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

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2
votes
1answer
51 views

Question about convergence in $L^2$

Assume we have a sequence of functions $\{f_n\}_{n\geq 0}\subset L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$, i.e. $$\lim_{n\to \infty} \int_0^1 |f_n(x)-f(x)|^2dx =0.$$ Is it then true ...
12
votes
1answer
178 views

Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ which can also ...
-1
votes
3answers
133 views

Mathematics Calculus Question [closed]

A rectangle is to be inscribed in a semicircle with radius 10.Plot the area S of the rectangle as a function of the length of its sides. Find the maximum for S and for what sides it is achieved.
9
votes
2answers
811 views

How Differential got into calculus

I am confused what differentials are and how they become related to derivatives and integrals, as neither my textbook explains them nor my teacher. What we do to solve differential equations is to ...
1
vote
2answers
124 views

Spivak's Calculus, Chapter 6 problem 16 d)

I don't understand how to solve this problem and the official solution does not make much sense to me either. The problem is: (d) Let $f$ be a function with the property that every point of ...
0
votes
1answer
25 views

(Dis)continuity of integral of Riemann integrable function

Let $F\left(x\right)=\int_{0}^{x}f\left(t\right)dt$. It's known that if f is Riemann integrable on [a,b], then F is continuous on [a,b]. However, consider $$f(x)=\begin{cases} 0, & ...
2
votes
2answers
85 views

Squeeze Theorem: $\lim_{x\to 0} \, \frac{x^2}{\sin ^2(x)}$

I'm having a hell of a time understanding how to apply the Squeeze Theorem and the corresponding theorems to solving problems like the following. $\lim_{x\to 0} \, \frac{x^2}{\sin ^2(x)}$ So I can ...
1
vote
2answers
45 views

Find the MacLaurin Series for $f(x)=\ln(√81-x^2)$

Find the MacLaurin Series for $f(x)=\ln(√81-x^2)$ I know that $\ln(\sqrt{81}-x^2) = 1/2(\ln(81-x^2)) = 1/2(\ln(9-x)) + 1/2(\ln(9+x))$ I need help finding the series, not the expansion.
0
votes
2answers
77 views

Find the derivative of absolute value using the chain rule

I need help solving this derivative using the chain rule. I have tried setting $u = -x^2$
1
vote
1answer
73 views

Hard Integral [Heat Equation + Fourier Sine Series]

I encountered this integral while doing a heat equation problem in Advanced Calculus. How does the person evaluate the integral involving $$\int_0^\pi \sin x \cos (nx) \: dx $$ Can someone ...
2
votes
1answer
50 views

Help with evaluating a double integral

How would I go about evaluating... $$\int_{0}^{2}\int_{0}^{4-x^2} \frac{xcos(3y)}{4-y}dydx$$ I don't really have any work to show, since I don't even know how to start this problem, but I'll offer ...
0
votes
1answer
56 views

Propose a change of variables to simplify the double integral of (y-2x)^2(x+y)^2dydx

$$\int_{-1}^1\int_{2x-2}^{2x+2}\ (y-2x)^2\ (x+y)^2\ dy\ dx$$ I began by proposing $u=y-2x$ and $v=x+y$ and then solving each for $x$ and $y$ then computing the Jacobian which came out to be $-1/3$. ...
1
vote
0answers
56 views

Parallelogram with vertices $\mathbf{0}$,$\mathbf{Xa}$,$\mathbf{Xb}$,$\mathbf{Xa+Xb}$ ($\mathbf{X}$ matrix, $\mathbf{a}$ and $\mathbf{b}$ vectors)

There is a paralellogram with vertices $\mathbf{0}$, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a+b}$, whose area is $34$. What is the area of the parallelogram which has vertices $\mathbf{0}$, ...
0
votes
0answers
55 views

Volume of an ellipse rotated about a line

The question is: Find the volume enclosed by the ellipse $$9x^2+4y^2=36$$ after it has been rotated about the line $$2x+y=1$$ Basically, I don't really know where to go. I tried rotating the ellipse ...
1
vote
4answers
47 views

$d_n=(1+(2/n))^n$ converge or diverge and find the limit?

I know the answer is $e^2$ and I'd like to use L'Hopital's rule because this is an indeterminate form. Can someone explain how to get there? $$d_n=\left(1+\frac{2}{n}\right)^n$$
0
votes
1answer
11 views

Set of points - exclude a point

I want to make a set of the points given out by x and y axis where 1 <= x <= 3 and 1 <= y <= 2. The point ...
1
vote
2answers
31 views

How would you use Fubini's theorem to solve $\int_{0}^{1}\int_{\arcsin(y)}^{\frac{\pi}{2}} e^{\cos(x)} \, dx \, dy$

I think I need to re-define the upper and lower limits, but in any case I end up with: $\int e^{\cos(x)} \, dx$ which doesn't allow me to solve the equation.
1
vote
1answer
48 views

Proving that these curves intersect

Let $\Gamma$, $\Sigma$ be two curves with ranges in $(\{0\}\cup\mathbb{R}_{+})^2$. $\Gamma$ starts on the $y$ and ends on the $x$ axis: $\Gamma(0)=(0,\gamma_2),\Gamma(1)=(\gamma_1,0)$. $\Sigma$ is a ...
1
vote
3answers
84 views

number of integrable functions is greater than number of differentiable functions

It is said that number of integrable functions is greater than number of differentiable functions.Why so?But in reality isn't it quite the opposite?
0
votes
1answer
112 views

Question on finding formula for a Sequences

Hello would appreciate any help, the sequence is given below $y_0 = 1,\,y_1= 2y_0+1 = 2+1 = 3,\,y_2= 2y_1+1 = 2^2 + 2 + 1 = 7,\,y_3= 2y_2 +1 = 2^3 +2^2 +2 +1 = 15$ The question is What is the ...
0
votes
1answer
58 views

$f$ satisfies $y''+by=0 \implies f$ is of class $C^{\infty}$

I know that the solutions are a linear polynomial (if $b=0$), a linear combination of exponencials ($b<0$) or linear combinations of sine and cosine ($b>0$). However, supposing I didn't know ...
-2
votes
2answers
483 views

Find The Critical Numbers: h(p)= (p-4)/(p^2+2)

I have found the derivative of this rational polynomial but I am stuck at this point. The numerator cannot be factored. I set $h'(x)=0$. What should I do? Is the answer DNE?
0
votes
2answers
75 views

Trisecting a line in the complex plane

We have $x = 11-13i$ and $y = 35-i$. $a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$. I'm not sure where to start. Would a ...
0
votes
1answer
28 views

Conditional (Truncated) Expectations > Unconditional Expectations

$x$ is a continuous random variable with $pdf$ given by $f$ in the interval $[0,1]$. There is a continuous function $\lambda(x):[0,1]\rightarrow[0,1]$ with $\lambda'(x)>0$ such that its ...
0
votes
2answers
291 views

Convergence of infinite series from 2 to infinity 1/(x((lnx)^2))

On a recent exam I was asked to test the following series for convergence From $2$ to $\infty$ $\frac{1}{x(lnx)^{2}}$ I blanked on the integral but set up a comparison test, saying that ...
1
vote
0answers
68 views

Solids of Revolution around other functions.

Recently I've been thinking about solids of revlution, and thought about an interesting experiment. Can you rotate functions around, for example, the line $f(x)=x$? And consequently, could you rotate ...
2
votes
2answers
191 views

Terms needed to approximate with given error?

How many terms of this series would one need to add to get an approximation of $\pi$ with error less than $10^{-4}$? $$ 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots $$ So far, I wrote the ...
5
votes
1answer
193 views

Find $\left \lfloor 1000S \right\rfloor$ where $S=\sum_{k=2}^\infty (-1)^k\log_k e$

$$S = \log_{2}{e} - \log_{3}{e} + \log_{4}{e} - \log_{5}{e} + \log_{6}{e}\cdots $$ Find the Value of $\left \lfloor 1000S \right\rfloor$. My Attempt I changed it into summation and tried to ...
3
votes
1answer
79 views

Improper Integral with trigonometric functions

Determine if the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^3+4x}dx.$$ So far I've thought about using the comparison test but I'm not sure how to implement it. My first ...
1
vote
0answers
24 views

How do you rearrange the terms of a lower and upper Darboux sum to yield another inequality?

It is known : $L(f,P) \leq \int_{a*}^{b}f \leq \int_{a}^{b*} f\leq U(f,P)$ where $\int_{a*}^{b}f$ is the lower integral $\int_{a}^{b*}f$ is the upper integral $L(f,P) $ is the lower Darboux ...
1
vote
5answers
147 views

Maximum value of $ x^2 + y^2 $ given $4 x^4 + 9 y^4 = 64$ [closed]

It is given that $4 x^4 + 9 y^4 = 64$. Then what will be the maximum value of $x^2 + y^2$? I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .
1
vote
2answers
97 views

Differential $dx$

I have some trouble understanding a thing. I will reproduce two texts from two different books. In the first, the author defines the function $T:\mathbb{R}\longrightarrow \mathbb{R}$, ...
1
vote
0answers
28 views

Region bounded (Positive x-axis and y-axis)

Curve A is defined as $\ln (x^2 + 4)$ Find the region bounded by curve A, positive X axis and Y axis. My attempts, I've drawn the function, split it into two parts and found out that it's the area ...
0
votes
1answer
77 views

Can anyone please help with this integral. Very much appreciated..

Now , i've tried a couple of different substitutions and integrating partially but unfortunately to no luck, was wondering on your thoughts on it. I'd also be very thankful if someone were to have a ...
0
votes
0answers
26 views

Formula for the area of the surface obtained by revolving the curve $y = f(x)$ about the line $y = -k$

Assume $f$ is continuous and $f(x)\geq 0$ for $a\leq x\leq b$. Derive a formula of the area of the surface obtained by revolving the curve $y=f(x)$ , $a\leq x\leq b$ about line $y=-k$ ($k>0$).
0
votes
2answers
71 views

Need help with Riemann sum

Find $$\int_{a}^{b} x^m dx$$ where $0<a<b$ and $m\neq -1$. The answer goes as follows : but I get lost in the calculations which I need help with pinning down precisely: We choose the points ...
0
votes
1answer
91 views

Extend ${\bigl(1+\frac1x\bigr)}^{{x}}$ to $\overline{\mathbb R}$

We can extend these functions to $\overline{\mathbb R}$ by taking limits says here. \begin{align} \mathrm e^{-\infty} &= 0 \\ \mathrm e^{+\infty} &= \infty \\ \ln{\left|0\right|} &= ...
12
votes
5answers
2k views

Are all continuous one one functions differentiable?

I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the ...
0
votes
1answer
29 views

Inequality involving probabilities

While working on stochastics processes, I have found the following inequality, which I have not been able to proof: Let $h>l>1$ and $0\leqslant p\leqslant 1$ (probability). Then ...
0
votes
1answer
31 views

Calculating sequence product

I need your help calculating the limit of: $((n+a_{1})(n+a_{2})...(n+a_{p}))^{1/p}-n$ I've tried to multiply by the conjugate but the expression isn't friendly. Also I've tried to decompose the ...
2
votes
0answers
55 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
1
vote
1answer
51 views

ODE arising in physics

I was solving a physics excercise that basically was about considering an object being gravitationally pulled from earth during a given time, but considering the variation of gravity along the way (it ...
1
vote
1answer
65 views

Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order.

Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order. I know that if there exists a constant a and positive constants $t_0$ and $M$ such that $|f(t)| \leq M e^{at}$ at for all $t > ...
-1
votes
1answer
36 views

The asymptote of $y=\mathrm{sinc}(t)$ as time increases

Is there any known approximate formula that maps decay percentage of $\mathrm{sinc}(t)$ with decaying time? Or in other words, is there a known asymptote of $y=\mathrm{sinc}(t)$ as time increases?
1
vote
1answer
89 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
0
votes
1answer
22 views

Having trouble setting up the limits of integration.

Let $E$ be the solid below the plane $z=8$ and above the cone $z=\sqrt{x^2+y^2}$. Find the mass of $E$ if the density $\rho(x,y,z)=z$. I'm supposed to use triple integrals with cylindrical ...
3
votes
1answer
81 views

Climbing a hill with increasing & concave marginals. As you climb, do all coordinates go to $\infty$?

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
1
vote
1answer
66 views

Taylor series expansion approximating an integral?

I need to use the Taylor series expansion of $$\frac{1}{1+3x^2} $$ to find a series approximating $$\int_0^1 \frac{1}{1+3x^2} \, dx $$ and $$\int_0^{1/3} \frac{1}{1+3x^2} \, dx $$ I tried to start the ...
0
votes
1answer
46 views

Antiderivatives: A car is traveling at $50 mi/h$ when the brakes are fully applied…

A car is traveling at $50 mi /h$ when the brakes are fully applied, producing a constant deceleration of $38 ft/s^2$. What is the distance covered before the car comes to a stop? Since the car is ...
0
votes
2answers
71 views

(Maple) Finding the number of terms of the 2014th Term of Fibonacci Sequence

The Fibonacci sequence is defined by F0 = F1 = 1 and Fn = Fn−2 + Fn−1, n ≥ 2. In Maple, we can define the Fibonacci sequence as procedure: F:=proc(n::nonnegint) option remember; if n<2 ...