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

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0
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0answers
18 views

$f(x) =\ ln(2x^2 + 1)$ is continuous on $R$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
0
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0answers
6 views

Prove of disprove: If f is twice differentiable on (-1,1) and f"(0)>0, then there is δ>0 such that f is convex on (- δ, δ) [duplicate]

First of all I feel like this statement is true. The way I'm thinking about it is that f"'(x)>0 implies convexity. The fact that they used f"(0)>0 makes no difference I think. Even if we consider it, ...
0
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0answers
9 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
0
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0answers
26 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
3
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1answer
21 views

Volume of a Solid of Revolution Rotated Around the Y-Axis

Sorry to post an obvious homework question here, but my daughter's calculus teacher isn't much on "teaching" and left a problem like this one out of the notes. I can't find much on the internet to ...
1
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2answers
45 views

Antiderivative of $ (x^2 + c)^{-3/2} $

What method should be used to determine the antiderivative of this expression? Edit: I have $ c > 0 $ in the problem I'm working on.
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1answer
23 views

For differentiable function where $f'(0)=a$ and $f'(1)=b$ we have that for all $c\in(a,b)$ there exists a $y$ such that $f'(y)=c$.

So what I'm trying to prove: Assume a function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(0)=a$ and $f'(1)=b$. Prove that for any $c\in(a,b)$ there exists a $t\in\mathbb{R}$ such that ...
0
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1answer
30 views

Fourier series of $\cos^2x$

This is my first step with Fourier series and I'm stuck at the beginning. So my solution: The function $f(x)=\cos^2x$ is an even function. Thus I use formulas: $a_0 = \frac{2}{\pi} \int _0 ^\pi ...
0
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2answers
30 views

Determine if the sequence- $a_1 = 2$ , $a_{n+1} = 72/(1+a_n)$ is convergent.

I've tried to prove that the sequence is convergent by using the monotonic sequence theorem but after computing the first few terms, I realized that the sequence is not monotonic thus, it isn't ...
-1
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2answers
22 views

Integrating to find deceleration, and finding ball height?

1) A ball is thrown straight up from a height of 8 ft with an initial velocity of 40 ft/sec. How high will the ball go? (Take g = 32 ft/sec2.) How would I do this? Wouldn't I need to find a velocity ...
0
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1answer
14 views

Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!
1
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2answers
19 views

Why is $\frac{d^m}{d(-x)^m}=(-1)^m \frac{d^m}{dx^m}$

Im not a mathematician so dont judge me with my following "proof". I want to show that a Legendre polynomial is odd if its index is odd, and even if its index is even. This in order to prove that ...
0
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0answers
14 views

Any formulas for $\sum_{k=1}^{\infty}e^{\frac{-t}{k}}$ for $t>0$

Any formulas for $\sum_{k=1}^{\infty}e^{\frac{-t}{k}}$ for $t>0$? In general if $a_{k}$ is a positive monotonically increasing sequence can we get an upper bound for ...
0
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0answers
53 views

prove properties of a power series [on hold]

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

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

How do I proceed to write a taylor series expansion for $e^{x^2-1}$? I know the series for $e^x$: it is $1+(x)+(x^2/2!)+\dots$ Edit: Would a Maclaurin series expansion be different?
0
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1answer
40 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
26 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
71 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
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2answers
58 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
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1answer
67 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
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0answers
24 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} + ...
-2
votes
2answers
53 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
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3answers
70 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
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2answers
38 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
32 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
56 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
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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
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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
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2answers
34 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
39 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
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1answer
54 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
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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
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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
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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
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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
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0answers
35 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ 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
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1answer
43 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
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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
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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
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1answer
44 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
96 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
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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 ...
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2answers
76 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
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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
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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. ...