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

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4
votes
3answers
57 views

Calculus and minimum values

This is a simple question but I think I don't understand exactly what the question is asking.
16
votes
6answers
846 views

What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
2
votes
2answers
175 views

Interchange limit and supremum

I simply don't get the following question answered: How can i proof the equality $\lim_{a\to 0}\sup_{z\in\mathbb{Z}}2-2\cos(2\pi a z)=0$? Or is it even false? Thanks in advance!
2
votes
3answers
755 views

Taylor and Maclaurin Series for $f(x)=e^x$

I just came from a final exam where in one question I was asked to derive the Taylor Series for $f(x)=e^{2x}$ centered at $x=1$. I came up with the following: ...
0
votes
0answers
59 views

Nonnegative series converges implies terms decay exponentially?

Let $\{a_n\}$ be a sequence with nonnegative terms ($a_n\geq 0$). If $\sum_{n=0}^\infty a_n < \infty$, does this imply that there exists $\alpha<1$ such that $a_n \leq \alpha^n$ for all but ...
1
vote
2answers
236 views

Differentiation of logarithmic functions using the chain rule

What's the derivative of $x^2(\ln(x^2))$? I'm having a really hard time with logarithmic differentiation. Can someone help rationalize it for me?
1
vote
1answer
477 views

Using the FTOC to find the derivative of the integral.

I'm apologizing ahead of time because I don't know how to format an integral. If I have the following integral: $$\int_{x^2}^5 (4x+2)\;dx.$$ I need to find the derivative, so could I do the ...
1
vote
1answer
38 views

an integral inequality about Lebsegue measurable functions

Let both $f:[0,1]\to (0,\infty)$ and $g:[0,1]\to (-\infty,0)$ be Lebsegue measurable functions.Show the following inequality: $$\int^1_0 f(s)ds\int^1_0 g(t)dt\leq\int^1_0 f(x)g(x)dx.$$ It can be ...
2
votes
1answer
182 views

Cardioid calculus: Problems with calculating the perimeter

I am having several problems with calculus dealing with cardioid so I guess I will ask several questions. This is the first one. I should get the formula for the perimeter of the cardioid given by: ...
0
votes
2answers
104 views

Basic Calculus Problems.

The line $y=1$ cuts the parabola $y=\frac{1}{4}x^2$ at the points $P , Q$. Find the coordinates $P,Q$. Determine the gradients of the tangents at $P,Q$ to the curve $y=\frac{1}{4}x^2$; what is the ...
5
votes
2answers
269 views

Definition of Convex Function

Spivak's book states first this definition of Convex Function: Definition 1: A function $f$ is convex on an interval, if for all $a$ and $b$ in the interval, the line segment joining $(a, f(a))$ ...
0
votes
1answer
110 views

truncation error - help

I am trying to understand the concept of local truncation error and came accross this in my lecture notes: what I don't understand here is where the term 'O' comes from and what it stands for in ...
4
votes
1answer
3k views

Work to pump water from a cylindrical tank

We have a cylindrical tank with radius 2m and length 10m filled with water. How much work does it tank to pump the water out of the tank from the top? My attempt at the problem goes as follow. $g$ ...
2
votes
1answer
148 views

Show that $\int_{0}^{1}x^{m}(1-x)^{n}dx=\int_{0}^{1}x^{n}(1-x)^{m}dx$

I am trying to show that: $$\int_{0}^{1}x^{m}(1-x)^{n}dx=\int_{0}^{1}x^{n}(1-x)^{m}dx$$ For any positive integers $n$ and $m$. Which is true if I try to evaluate it numerically. I tried to use the ...
0
votes
1answer
77 views

is the book wrong? solve $\int\frac{1}{X^2\sqrt{a^2-x^2}}$

$\int\frac{1}{X^2\sqrt{a^2-x^2}}$ i solved it by substituting $x=asec\theta$ i ended up with $\frac{a}{a^3}\int\frac{\sec\theta tan\theta}{sec^2\theta tan\theta}$ so after the cancellation it ...
1
vote
2answers
127 views

Annuity rate differnetial equation

I am trying to find the rate of interest to keep a balance going forever. Initial is 100,000 and withdraw is 8000 So I know I have the general solution $$p(t) = \frac{8000}{r} + C e^{rt}$$ I know ...
1
vote
1answer
105 views

What is the $n$th derivative of $e^{-1/x^2}$?

How can I calculate the $n$th derivative of $e^{-1/x^2}$? I think I need to use chain rule but I'm not sure.
2
votes
3answers
67 views

Simple question about integrals?

simple question , if we define $\displaystyle F(x) = \int_a^x f(t) \, \mathrm{d}t$ does that mean $x>a$? or $x$ could be smaller than a even though that expressoin means it's an upper bound?, ...
2
votes
1answer
334 views

How to restrict Lagrange multiplier on positive values?

Here's the function that i want to optimize: $$f(x,y) = x-2y$$ and the constraint is: $$g(x,y) = x^2 + y - 10 = 0$$ Solving with Lagrange multiplier I get: $$F(x,y) = x-2y - x^2\lambda - y\lambda ...
3
votes
1answer
60 views

Integral solution of a differential equation (verification)

I have to verify that the integral $$y(x) = \int_0^\infty \exp\left(-t - \frac{x}{\sqrt{t}}\right) dt$$ satisfies the ODE $$xy''' + 2y = 0$$ ($x > 0$). Differentiating under the integral sign three ...
1
vote
2answers
458 views

Differential equation word problem water leaking $y=x^2$

A tank has the shape of a parabola $y=x^2$ revolved around the y-axis. Water leaks from a hole area $B= .0005 m^2$ at the bottom, let $y(t)$ be the water level at time $t$. How long does it take for ...
74
votes
10answers
3k views

Why does factoring eliminate a hole in the limit?

$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$ I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and ...
2
votes
1answer
38 views

Net Outward Flux

I have what seems to be a simple problem, but my answer is not matching the book's. Find the net outward flux of the vector field $F$ across the boundary $D$. $F = (x^2, -y^2, z^2)$ $D$ is the ...
3
votes
1answer
46 views

A question of Integration by parts

Consider $\int_a^b f'(x)g(x)dx$. Then the integration by parts gives $$ \int_a^b f'(x)g(x)dx = \left[ f(x)g(x) \right]_{a}^b - \int_a^b f(x) g'(x) dx.$$ In the case that $f(a), g(a), f(b), g(b)$ are ...
3
votes
3answers
222 views

Necessary condition for an improper integral to converge

I am working on this problem from a past examination: Let $f:[0,\infty)\rightarrow\mathbb R$ be a continuous, non-negative and non-increasing function such that the improper integral ...
3
votes
1answer
254 views

A tricky integral (flux of a point charge through a disk)

The integrals: $$ \oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}} $$ If we have a point charge at the origin and we ...
0
votes
1answer
72 views

$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$

I have worked up to this stage of the question : $$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$$ So that's where I worked up to. Can someone please show me how to finish it off?
0
votes
2answers
83 views

Trigonometric function, with integration of definited integrals

I have worked up to this stage of the question : $$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$$ so that's where I worked up to. can someone please show me how to finish it off
1
vote
4answers
1k views

Integral of $\sin^2 \pi x$

Evaluate $$\int_0^{1/4} \sin^2 \pi x \; dx$$ Can someone please explain what to do if theres a power and how to do it in general thanks
2
votes
2answers
204 views

Injective function $f(x) = x + \sin x$

How can I prove that $f(x) = x + \sin x$ is injective function on set $x \in [0,8]$? I think that I should show that for any $x_1, x_2 \in [0,8]$ such that $x_1 \neq x_2$ we have $f(x_1) \neq f(x_2)$ ...
0
votes
0answers
310 views

Derivative Riccati-Bessel function

I have found two derivatives of the so-called Riccati-Bessel functions in a textbook $$ (x j_n(x))'=xj_{n-1}(x)-nj_{n}(x)$$ and $$ (x h_n^{(1)}(x))'=x h_{n-1}^{(1)}(x)-n h_n^{(1)}(x)$$ so $j_n$ is ...
1
vote
1answer
131 views

How to simplify $1-\frac1{a+b+1} $

I have this term with two factors a and b. a and b are positive integer numbers. $$1-\frac1{a+b+1}$$ b is an error in the problem that I want to separate it from the problem. For example such that ...
1
vote
1answer
33 views

Question involving PDE's

Suppose $f$ is a differentiable function of a single variable and $F(x,y)$ is defined by $F(x,y) = f(x^2-y)$. a) show that F satisifies the PDE $\frac{\partial F}{\partial x} + 2x \frac{\partial ...
4
votes
2answers
127 views

what fails in this proof of limits?

I always thought that this theorem was true, but today when asking about the proof of the chain rule in calculus I realized that it is false. I understand the counterexample, but now I don't ...
1
vote
0answers
27 views

How find this $m$ Value range

let $a\ge\dfrac{2^{m-1}-1}{m-1}$and such $$\left(\dfrac{\dfrac{3}{4}(\dfrac{3}{4}+a)(\dfrac{3}{4}+2a)\cdots(\dfrac{3}{4}+(m-1)a)}{(1+a)(1+2a)\cdots ...
1
vote
1answer
95 views

Existence of a function in one real variable

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) \neq x$ for all $x \in \mathbb{R}$ and for every $a \in \mathbb{R}$ there exists a sequence $\{x_n\}$ such that $$\lim_{n ...
2
votes
2answers
147 views

Is there a general family of curves that satisfies the following conditions?

Is there a general family of curves $f(x,c)$ that satisfies the following conditions? $f(x,c)$ is strictly increasing for $x \ge 0$ $f(0,c) = 0$ $f(1,c) = 1$ $f(x,c) \to \infty$ as $x \to \infty$ ...
2
votes
2answers
111 views

2 limits logarithms $ \lim_{t\to \infty} t-t^2\ln(\frac{t+1}{t}) $

I have problems with the following limits logarithms, and I can not use L'Hopital or power series (I know the results of the problems, using these methods), so I need solutions that do not occupy ...
1
vote
4answers
260 views

Differentiate $\ln(\cos2x)$ With respect to $x$.

I need to differentiate $\,\ln(\cos2x)$. Can someone please explain how to do this question? Thank you.
0
votes
1answer
123 views

Berkeley summer '81

Let $$y(h) = 1 - 2 \sin ^2 (2 \pi h), \quad f(y)= \dfrac{2}{1+\sqrt{1-y^2}} .$$ Justify the statement $$ f(y(h)) = 2 - 4 \sqrt 2 \pi + O(h^2) $$ where $$\limsup _{h \rightarrow 0} ...
5
votes
2answers
293 views

Is there any notable difference between studying the Riemann integral over open intervals and studying it over closed intervals?

(1) A function $f:[a,b]\to\mathbb{R}$ is said to be Riemann integrable on $[a,b]$ . . . (2) A function $f:(a,b)\to\mathbb{R}$ is said to be Riemann integrable on ...
1
vote
2answers
208 views

improper (double) integral: $\int_0^\infty\int_x^\infty\frac{1}{\sqrt{t^{3}+1}}\,\mathrm{d}t\,\mathrm{d}x$

I want to determine if the integral $\,\displaystyle\int_0^\infty\displaystyle\int_x^\infty\frac{1}{\sqrt{t^{3}+1}}\,\mathrm{d}t\,\mathrm{d}x$ converges. I know that ...
0
votes
2answers
251 views

Indefinite integral of normal distribution

How does one calculate the indefinite integral? $$\int\frac1{\sigma\sqrt{2\pi}}\exp\left(-\frac{x^2}{2\sigma^2}\right)dx$$ Where $\sigma$ is some constant. Work so far: Integrating from exp as ...
3
votes
1answer
639 views

How to prepare for Integral Calculus (Calculus 2)

I'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I've ...
1
vote
1answer
112 views

Question about lemma (2) - Spivak' calculus - page 89

I have a question about the proof of lemma (2) in Spivak's calculus, page 89. How does he simplifies $$ \ |y_0|\frac{\epsilon}{2(|y_0|+1)} $$ to get $$ \ \frac{\epsilon}2 $$ Thanks.
1
vote
2answers
123 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
6
votes
3answers
379 views

Prove the sum of squares of two functions equals 1

If you have $f'(x)=g(x)$, $g'(x) = -f(x)$, $f(0)=0$ and $g(0)=1$, how do you prove that $f^2(x)+g^2(x) = 1$?
2
votes
4answers
174 views

a dog tied to a pole by a rope

A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L>\sqrt{2a^2+h^2}$ ,and walks on the ...
4
votes
2answers
397 views

proof of the chain rule for calculus

I was comparing my attempt to prove the chain rule by my own and the proof given in Spivak's book but they seems to be rather different. Please tell me if I'm wrong or if I'm missing something. I ...
1
vote
2answers
210 views

Differential of normal distribution

Let $$f(x)=\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sigma\sqrt{2\pi}}$$ (Normal distribution curve) Where $\sigma$ is constant. Is my derivative correct and can it be simplified further? ...