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

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3
votes
3answers
133 views

How do you integrate the following trigonometric function involving sin and cos?

How do you integrate the following functions: $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta $$ ...
2
votes
3answers
2k views

Second Derivative of basic fraction using quotient rule

I know this is a very basic question but I need some help. I have to find the second derivative of: $$\frac{1}{3x^2 + 4}$$ I start by using the Quotient Rule and get the first derivative to be: ...
9
votes
3answers
1k views

Evaluate $\int \cos(\cos x) dx$

Evaluate $\int \cos(\cos x) dx$ I tried to use chain rule but failed. Can anyone help me please?
1
vote
3answers
93 views

Integration of a rational function from +/- infinity

I am trying to calculate the integral $$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$ where $\{a, b, c\}\in \mathbb{R}$. I have looked in a table of integrals for ...
3
votes
2answers
192 views

Computing a derivative using logarithmic differentiation

Using the Logarithmic Differentiation find the derivative of $y=\sqrt{x(x-1)/(x-2)}$...so I tried,but the result is not correct..can you show me a hint? so $\ln y= 0.5\ln[x(x-1)/(x-2)]$ $$\ln ...
26
votes
2answers
466 views

Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$

Here is another infinite sum I need you help with: $$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$ I was told it could be represented in terms of elementary functions and integers.
0
votes
1answer
60 views

$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $

I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct. $$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$ For this one, ...
0
votes
2answers
66 views

How to solve this integral $\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$

Problem : How to solve this integral $\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$ I thought it should be $ x + 3x^2$ in the numerator so that I will take $x+x^3$ = u then taking derivative both sides and ...
2
votes
1answer
130 views

Derivative of Linear Map

I'm reading Allan Pollack's Differential Topology and got stuck on this argument: In the second paragraph of page 9, section 1.2 he said "Note that if $f:U\to \mathbf{R^m}$ is itself a linear map ...
1
vote
4answers
96 views

What is the derivative of $\ln(4^x)$?

What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)? Is it $\dfrac{1}{x\ln4}$?
1
vote
2answers
37 views

Why does $7^{2\ln x}\cdot \ln(7) \cdot (2/x)$ equal to $7^{2\ln x}\cdot \ln(49) /x$?

While reviewing, I came upon this problem which has the derivative $7^{2\ln x}\cdot \ln(7) \cdot (2/x)$ simplified to $7^{2\ln x}\cdot \ln(49) /x$ How/why is it simplified like that?
1
vote
2answers
49 views

Values of a parameter $x$ in an infinite series that makes it converge

I am required to find the values of $x$ in the following infinite series, which cause the series to converge. $$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$ I tried to use the ratio test, and found that ...
0
votes
2answers
173 views

Integrating $\int{\frac{1}{1+e^{x}}}dx$, Partial Fractions(?)

I need help with this integral: $$H(x) = \int{\frac{1}{1+e^{x}}}dx$$ It should be easy, but I'm stuck. I thought about using a u-substitution but I didn't get any further. Am I meant to use partial ...
0
votes
1answer
49 views

Using complex logarithms to solve equations

Could someone please just explain the formula/method for solving the complex equation $$e^{iω}=k$$ where $k∈C$. As an example, I know that when $ω=x+iy$, $e^{2iω}=1$ has solutions $ω = n\pi$ for ...
4
votes
4answers
147 views

Differentiate $\log_{10}x$

My attempt: $\eqalign{ & \log_{10}x = {{\ln x} \over {\ln 10}} \cr & u = \ln x \cr & v = \ln 10 \cr & {{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} ...
0
votes
4answers
394 views

How to integrate $\int_0^\infty e^{-ty^2} \sin t dt$

My book suggests that I do some sort of limiting $\lim_{A \to \infty} \int_0^A e^{-ty^2} \sin t d t$ But I'm not getting anywhere.
2
votes
1answer
172 views

How to place a limit that it's inside the integral, outside.

I did this: $$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
1
vote
2answers
203 views

Spivak problem on Schwarz inequality

I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality: $$ x_1y_1 + x_2y_2 \leq ...
0
votes
5answers
137 views

Parametric equations for given line

How would you find the parametric equations for: 1) a line through $(3,1)$ and $(-5,4)$. 2) a segment joining $(1,1)$ and $(2,3)$. Can anyone show me the steps of doing it cause the way my textbook ...
0
votes
1answer
37 views

Suppose you invest \$10 at 10.2% per annum compounded annually. How many years would it take for your investment to grow to \$15 000?

I'am solving a simlar equation to this and just trying to figure out how they did it? the only part I don't understand is how they got the number.... 1.102 15000 = 10(1.102)n ¬1 mark 1500 = 1.102n
0
votes
1answer
55 views

How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?

Short Version of the Question: How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$? Long Version of the Question: I'm currently attempting ...
1
vote
4answers
145 views

How to integrate $\int_0^\infty \frac{1}{1+y^4} dy$ [duplicate]

I tried the trigonometric substitution $y^2 = \tan \theta, sec^2\theta = 1 + y^4$ But now I'm stuck with $\frac12 \int \frac{\sqrt{\sin \theta}}{(\cos\theta)^{\frac92} } d \theta$ I ran out of ...
2
votes
1answer
164 views

Solving for a matrix from its quadratic form

I have a set of vectors that I am trying to predict from another set of vectors using a matrix $W$. To find this matrix, I decide I want to minimize the $\ell^2$ norm of the error, e.g.: $$ ...
10
votes
1answer
206 views

Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that $$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$ Prove that $$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$ Where should ...
1
vote
2answers
443 views

Calculus Reduction Formula

For any integer $k > 0$, show the reduction formula $$\int^{2}_{-2} x^{2k} \sqrt{4-x^2} \, dx = C_k \int^{2}_{-2} x^{2k-2} \sqrt{4-x^2} \, dx$$ for some constant $C_{k}$. (original image) ...
0
votes
1answer
29 views

Integral, set and parametric representation

I am to compute the following: $\displaystyle\iiint\limits_V 1\, dx\, dy\, dz$, where $V= \{{(x,y,z) \in \mathbb R^3 : (x-z)^2 +4y^2 < (1-z)^2} \text{ and } 0<z<1\}.$ Does anyone have idea ...
1
vote
1answer
369 views

Finding the value of t where tangent line is perpendicular to x axis

For the curve x = t$^2 - 1, y = t^2 - t$, the tangent line is perpendicular to x-axis, where Options are : a ) t = 0 b) $t \to \infty$ c) $t = \frac{1}{\sqrt{3}}$ d) $t = \frac{-1}{\sqrt{3}}$ ...
3
votes
1answer
96 views

Derivatives using the Limit Definition

How do I find the derivative of $\sqrt{x^2+3}$? I plugged everything into the formula but now I'm having trouble simplifying. $$\frac{\sqrt{(x+h)^2+3}-\sqrt{x^2+3}}{h}$$
2
votes
3answers
113 views

Linear approximation to 1/0.254

The question says: Use linear approximation to approximate $1/0.254$. I know that $1/0.25 = 4$. Where do I proceed from next. Do I subtract $0.004$ from the answer, or what else could I do?
1
vote
1answer
33 views

how to calculate $d\Omega(f)$ here

the question was to find $d \Omega(f)$ with : $$ \Omega : (E,[.]) \to (F,||.||) \\f \to -f'' +f^3$$ $ [f] = |f'(0)| + ||f''|| $ ; $ ||f|| = Sup_{[0,1]}|f(x)| $ the answer is given to me like this ...
2
votes
5answers
283 views

Can someone please explain $e$ in layman's term? [duplicate]

I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?
0
votes
2answers
132 views

Perimeter or Calculus Word problem

A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single strand of electric fence. With 1400m of wire at your disposal, what is the largest ...
2
votes
4answers
94 views

Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ [duplicate]

I am trying to solve this integral and I need your suggestions. $$\int \frac{x^4+2x+4}{x^4-1}dx$$ Thanks
1
vote
0answers
66 views

Continuity of integer part function of $1/x^3$

Check the continuity of $$f(x)=\left[\dfrac{1}{x^3}\right]\mathrm{Sgn}\big(\sin(\pi x/2)\big)$$ where $[\cdots]$ is the integer part of $\dfrac{1}{x^3}$ Is there someone who knows the answer?
0
votes
2answers
263 views

Finding the indefinite integral $\int \frac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$

I'm not too familiar with how to solve this. Could anyone present a step by step guide on how to get the answer? $$\int \dfrac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$$
0
votes
1answer
63 views

Theorem or just a change of varibles?

I have a formula in my text: $$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$ I am really lazy and hate remembering formulas to me this looks like a ...
2
votes
0answers
58 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
1
vote
2answers
50 views

Help making Lipschitz proof rigorous

A function $f:R→R$ is defined to be Lipschitz if there is a constant $K>0$ such that for all $a,b∈R$ $$|f(a)−f(b)|≤K|a−b|$$ Suppose $f:R→R$ is Lipschitz. Prove that $f$ is continuous. Could ...
0
votes
1answer
50 views

How to express the sum of a set?

Suppose I have a set of numbers. How can I express in set-theory terms the sum of the elements in that set?
0
votes
1answer
72 views

How to calculate $\sum_{i=2}^n {\frac 1{\log_2 i}}$

How to calculate $$\sum_{i=2}^n {\frac 1{\log_2 i}}$$
0
votes
1answer
234 views

making the domain of $z ↦\tan(z)$ injective

Given the following: $\sin(z)$ = ($e^i$$^z$ - $e^-$$^i$$^z$)/$2i$ $\cos(z)$ = ($e^i$$^z$ + $e^-$$^i$$^z$)/$2$ $\sin(z)\cos(w) - \cos(z)\sin(w) = \sin(z-w)$ $\sin(z) = 0$ has solution $z = kπ$ for ...
1
vote
1answer
135 views

Find local maxima of this quadratic function

How can I find local maxima of this quadratic function? $$f(x) = \sum _{i=1}^n -\frac{(z_i - x)_+^2}{2} - \left\{((\frac{(z_i - x)_+^2}{2})-(\frac{(y_i - x)_+^2}{2}) ) * c_i\right\} $$ which ...
3
votes
1answer
73 views

Calculus II, Curve length question.

Find the length of the curve $x= \int_0^y\sqrt{\sec ^4(3 t)-1}dt, \quad 0\le y\le 9$ A bit stumped, without the 'y' in the upper limit it'd make a lot more sense to me. Advice or solutions with ...
3
votes
3answers
349 views

Using the Intermediate Value Theorem and Rolle's theorem to determine number of roots

Use the Intermediate Value Theorem and Rolle's Theorem to show the that the polynomial $$p(x) = x^{5} + x^{3} + 7x - 2$$ has a unique real root. Can someone please give some hints on how to do this ...
1
vote
3answers
327 views

Limit as N goes to Infinity

Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
2
votes
2answers
130 views

Without calculating limit directly show that it is equal to zero

$$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\frac{1}{3^n}=0$$ I am not really sure what it means by "without calculating limit" and I don't really have ideas how to do it.
0
votes
0answers
30 views

Bounding the partial derivatives of $\left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}$.

Let $\tau = (\tau_1,\ldots,\tau_n) \in \mathbf{R}^n$, $s\geq 0$, and define the function $f : \mathbf{R}^n \to \mathbf{R}$ by $$ f(\xi) = \left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}. $$ ...
8
votes
2answers
165 views

How can I see if this integral is convergent or not $\int_0^\infty \ \frac{1}{1 + x^4\sin x} \,dx $

I think the integral is convergent, but I don't know how to prove it. $\int_0^\infty \ \frac{1}{1 + x^4\sin x} \,dx $
2
votes
3answers
120 views

How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$?

How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ? i know that its "1", but why? How can i calculate this? Thank you very very much =)
0
votes
2answers
150 views

A number theory question about a “double infimum”

Let $x_1,x_2,x_3,\ldots,x_S$ be numbers with $x_i>-1$ for all $i$ and $x_k<0$ for some $k$. How can one show that \begin{equation} \inf_{s\in[1,S]}\inf_{t\in[1,s]}\prod_{i=t}^s ...