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

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0
votes
0answers
49 views

prove properties of a power series

i really have no idea how to finish this question. Thanks for help.
0
votes
2answers
50 views

How to find a Taylor series for $e^{x^2-1}$?

How do I proceed to write a taylor series expansion for $e^{x^2-1}$? I know the series for $e^x$: it is $1+(x)+(x^2/2!)+\dots$
0
votes
1answer
34 views

What is the limit of the following function, as x approaches 0.

What is the limit of the following function, as x approaches 0. $$\displaystyle{\lim_{x \rightarrow 0} \frac{26x^3}{3}(ln(x)-\frac{1}{3}})$$ I try putting the equation in another way. ...
1
vote
1answer
19 views

Evaluate $∫_C(4y^3+\cos x^2 )\,dx-(4x^3+\sin y^2)\,dy$ where $C$ is the boundary of the disk centered at the origin of radius $2$, oriented clockwise.

I'm not sure how to begin with this equations, doesn't really understand what does "$C$ is the boundary of the disk centered at the origin of radius $2$, oriented clockwise." mean.
3
votes
1answer
68 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
2
votes
2answers
56 views

Differentiate $(x + 1)(x + 2)^2(x + 3)^3$

Obviously we can "brute force" this by multiplying the various terms and differentiating from there. But based upon the solution provided in the text where I found this problem, it looks like there's ...
2
votes
1answer
62 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
0
votes
0answers
22 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
-1
votes
2answers
44 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$?

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
0
votes
3answers
68 views

Integrate $\frac{1}{1+\cos^2x}$. Probably need using some trigonometric identity I don't know

Integrate $\frac{1}{1+\cos^2x}$ I probably need using some trigonometric identity I don't know. I tried all methods I'm familiar with. Any assistance will be great. Thank you!
0
votes
2answers
37 views

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

First fundamental theorem of calculus: $$g(x) = \int_a^xf(t)dt$$ then $$g'(x) = f(x)$$ But how does this guarantee the existence of antiderivatives of functions? Tutorials always state it does, but ...
0
votes
1answer
29 views

Decay formula problem

I have the problem, A radioactive substance has a half-life of $10$ days. The initial amount of the substance is $100$ milligrams. (a) Determine the decay rate of the substance. (b) How much of ...
3
votes
3answers
52 views

Integral of trig fraction using substitution?

I'm chewing on an integral problem and don't have a clue where to begin. If someone could assist by suggesting a good starting point, I'd really appreciate it! Not asking for anyone to solve the ...
0
votes
1answer
36 views

Convergence of the limit of the $n$th root of a term

In my course I often see questions that ask me to calculate the limit of sequences such as: $$\lim\limits_{n \to \infty}{\sqrt [n]{a_n}} $$ How do I handle these questions? A related question is to ...
-4
votes
2answers
54 views

Does Limit exist or not?

Does $\lim_{x \to 0}$ $\frac {1}{x^3}$ exist ? How can I prove existence or otherwise of the above limit? I am finding difficulty in either $-\infty$ and $+\infty$ are equal or not. Can we tak both ...
0
votes
2answers
33 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
4
votes
1answer
36 views

If $f\in L^2[0,1]^2$, do we have $\int_0^1|f(x,x)|dx<\infty$?

Let $f\in L^2[0,1]^2$. Does it follow that $$\int_0^1|f(x,x)|dx<\infty$$ By Cauchy-Schwartz inequality $$\int_0^1|f(x,x)|dx\leq \int_0^1|f(x,x)|^2dx = \int_0^1\int_0^1|f(x,x)|^2dxdy$$ So I need ...
0
votes
1answer
53 views

Prove the Maclaurin-series representation of $ \sqrt{1 + x} $.

I have the function: I have to prove following statement: ( mit = with) b) The series representation: I hope somebody could follow up on this task. Best regards
-1
votes
1answer
22 views

Solve the integral , only by using the table of derivatives [on hold]

Can we solve this integral only by using the table ?
0
votes
2answers
27 views

Finding the asymptote of $\tan(x)$

Using limits to find the asymptote of a function $y=f(x)$ is usually done with limits as : if the asymptote is of the form $y=mx+c$ then : $m=\lim\limits_{x\to\infty} \dfrac{f(x)}{x}$ ...
-3
votes
0answers
15 views

Complex Analyse of a fuction [on hold]

I have this function: I have to prove the following properties: a) There exists a sequence of polynomials such that for the n-th derivation from f obtains: b) The taylor series T(f;x) from f ...
1
vote
2answers
35 views

Evaluate the integral without substitution 2

So I am trying to solve this integral only by knowing the table of derivatives, can this be solved that way, can someone give me a hint ?
0
votes
0answers
30 views

Find the function satisfying the given condition

If $f(x,y)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
0
votes
1answer
42 views

Evaluate the integral without substitution

I think that from this conclusion I suppose I have done something wrong at algebraic modification or I have chosen the worst way around this integral, can someone help me
0
votes
1answer
23 views

How do you integrate $\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$

Given $p(s)$ some single valued function How can I show that $$\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$$ has resulting in something along the line of $$\frac{p^2(s)}{2}$$ note $\dot p(s)$ signifies ...
0
votes
0answers
33 views

Solve the complex euqtions

I have a question from complex calculus. How to solve this two equations: a) $$ sin(z)=2015 $$ I know that sin(z) equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
1
vote
1answer
43 views

What is the formal problem caused by interpreting dx as an infinitesimal?

My mother is a physicist. One evening, I told her $dx$ is a linear mapp $\mathbb{R}^3\to\mathbb{R}$ taking $\hat u_x=e_1$ to 1 and the other canonical vectors to 0 (if considered on $\mathbb{R}^3$). ...
3
votes
3answers
215 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
1
vote
1answer
45 views

Gradient of a vector function

I have a vectorial function $f$, defined on the set of all $n$-dimensional vectors. $f(x) = \log(x^TAx)$, where $\log$ is the natural logarithm, $x^T$ is $x$ transpose and $A$ is a symmetric $n \times ...
2
votes
4answers
94 views

Compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$

We have to compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$ where $f\left(x\right)=\frac{-x^3+2x^2-5x+8}{x^2+4},\:x\in \mathbb{R}$ is an bijective function. How help if we kno![enter image ...
0
votes
0answers
28 views

Dirac delta question from “Classical covariant fields” by Burgess

If you have the book with you. Kindly tell me how did he reach equation 2.54 from equation 2.52. I tried to solve the delta function according to given instruction but I am making some mistake. Kindly ...
1
vote
2answers
74 views

Finding the fallacy in this wrong limit computing result

for this limit $$\lim_{{{k}\to\infty}}{\left(-{\left({2}{k}+{1}\right)}\right)}^{{\frac{{1}}{{{2}{k}+{1}}}}}$$ a friend gives the computing process like this ...
0
votes
3answers
43 views

Taking the derivative of $f(x) = (x+1)/\sqrt{x^2+1}$

This is what I have so far. $$((x^2+1)^{1/2} - (x+1) * 1/2(x^2+1)^{-1/2} * 2x)/({x^2} + 1)$$ ((x^2+1)^(1/2) -(x+1) * x(x^2+1)^(-1/2)) /(x^2+1)
0
votes
1answer
57 views

I can use MVT on $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$?

If I can use MVT: $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt=x\cdot f\left(c\right)$ when $x\rightarrow \infty ,\:c\rightarrow \infty $ so we'll have to evaluate $\lim _{x\to \infty }x\cdot ...
-1
votes
1answer
69 views

Find the max and min of $f(x) = x^5 -x^4+x^2-x$ [duplicate]

I get $5x^4-4x^3+2x-1$ for the derivative but I am not sure what to do after. The teacher told us that we would have to use Newton's method to solve the problem.
0
votes
0answers
75 views

Can we integrate less than 0.00001% of functions? [on hold]

I'm told that we can integrate less than 0.00001% of functions. Is this true? Any proof?
0
votes
1answer
57 views

Finding local maximum and minimum

$f(x) = x^5-x^4+x^2-x$ on the interval $(-\infty,\infty)$ The teacher told us that we would have to use Newton's method to solve this problem but I am not sure what to do after taking the derivative. ...
6
votes
4answers
196 views

determination of the volume of a parallelepiped

Here is a parallelepiped.I want to determine the volume of the parallelepiped. One of my friends said to me that the volume of the parallelepiped can be found out by the following formula. ...
1
vote
3answers
146 views

How to solve this integral by a simple way?

I'm given $$\int \frac{x^3}{\sqrt{x^4+x^2+1}}dx$$ My attempt, Let $u=x^2$, $du=2xdx$ $$=\frac{1}{2}\int \frac{u}{\sqrt{u^2+u+1}}du = \frac{1}{2}\int ...
1
vote
3answers
82 views

$f(x)$ is Riemann integrable $\Rightarrow$ $\frac{1}{1 + f^2(x)}$ is Riemann integrable

Let f(x) be Riemann integrable on [a,b]. Then there exist $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ f(x) has only removable or jump discontinuities. The set of ...
0
votes
0answers
17 views

Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
-4
votes
0answers
14 views

Contours of Some Fnctions [on hold]

I'm trying to sketch the contours of the following functions and I'm having problems. Could you help me explaining step-by-step about how to work on this... ? I don't need the answers, just hints... ...
1
vote
1answer
22 views

The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] …

Problem : The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] denotes the greatest integer function and $\{.\}$ denotes the fractional part function. ...
1
vote
1answer
20 views

Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: ...
2
votes
2answers
25 views

Series Radius/interval of convergence

Help please! I have no idea how to do this question. I tried using the ratio test
1
vote
1answer
18 views

How to show this limit tends to 0 as approaches infinity

$$\lim_{n\to \infty} \frac{{(2)}^{2n}{0.5}^{(4n+3)}}{(4n+3)(2n+1)!} $$ Anyone can suggest me any larger f(x) to be used for squeeze theorem to conclude this limit is 0? And please explain how you get ...
1
vote
5answers
28 views

Infinite sequence and power series

infinite sequence $a_{n}$ where $$\lim_{n\to \infty} |na_{n}|=1101 $$ Find R of convergence of the power series $$\sum_{n=1}^\infty a_{n}x^n$$ Anyone can guide me for this question? Thank you so ...
0
votes
2answers
20 views

Is variable substitution $x-h=a$ appropriate under limit?

Originally, I have $$ \lim_{h\to 0}(f(x+h)-f(x-h))=0$$ If I let $x-h=a$, will I get $$\lim_{h\to 0}(f(a+2h)-f(a))=0\quad ?$$ I feel confused because $a$ is related to $h$ somehow, but doing this ...
0
votes
0answers
17 views

Change of variables in double integration

I was trying to solve this double integral $\int_{0}^{1}\int_{0}^{y}(1-x)^{59}(y-x)^{27}dxdy$, I could do this by taking binomial expansion but that would be very painful. So a sure thing here is a ...
0
votes
0answers
6 views

Finding the maximal superior error while approximating f(x,y) by Taylor of degree 1

I have $$f_{(x,y)} = ln(2x + 2y)$$ Defined over the domain $$D=\left \{ (x,y) \in \mathbb{R} ^2 : |x - 2| \leqslant 1 \ \ and \ \ |y - 3| \leqslant 1 \right \}$$ The question ask for Taylor ...