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

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votes
5answers
183 views

Why is $\lim \limits_{h \to 0} \frac{e^{h} - 1}{h} = 1$? [duplicate]

Can anybody show me how $\lim \limits_{h \to 0} \dfrac{e^{h} - 1}{h} = 1$?
3
votes
4answers
535 views

Calculus books recommendation (intermediate level)

I would like to ask for some intermediate level textbook for calculus (single variable), or, at least, some supplement to Spivak's Calculus for better understanding on how to approach and solve his ...
2
votes
4answers
168 views

the relationship between $f^{-1}(x)$ and $x$

A elementary question here. If I have $$f(x)<x\quad\forall\,\,x\in (0,1).$$ Can I deduce $f^{-1}(x)>x$ for $x\ne 0,x\ne 1$? If yes, could anyone show me how?
0
votes
1answer
66 views

Deduce alternate series test from Abel’s theorem

Show that the alternate series test can be deduced from Abel’s theorem. I know that Abel's theorem is Abel's Theorem Let $(a_n)$ and $(b_n)$ be two sequences of real numbers such that • $(a_n)$ ...
4
votes
2answers
181 views

Evaluating a definite integral by definition

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
1
vote
1answer
49 views

How to write the equation of this graph in the given figure?

By looking at the graph we come to know that whether the graph is linear or exponential or quadratic etc. If the graph is a straight line then we write the equation of the straight line for the ...
1
vote
0answers
68 views

Calculus book with interesting examples [duplicate]

I need to prepare presentation about calculus, paying special attention for examination of function of one variable. I am not very advances in the topic, but I'm looking for book which contains ...
1
vote
1answer
19 views

A question regarding the order of an asymptotic estimate

Suppose that $m, n \in \mathbb{N}$ such that \begin{equation} m \cdot \log m = n, \end{equation} where the logarithm is in the natural base. How can we estimate the solution $m = m(n)$ ...
0
votes
1answer
29 views

Change the order of integration:$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$

$$\int_{0}^{1} dx \int_{0}^{1}dy \int_{0}^{x^2+y^2}f(x,y,z)dz$$ in the order : $$\int dz \int dx \int f(x,y,z)dy$$ I don't think it needs to be divided into two seperate integrals, but the professor ...
3
votes
0answers
137 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
1
vote
2answers
363 views

Is there another simple way to solve this integral?

$$\int \frac{x(2-x^3)}{(x^3+1)^2}dx$$ Is there some simple ways to solve this integral? As my solution for this integral is very long. It's not suitable for my student.
1
vote
2answers
52 views

Convergence of the sequence $\frac{x_{n+1}}{x_n}$

Let $x_n$ be a positive sequence such that the sequence $(\displaystyle\frac{x_{n+1}}{x_n})$ converges to $\lambda<1$. Show that $x_n$ converges to $0$. Hint: Show that there exists $c,r$ such ...
1
vote
3answers
97 views

Convergence of an infinite series involving conjugates

I have the infinite series $$\sum_{n=1}^\infty \left(1-\cos\frac{1}{n}\right) $$ I have to find if it converges or not, and I know I have to use the conjugate find it. So I get ...
1
vote
1answer
17 views

if $w$ is a normal distribution where $n(0,1)$, then find the mgf of $w^2$.

if $w$ is a normal distribution where $n(0,1)$, then find the mgf of $w^2$. I have looked it up and the answer is chi squared but i cannot seem to find a way to integrate this correctly. I start the ...
1
vote
1answer
83 views

integrate $dx/(a^2 \cos^2x+b^2 \sin^2x)^2$

Integrate $\dfrac{dx}{(a^2 \cos^2x+b^2 \sin^2x)^2}$. I can go up to the residue formula like in this example here but then I just can't end up with the result for when $n=2$. I keep messing up my ...
5
votes
1answer
269 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
0
votes
3answers
856 views

Norm of the sum of two vectors

This problem has two parts. Part a): $x$ and $y$ are vectors. If $||x|| = 7, ||y|| = 11$, what is the smallest value possible for $||x+y||$? (Note: the || || denotes the norm of a vector). This is ...
0
votes
4answers
49 views

When does $ \sum_{n=1}^{\infty} (-8)^nx^n $ converge?

I am trying to determine when the following series converges: $$ \sum_{n=1}^{\infty} (-8)^nx^n $$ When approaching these problems, do I simply just have to guess-and-check with cases or is there a ...
-3
votes
2answers
75 views

(SOLVED) Calculus Integration $\int_1^2 \frac{2}{((x^2)(x+1))}dx $, where's my mistake?

Integrate (upper limit =2, lower limit = 1) $\int_1^2 \frac{2}{((x^2)(x+1))}dx $ My ans: $1+2\ln(3/2)$ Lecturer ans: $1+2\ln(3/4)$ Checked multiple times, can't find out where's the careless ...
0
votes
1answer
30 views

How to stabilize cyclic tridiagonal matrix algorithm?

I've received a task which is: Solve equation by cyclic tridiagonal matrix algorithm: $$ \frac{\partial{f}}{\partial{t}} = \lambda*\frac{\partial{f}}{\partial{x}}, \\ x\in[0,1]\ t\in[0,1] \\ ...
8
votes
2answers
167 views

How find limit $\displaystyle \lim_{n\to\infty}n\left(1-\tfrac{\ln n}{n}\right)^n$

How find this limit $$\displaystyle \lim_{n\to\infty}n\left(1-\dfrac{\ln n}{n}\right)^n$$
0
votes
2answers
48 views

Evaluating a limit using Squeeze Theorem. [duplicate]

Consider a function $f (x)$ defined on $\mathbb{R}$ satisfying: $$ \left|f(x) - \frac{7^2 + 5x|x|+2}{x^2+16}\right| \leq \frac{1}{x^2} $$ for all $x\neq0$. Calculate: $\displaystyle \lim_{x \to ...
0
votes
1answer
36 views

Change the order of integrals..

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ to$$\int dz\int dx \int f(x,y,z) dy$$ Not sure if to split the shape into two shapes, or do it directly. Either way i would like to know how its ...
-1
votes
2answers
103 views

Velocity of a curve given by parametric equations

In standard Cartesian equations, $\frac{dy}{dx}$ is the velocity function because it's the derivative of position. $$\frac{dx}{dt} = \sin^{-1}\left(\frac{t}{1 + t}\right) ...
1
vote
1answer
31 views

Using chain rule for partial derivatives

If $u=f(x,y)$, where $x=e^{5s}\cos(2t)$ and $y=e^{5s}\sin(2t)$, I must find $g(s,t)$ and $h(s,t)$ in the following equation: $$\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial ...
0
votes
1answer
45 views

Law of Cosines, Trigonometric Angle Addition Theorems, and Dot Product Relations

Just as the derivative, slope, and gradient are essentially the same thing I've realized that the Law of Cosines, trigonometric angle addition, and dot product are saying the same thing. My question ...
1
vote
0answers
14 views

Calculate $I=\int d^{n}q\frac{(q^2)^{a}}{(q^2+D)^b}$

I want to evaluate this integral : $I=\int d^{n}q\frac{(q^2)^{a}}{(q^2+D)^b}$ It is a n-dimensional multiple integral in cartesian coordinates where $q=\sqrt{\sum_{1}^{n}q_{i}^2}$ is the Euclidean ...
2
votes
1answer
302 views

True or false statement about a simple limit of product

I have to determine whether the following statement is true or not: Let $(a_n)$ and $(b_n)$ be two sequences such that $$\lim_{n\to\infty}(a_nb_n)=0$$ then either $\lim_{n\to\infty}(a_n)=0$ or ...
3
votes
3answers
158 views

How to integrate $ \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$

Evaluate $$\displaystyle \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$$ Can someone just tell me the necessary manipulations? Hints will be enough. Can it be done by integration by ...
4
votes
3answers
458 views

Calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$.

I need to calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$. I defined : $$f(x)=\sqrt{x}$$ Therefore : $$f'(x)=\frac{1}{2\sqrt{x}}$$ ...
0
votes
1answer
98 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
0
votes
3answers
385 views

Are curves in a level set continuous?

Wikipedia defines a level set as a level set of a real-valued function of $n$ real variables $f$ is a set of the form $$L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c ...
0
votes
2answers
40 views

Power series representation of xln(3x+5)

I get to the point $\sum_{n=0}^{\infty }\frac{(-1)^n3^{n+1}x^{n+2}}{(n+1)5^{n+1}}$ by using the geometric series and integrating etc. But I looked up the answer and it is what I have plus the term ...
3
votes
1answer
97 views

Continuity of left derivative implies differentiability?

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and has a left derivative, $f^-$, everywhere in a neighborhood of $x.$ Suppose $f^-$ is continuous at $x.$ Does this imply that $f$ is ...
2
votes
2answers
71 views

Divergence to $\infty$ basic questions

i was given this exercises: 1) Show that $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{k(k+1)} = 1$$ 2) Show that $\exists L\in\Bbb R $ such that $$\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{k^2} = L$$ ...
2
votes
2answers
58 views

How do I find a formulae for $S_n$ for the sequence 3/16, 3/64,3/256, 3/1024?

I have this homework and I need to find the formulae for S_n in function of "n" The sum is sum (3)/4^(n+1) n=1 to infinity. I have calculated the first four terms of the series. These are the ...
1
vote
1answer
54 views

Proving convolution identity

I am trying to prove the following identity: $$\int_0^x(f*g)(y)dy = (\int_0^xf(y)dy)*g(x) = f(x)*(\int_0^xg(y)dy)$$, where $(p*q)(t) = \int_0^tp(x)q(t-x)dx$. I thought that since I already know that ...
1
vote
1answer
34 views

Proving a statement about a sequence (using limit)

Suppose $\{a_n\}$ is a sequence such that $\lim_{n\to\infty} (a_{n+1}-a_n)=0$. Prove that if there exists $c>0$ such that for all $n$, $|a_n|\geq c$ then either for all but a finite number of ...
2
votes
2answers
191 views

Interesting calculus problems for beginner

Recently I started learning calculus and I think I have grasped the basics. However when calculating examples I tend to drift away and not put much effort in it. When I was learning programming in ...
0
votes
1answer
75 views

Calculating $f(x,y)=3xy +x^2$ on the unit circle

$f:\mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \mapsto 3xy +x^2$ $D:=\{(x,y)\in \mathbb{R^2}: x^2+y^2 \leq 1\}$ $\iint_Df(x,y)dxdy=\int_0^1 ...
5
votes
1answer
218 views

$f ' (x) = f(x - (x+1)^t + 1)$

Let $x > 0 $ and $c $ a given real $> 0.$ Let $t $ be between $0 $ and $1.$ How to find $f(x)$ or good asymptotics for $ f(x)$ such that $$ f ' (x) = f(x - (x+1)^t + 1) $$ And $ f(1) = 1 + ...
0
votes
1answer
27 views

Finding derivatives for a Cauchy-Euler ODE

I'm having some trouble following along with the reduction of the Cauchy-Euler equation into a linear one with constant coefficients. I've been trying to follow along with the work here, but I don't ...
1
vote
1answer
60 views

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method ...
5
votes
3answers
685 views

Improper integral of a rational function!

Find the value of the integral $$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx$$ I tried the substitution $x=t^5$ to obtain $$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$$ Now we can factor the denominator to ...
4
votes
3answers
136 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series ...
5
votes
3answers
277 views

If $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$

Assume $f$ is continuous on $[a,b]$, if $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$. The problem didn't state anything about the function $f$, is it safe to assume ...
4
votes
2answers
69 views

Is $dy = -\frac{1}{x^2}dx$ a valid rearrangement of $\frac{dy}{dx} = -\frac{1}{x^2}$?

$$ y = \frac{1}{x}$$ $$\frac{dy}{dx} = -\frac{1}{x^2}$$ Is $$dy = -\frac{1}{x^2}dx$$ a valid rearrangement of $\frac{dy}{dx} = -\frac{1}{x^2}$? That is, is that a mathematically meaningful/legitimate ...
2
votes
2answers
64 views

Evaluating integral using result

The problem is this. Given that $\int_0^a f(x) dx = \int_0^a f(a-x)dx$, evaluate $$\int_0^\pi \frac{x\sin x}{1+\cos^2x} dx$$ I write the integral as $$\int_0^\pi ...
4
votes
3answers
181 views

Integration by substitution notation question

Often with integration by substitution I see (and use) the notation $ x \to \frac{\pi}{2} - x $, for the simple reason that I don't have to rename the variable that I am integrating with respect to, ...
0
votes
2answers
50 views

Equations of the tangent lines of $y=x^4$ at the point $(2,0)$?

Consider the curve $y=x^4$. $(A)$ - The item $A$ was asked yesterday, I put it here in case it is useful. $(B)$ - Determine the equations of the tangent lines to the curve that pass ...