2 = 2(x -1)$ Using the chain rule, I was expecting to get: $2(1 - x)$ Instead. I ...
13
votes
1answer
178 views

The limit of $ \displaystyle\lim_{n\to\infty} \exp(-1+\exp(-2+\exp(-3+\ldots\exp(-n)…)))$.

Does the following limit exist ? $$\lim_{n\to\infty} \exp(-1+\exp(-2+\exp(-3+\ldots\exp(-n)...)))$$ If yes, can it be expressed in a closed form ? PARI shows the following numerical value : ...
1
vote
2answers
63 views

How to find the limit of this equation?

I have a Calc final coming up soon and am currently reviewing. I am currently stumped on finding the limit of as $x \rightarrow 1$, of $(1-\sqrt{2x^2 -1}) / (x-1)$. Don't you have to multiply by the ...
2
votes
2answers
4k views

Find rates of change of surface area and volume of a cube.

The original 24 m edge length x of a cube decreases at the rate of 2 m/min. Find rates of change of surface area and volume when x = 6 m. I don't even know where to begin.
0
votes
1answer
93 views

Using definition of derivative to find $\sqrt{11\theta}$

$$\lim_{h \to 0} \frac{f(\theta+h)-f(\theta)}{h} \;\;\;\;\;\;\;\;\;\;\; \textbf{(1)}$$ $$\lim_{z \to \theta} \frac{p(z)-p(\theta)}{z-\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \textbf{(2)}$$ ...
0
votes
1answer
36 views

Transform this expression in something useful for me

I have $A \subset \Omega$ and $E \subset \Omega$. Now I have $A\cap E^C$. But I do not want to work with the complement, I am rather looking for an expression that somehow contains $A \cap E$. Does ...
3
votes
1answer
46 views

power sum of $\sum_{n=1}^\infty 2^n x^{n^2}$

How do I calculate the radius of the power sum of: $$\sum_{n=1}^\infty 2^n x^{n^2}$$ having a little trouble with the second power(of $x$) thank you
1
vote
1answer
179 views

Why taking derivative of equation sometimes gives gradient and other times tangent?

Just as the title says, I am confused by this. For example: For 3D curve $c(t)=(x(t),y(t),z(t)$, tangent is $(\frac {dx(t)}{dt},\frac {dy(t)}{dt},\frac {dz(t)}{dt})$. surface $S(a,b)$, tangents ...
1
vote
1answer
148 views

Values of a, b, and c that the curve $y = ax^3 + 3x^2 + bx + cx + e^x$ has one point of inflection?

For what values of a, b and c does the curve $y = ax^3 + 3x^2 + bx + cx + e^x$ have exactly one point of inflection? Two points of inflection? No points of inflection? Provide a numerical ...
2
votes
7answers
2k views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...
0
votes
3answers
77 views

Prove that if $f$ is differentiable at $a$ but $g$ is not, then $f$ + $g$ is not differentiable at $a$.

I know if two functions, $f$ and $g$, are differentiable at $a$, then so is $f + g$. However, I'm not quite sure how to prove the above. My first thought was proof by contradiction. So I would ...
0
votes
1answer
82 views

Function $\mathbb{R} \rightarrow \mathbb{R}$ local minimum is global

Could you tell me how to prove or disprove that if a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ has only one critical point and it is a local minimum, then it is a global minimum? ...
1
vote
1answer
184 views

Showing there does not exist a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f'(x)=h(x)$

If $h(x)=0$ if $x<0$ and $h(x)=1$ if $x\geq 0$, prove there exists does there does not exist a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f'(x)=h(x)$. Proof: We will show that $h$ is ...
0
votes
1answer
40 views

Find a length of a line

Find the lenght of a line presented in polar coordinates $r=e^{2\varphi}, (-1\leq\varphi\leq1)$ Sorry for the probable mistranslation Am I supposed to use ...
0
votes
1answer
58 views

For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.

If $f:[0,1]\to\mathbb R$ is continuous and $\int_0^x f(t)dt = ∫_x^1f(t)dt$ then prove that $f(x)=0$. I suppose that $f(x)$ isn't zero, set $g(x)= \int_y^xf(t)dt$ with $y\in[0,1]$ and use Bolzano but ...
1
vote
1answer
41 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
1
vote
0answers
97 views

Remainders in Alternating Series / Remainders in general

So there's a little part I get stuck on when I'm trying to find the remainders, i'll post two simple problems with my work and my answer compared to the book answer. Please help me out and tell me ...
1
vote
1answer
54 views

Partial derivative at (0,0).

Let $f:\mathbb{R}^2\to\mathbb{R}$ be defined by: $$f(x,y) = \left\lbrace\begin{array}{@{}l@{}l@{}} 0 & \text{if }xy = 0 \\ 1 & \text{if }xy\neq0 \end{array}\right..$$ Show that both partial ...
0
votes
1answer
113 views

Cauchy convergence test of a sequence

I need to prove that the sequence $ x_n = \frac{1}{n+1} $ Converges using the Cauchy convergence test. Now I know the following $ \forall \epsilon > 0 \enspace \exists N \enspace \forall n \geq ...
1
vote
2answers
40 views

Trying to compute area inside a region

question: Use double integral to compute the area of the region inside the cardioid $r=1+\cos(\theta)$ and to the right of the line $x=3/4$. So my first integral will be from $3/4$ to $2\pi$ and my ...
2
votes
1answer
85 views

conditions for the existence of complex roots:

find the necessary conditions under which the following polynomial will have non-real roots: $P(x)=Ax^3+Bx^2+x-D$ where $A>0$ and $D>0$. well if it has a+ib and a-ib as conjugate root then the ...
11
votes
1answer
143 views

Proving that $\sum_{k=0}^{\infty}\frac{1}{(k+1)(2k+1)(4k+1)}=\frac{\pi}{3}$

The following problem(p.668, 7) is from Integrals and Series [ Интегралы и ряды, А.П. Прудников, Ю.А. Брычков, О.И. Маричев.] states that ...
2
votes
1answer
38 views

Proving that 2 functions are equal/not equal

Prove the equality of $f_1$ and $f_2$ given the following conditions: Problem 1 $f_1(x)$ and $f_2(x)$ are functions of finitely summed sine and cosine functions (e.g. $3\cos2x+\sin5x$), any ...
1
vote
1answer
85 views

Prove that the $\lim_{x \rightarrow 0}\frac1x$ does not exist

we have been doing epsilon and delta limit proofs in my first year calculus class and I can't seem to wrap my around this stuff. We have not done an example in class of disproving a limit, and this ...
8
votes
3answers
280 views

How prove this limit $\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha \pi)}-J_{-\alpha}(x)}{\sin{\alpha\pi}}$

let $$J_{\alpha}(x)=\sum_{m=0}^{\infty}\dfrac{(-1)^m}{m!\Gamma{(m+\alpha+1)}}\left(\dfrac{x}{2}\right)^{2m+\alpha}$$ show that: \begin{align*}&\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha ...
2
votes
2answers
92 views

The limit $\lim_{x\to \infty}\frac{x-\frac{1}{2}\sin x}{x+\frac{1}{2}\sin x}$

How to find the value of the limit: $$ \lim_{x\to\infty}\frac{x-\frac{1}{2}\sin x}{x+\frac{1}{2}\sin x} $$ (l'Hopital not working here, right?)
5
votes
6answers
245 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
0
votes
1answer
491 views

Volume using cross sections

Can someone please help me solve this problem? Q: The base of S is an elliptical region with boundary curve 9x^2 +4y^2 = 36. Cross-sections perpendicular to the x-axis are isosceles right ...
9
votes
5answers
170 views

$\sqrt{2\sqrt{2\sqrt{2\cdots}}}=2$

Show that $$\sqrt{2\sqrt{2\sqrt{2\cdots}}}=2$$ $$\sqrt{2}=\mathbf{2}^{1/2}$$ $$\sqrt{2\sqrt{2}}=\mathbf{2}^{1/2+1/2^2}$$ $$\sqrt{2\sqrt{2\sqrt{2}}}=\mathbf{2}^{1/2+1/2^2+1/2^3}$$ Show the limit of ...
0
votes
0answers
35 views

Proving a DTFT relation

I'm trying to prove that the inverse DTFT of: $$\frac{1}{1-ae^{-j\Omega }} $$where |a|<1 is: $$a^{n}u[n]$$ The way to prove it is by the integral below but I'm not sure how to proceed: $$ ...
2
votes
1answer
64 views

Can someone show me step by step how to handle this convergence problem?

I just took my final and one of the questions read: Use the integral test to determine the convergence or divergence of the series $$\sum_{n=1}^{\infty} \frac{e^n}{1+e^{2n}}.$$ I struggled ...
1
vote
2answers
78 views

Solve Double Integral, likely using Fubini's Theorem

The second form throws me, that the dy appears before the second integral. I have unfortunately little skill for Calculus, and I can't seem to find a similar question online to take example from. ...
0
votes
1answer
58 views

Vector functions and motion along a curve

A particle moves along the curve $x=\ln y$ with a constant speed of $4$ units per second. Find the normal scalar component of acceleration as a function of $x$. Honestly, what I don't understand ...
1
vote
1answer
160 views

Telescoping Series Question

$$\sum_{k=0}^\infty\frac1{16k^2+8k-3}$$ I did this and got $-1$, the book says $-1/4$, and for some reason they are pulling out a $1/4$ early on in the steps and i do not understand why the $1/4$ is ...
0
votes
1answer
48 views

Integrating over or under this object…

There is clearly a kink in my understanding of double integrals with polar coordinates. Our problem is to find the volume enclosed by the hyperboloid $-x^2-y^2+z^2=1$ and plane $z=2$. I correctly ...
1
vote
1answer
95 views

Exercice 4 , p.965 from Stewart's Calculus: Concepts and Contexts

Use Stokes' Theorem to evaluate : $$\int \int_S curl \ \vec F \cdot d \vec S $$ $F(x,y,z) = x^2 \ y^3\ z \ \vec i + \sin(x\ y\ z)\ \vec j + x\ y\ z\ \vec k$ , $S$ is the part of the cone $y^2 =x^2 ...
1
vote
1answer
167 views

How to integrate $\int{\frac{6x}{x^3+8}dx}$

I'm having some trouble solving this integral using partial fraction method: $$\int{\frac{6x}{x^3+8}dx}.$$ After expanding $x^3+8$ into $(x-2)(x^2+2x+4)$ and expanding the original integral into ...
0
votes
0answers
117 views

Changing the Order of Integration in this Triple Integral

Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$. So after over at least an hour of thinking, I might have all 6 ...
2
votes
2answers
82 views

Is this an accepted/given form of the definition of derivative?

$$\lim_{h\to 0}\frac{\int_x^{x+h}f(x)dx}{h}=f'(x)$$ Is this an accepted/given/understood form of the definition of derivative? Or does one need to work this into the traditional difference quotient ...
0
votes
1answer
52 views

Does this constitute sufficient proof? [duplicate]

Task I have the following function $f(x)=x^2+1$ I need to prove, according to the $\epsilon - \delta$ definition of a limit, that $f(x)$ is continuous at $x = 2$. Step 1 $\forall \epsilon > 0 ...
1
vote
1answer
41 views

Where did I do this min/max problem incorrectly?

Current price of a car is \$20,000 The current price changes at a rate of $50-50\sqrt{t}$ When will the price of a new car be at a maximum? So.... $$\frac{dp}{dt}=p'(t)=50-50\sqrt{t}$$ First, to ...
1
vote
1answer
387 views

Proof (epsilon delta) for the continuity of a function at a point

Task I have the following function $f(x)=x^2+1$ I need to prove, according to the $\epsilon - \delta$ definition of a limit, that $f(x)$ is continuous at $x = 2$. Step 1 $\forall \epsilon > 0 ...
4
votes
1answer
66 views

Is $\dot{f}(0)$ a function or a point?

Say \begin{align} g: & \mathbb R^m \to \mathbb R^n \\ \implies g': & \mathbb R^m \to \mathcal{L}(\mathbb R^m,\mathbb R^n) \\ \implies g'(0): & \mathbb R^m \to \mathbb R^n \end{align} ...
0
votes
3answers
77 views

Standard Triple Integral Problem

Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$. How would I do this problem? I can't even visualize the region D. ...
2
votes
2answers
26 views

Trouble understanding Riemann Integral and Limits

Let p>0. Using Riemann's Integral find the limit. Here is the problem : The problem is , i don't know how to start or what should i do. I can't really understand this chapter from my book and ...