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

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0
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0answers
8 views

Angles and tangents finding specific points

Find any points on the graph $y=x^3$ where the tangent makes an angle of $45$ degrees with the $x$ axis in the positive direction I don't understand the $45$ degree bit, I differentiated but I'm ...
3
votes
0answers
11 views

Conceptual question on differentiation in calculus?

In Calc class, my teacher told us that the only solution to y' = y is y = c*e^x, with c being a real number. I am having difficulty understanding the only part. Is there a proof of this? Or am I ...
0
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1answer
30 views

Convergence of $ \sum_{n=1} ^\infty \frac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$

Convergence of $$ \sum_{n=1} ^\infty \dfrac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$$ Attempt: I believe not a nice attempt: $ n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} ...
0
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0answers
21 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
2
votes
2answers
13 views

Limit of non-linear multi-variable function

I'm trying to prove the limit of the following function is $0$: $\lim_{(x,y) \to (1,-1)} {x^3} - {2xy^2} + 1$ I know that I'm trying to find a $\delta$ s.t $ 0 < \sqrt{(x - 1)^2 + (y + 1)^2} < ...
0
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1answer
23 views

How to integrate $1/\sqrt{(1+x^2)^3}$?

Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem. Is anyone able to assist me with ...
1
vote
1answer
27 views

Simple Trig Integration. Why is my answer wrong?

$$\int \dfrac{\cos x+\sin 2x}{\sin x}dx=\int \dfrac{\cos x+2\cos x\sin x}{\sin x}dx=\int \dfrac{\cos x\left(1+2\sin x\right)}{\sin x}dx$$ Substitute $u=\sin x$ and $du=\cos x\ dx$: ...
0
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3answers
12 views

Finding a point on a curve where the tangent is parallel to another line noted

Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$ I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks
1
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2answers
46 views

Is this identify valid?

$$\sin(t) \dot{}e^{if(t)} = \sin(t+f(t))$$ I've never seen this identify before but it follows directly from the relation between complex exponentials and the trigonometric functions.
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2answers
63 views

Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$

Find $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$ My professor showed us a few ways to compute the limit 1) Factor the numerator $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3} = \lim_{x ...
0
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1answer
17 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
0
votes
1answer
17 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
2
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5answers
51 views

Convergence of $\sum_{n=1}^{\infty} \log~ ( n ~\sin \frac {1 }{ n })$

Convergence of $$\sum_{n=1}^{\infty} \log~ ( n ~\sin \dfrac {1 }{ n })$$ Attempt: Initial Check : $\lim_{n \rightarrow \infty } \log~ ( n ~\sin \dfrac {1 }{ n }) = 0$ $\log~ ( n ~\sin \dfrac ...
0
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3answers
32 views

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$?

Does $f_n(x) = \frac{x^n}{1+x^n}$ converges uniformly on $[0,1]$? My answer is: No because obviously $f_n(0) = 0$ and $f_n(1)=\frac{1}{2}$, so for every $n\in\mathbb{N}$ it's true that for ...
-1
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0answers
15 views

The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles. [on hold]

Find the volume of the following solid S: The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.
2
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1answer
37 views

Is it true that $\lim_{x \to 0} \frac{f(x)}{g(x)} = 0 \implies \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$?

Is it true that $$\lim_{x \to 0} \frac{f(x)}{g(x)} = 0 \implies \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$$ ? Assume that $f$ and $g$ admit derivatives $f'$ and $g'$ at every point in an open interval ...
1
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3answers
48 views

convergence of $ \sum_{n=1}^\infty \frac {1}{\log (1 +\frac {1}{n})}$

Test convergence of $$ \sum_{n=2}^\infty \dfrac {1}{\log (1 +\frac {1}{n})}$$ I am not really sure how to move forward. Could anyone give me a direction to proceed please. EDIT" The only part I ...
1
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2answers
33 views

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$ So first we have two potentially problematic points which are $1,\infty$ We split the integral to $$\int_1^2 \frac{x\ln x}{x^4-1} ...
2
votes
3answers
59 views

Convergence of $\sum_{n=1}^{\infty} \frac {1}{n\log^2(n+1)}$

Convergence of $$\sum_{n=1}^{\infty} \dfrac {1}{n\log^2(n+1)}$$ Attempt: We note that $\lim_{n\rightarrow \infty} \dfrac {n}{ \log^2(n+1)} = \infty$ Hence, for a sufficiently large $n: \dfrac {n}{ ...
1
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3answers
45 views

convergence of $\int_a^b \frac{1}{x^2} dx$

Why is it true that $\int_0^a \frac{1}{x^2} = \infty$ but $\int_a^\infty \frac{1}{x^2} \lt \infty$? Shouldn't it be symmetric?
0
votes
2answers
55 views

convergence of $\sum_{n=1}^{\infty} \frac {1}{\log(e^n+e^{-n})}$?

Test convergence of $\sum_{n=1}^{\infty} \dfrac {1}{\log(e^n+e^{-n})}$ Attempt: I have tried the integral test, the comparison test ( for which I couldn't find a suitable comparator). However, I ...
0
votes
1answer
18 views

Find the change of electromotive-force per degree, at 15 degrees, 20 degrees, and 25 degrees.

I started to read "Calculus Made Easy", by Silvanus P. Thompson, and i can't figure out how to solve one problem E = 1.4340[1 - 0.000814(t-15)+0.0000007(t-15)^2] volts Find the change of ...
0
votes
2answers
21 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
votes
2answers
27 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
1
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1answer
35 views

Prove $\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}$

In the book "Lehrbuch der Analysis Teil I" of Heuser page 303, there was a task: Prove $$\frac{d}{dx}{\rm arctanh}(\ln \cosh x) = \frac{\tanh x}{1-(\ln \cosh x)^2}.$$ When I tried, I ended up with ...
-1
votes
1answer
29 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. How one can determine $a, b$ and $p$ such that 1) I ...
0
votes
1answer
32 views

How to calculate the multiplication result in two different ways?

First I want to apologize for a possibly ambiguous title, but don't know how to expose clearly the problem through the title because the speech is much more structured. I'm working on a system that ...
1
vote
1answer
25 views

Area between two functions

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 16. Let $f(x)=x-x^2$, $g(x)=ax$. Determine $a$ so that the region above ...
0
votes
2answers
25 views

Number of zeros of $ f^n $

Let $f:\Bbb R\to \Bbb R$ be infinetly differentiable function that vanishes at $10$ distinct points in $\Bbb R$.suppose $ f^{n} $ denote $n$-th derivate of $f$, for $n \ge 1$. Then which of following ...
2
votes
2answers
37 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
1
vote
0answers
23 views

Using Intermediate value theorem and Rolle's theorem

Find how many solutions $2\ln x+2x^2+7=0$ has. Define: $f(x)=2\ln x+2x^2+7$, derive it and equate to $0$: $f'(x)=0 \\ 2+4x^2=0$ The discriminant is negative so there are no solutions, so from ...
1
vote
1answer
19 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
0
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1answer
16 views

Which one of given set is connected…

Which of the following are connected? (Notation: $c(a, r) =\{(x, y) \in\mathbb R^2: (x-a)^2+(y-b)^2=r^2\}$) $c(0,1) \cup c(0,2)$ $c(0,1) \cup c(1,3)$ $c(0,1) \cup c(1,1)$ $c(0,1) \cup c(2,1)$
2
votes
3answers
33 views

Image of (0,1] under continous function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Which of the following sets cannot be image of $(0,1]$ under $f$. {$0$} $(0,1)$ $[0,1)$ $[0,1]$ My initial guess was using intermediate ...
0
votes
1answer
35 views

Find a Taylor series around $x=0$ [on hold]

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
2
votes
4answers
49 views

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$. What I have: Since $\{a_n\}\rightarrow \alpha$ we know that ...
0
votes
0answers
33 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
votes
2answers
67 views

how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$?

would someone give me a hint or a solution ? how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$? Thanks a lot.
0
votes
4answers
73 views

Limit $\lim \limits_{x\to0}{\frac{\ln(x+1)}{2^x-1}}$ wihout LHospital

I want to find this limit but without using L'Hospital, with which is one-liner. $$\lim \limits_{x\to0}{\frac{\ln(x+1)}{2^x-1}}$$
0
votes
2answers
97 views

Find the antiderivative of $(x^2+x+1)^{20}$ [on hold]

How do I find the antiderivative for that? The online calculators say that there's no solution. **NVM, the little scratch on my paper turned out to be a faintly copied handwritten $'2'$, turning it ...
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votes
0answers
17 views

question from Radius of curvature [on hold]

prove that for the ellipse x^2/a^2 + y^2/b^2 =1, Radius of curvature =a^2b^2/p^3, p being the perpendicular from the centre upon the tangent at (x,y)
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
0
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0answers
34 views

Prove that exists $\delta>0$ such that, if $(x,y)\in S$ satisfies $\lVert(x,y) \rVert < \delta$, then $f(x,y) \leq f(0,0)$.

This exercise appeared on my Calculus II exam, and I didn't know even how to start doing it. Any hint is appreciated. Let $\ f, \ g : \mathbb{R^2}\to \mathbb{R}$ two $C^2$functions over the plane. ...
4
votes
2answers
97 views

Who first found the value of $\int_{-\infty}^{+\infty}e^{-x^2}dx$?

A fairly pretty technique of showing that $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables ...
0
votes
1answer
53 views

Prove statement related to dot product

$a, b$ and $x$ are vectors in $\mathbb R^3$ and satisfy $$a\cdot x=b\cdot x$$ Prove $$ a=b $$ By using the definition of dot product, I come up with something like ...
1
vote
2answers
46 views

How do I evaluate $\int \cot^2x$? [duplicate]

I have an integral with $\frac{1}{\tan^2x}$ needed to be evaluated. But instead of searching online for the antiderivative of $\cot^2x$, how would i find it from first principles?
7
votes
3answers
365 views

Trig substitution; why can we ignore the absolute value?

If we have to integrate $$f(x)=\frac{x}{\sqrt{1-x^2}}$$ and we substitute $x=\sin \theta$ then we eventually have to take the square root of $\cos^2x$ which is equal to $|\cos x|$. But in my textbook ...
0
votes
2answers
23 views

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$. What I have: Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ ...
5
votes
1answer
179 views

Why do we put absolute brackets for ln?

When writing out the final answer in $\ln$ form, why is it necessary to put absolute brackets? How does it affect the answer? I have this answer of $-3\ln|\frac{3+\sqrt{9-x^2}}{x}|$, but why does it ...
0
votes
0answers
62 views

Prove the limit as $x$ approaches $0$, $\frac{\sin(x)}{x}$ approaches $1$ using the epsilon delta definition [duplicate]

Prove that $\lim_{x\to0}\frac{\sin(x)}{x} = 1$. So far i have things such as $|\sin(x)|\leq|x|$ for small $x$ and $|\sin(x)|\leq1$ so it is bounded but I'm rather stuck, Also I am not looking for a ...