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

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3
votes
2answers
54 views

Every open map $ f : \mathbb R \rightarrow \mathbb R$ implies

Let $ f : \mathbb R \rightarrow \mathbb R$ be an open map. Then $f$ is one-one $f$ is onto. $f$ is bounded $f$ has exactly two zeroes. if $f(x) = x $ which ...
0
votes
0answers
18 views

$|x^2y|+|ay \cos x|=O(x^2+y^2)$

Prove that $|x^2y|+|ay \cos x|=O(x^2+y^2)$. I have tried to expand $\cos x$ as Taylor series but I didnit see how this help, please helps.
0
votes
0answers
22 views

How can I solve differential equation near point that is not normal

Let we have the following differential equation : $$2z(z+1)w''+z(z+1)w'-w=0$$ By power series near the point $z_0=0$ the problem that the point $z_0$ isn't normal point for this equation , so how can ...
0
votes
2answers
18 views

Equating Coefficients in Partial fractions

I'm having a hard time figuring out A, B, and C for this problem. $$ \frac{8}{y^{3} - 4y} $$ All I've got so far is $$ \frac{A}{y} + \frac{B}{(y+2)} + \frac{C}{(y-2)} $$ $$ 8 = A(y+2)(y-2) + ...
0
votes
0answers
29 views

Proof with constant magnitude vector functions

One of the questions on a quiz I just took was to show that if the magnitude of a parametric function $X(t)$ for all $t \in \mathbb{R}$ is constant then $X''(t)\cdot X(t) = - \left|X'(t)\right|^2$. ...
0
votes
1answer
30 views

Show that the function is differentiable

I have to prove that the following function is differentiable and to find its derivatives at any point. $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \rightarrow x^2+y^2$$ In my book there is a ...
1
vote
1answer
50 views

Trigonometric integral evaluates to factorial

I would like to prove the integral identity $$\int_{0}^{2\pi} e^{\cos(x)} \cos(nx - \sin(x)) \, dx = \frac{2\pi}{n!}$$ One approach is to interpret this as the real part of a complex exponential ...
2
votes
2answers
56 views

Show that $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$.

Prove that for $x\in (0,\infty)$, $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$. I'm a little bit stuck, but I think I have the right idea. Any hints or solutions are greatly appreciated. Here is what ...
3
votes
6answers
187 views

How to Find the Function of a Given Power Series?

(Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.) Can someone please explain how to find the ...
1
vote
0answers
51 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
0
votes
1answer
36 views

Expansion of $(1+x)^\alpha$.

Let $\alpha\in\mathbb{R}$. Prove that, for all $x \in [0, 1)$ we have $(1 + x)^\alpha=1+{\alpha x\over 1!}+{\alpha(\alpha-1)x^2\over 2!}+\cdots+{\alpha(\alpha-1)\cdots(\alpha-k+1)x^k\over k!}+\cdots$. ...
2
votes
1answer
42 views

Prove that $\lim_{x \to a} \big[ f(x)+g(x)\big] = p+q$

Let $f: A \to \mathbb{R}^m$ and $g: B \to \mathbb{R}^m$ be functions such that $A,B \subseteq \mathbb{R}^p$, $a \in \mathbb{R}^p$ and let $\beta$ be a fixed number. Furthermore let $f(x) = ...
2
votes
3answers
73 views

Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour.

I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...
0
votes
0answers
36 views

Stokes Theorem on a sphere problem

I'm looking through my multivariable calculus notes and have come across a question I'm not sure I fully understand. It reads, "If $\omega$ is a differential form on $\mathbb{R}^3$ and $M$ is a sphere ...
1
vote
1answer
29 views

Trying to understand Spivak's answer for limit proof (Chapter 5 problem 3v)

Prove the limit l for the function at a: $$f(x)= x^4 + \frac1x, a =1$$ I have successfully found a $\delta$ in terms of $\epsilon$, and here is how I did it: Since we can see the limit is 2 at a = ...
0
votes
1answer
53 views

How can I integrate $\frac{1}{x^2-x-1}$?

I need to find $\int\frac{1}{x^2-x-1}dx$ and I don't know what to do. I've thought about substitution or partial fractions but neither has worked.
1
vote
2answers
32 views

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge?

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge? My try:As $n$ approaches zero, $\sec(\frac 1n)$ gets close to $\frac 1{1-0.5\frac 1{n^2}}=\frac{2n^2}{2n^2-1}=1+\frac ...
4
votes
3answers
57 views

If $ f'(c) > 0 $, then there is an $ x $ such that $ f(x) > f(c) $.

Here is the homework question that I have: If $ f: [a,b] \to \Bbb{R} $ is differentiable at $ c $, where $ a < c < b $ and $ f^{\prime}(c) > 0 $, prove that there exists an $ x $ such ...
0
votes
1answer
36 views

Derivatives - Show equality

Let $y(x)$ be defined implicitly by $G(x,y(x))=0$, where $G$ is a given two-variable function. Show that if $y(x)$ and $G$ are differentiable, then ...
1
vote
2answers
51 views

Evaluating $ \int \frac{1}{5 + 3 \sin(x)} ~ \mathrm{d}{x} $.

What is the integral of: $\int \frac{1}{5+3\sin x}dx$ My attempt: Using: $\tan \frac x 2=t$, $\sin x = \frac {2t}{1+t^2}$, $dx=\frac {2dt}{1+t^2}$ we have: $\int \frac{1}{5+3\sin x}dx= 2\int ...
2
votes
3answers
65 views

Limit $(0÷0)$ and use L'hopital Rule

find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
0
votes
1answer
13 views

Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing. $a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$. On the other ...
2
votes
0answers
12 views

uniqueness of solution of a nonlinear equation system

$f(x,y):\mathbb{R}_+\times \mathbb{R}_+\rightarrow\mathbb{R}_+$, is a differentiable function with partial derivatives $f_1(x,y)<f_2(x,y)<0$. $z_1,z_2,...z_n$ are n given positive numbers. ...
0
votes
1answer
26 views

Integration: Finding area, volume and arc length

I am new to integration, so please do not mark this question as "not enough research done" Here is the question (please open image in new tab to see it clearly) - I am getting stuck with the ...
-2
votes
0answers
38 views

Integration help needed with two problems

Here are 2 problems that are confusing me to a great extent - I have no issues with integration, but I'm unable to figure out the limits and the equation to be integrated in these problems. The ...
1
vote
1answer
18 views

Finding interval of convergence for complicated sum

I'm going through old exams for my Calc III course and came across a problem that I did not know how to do. The problem is: Find the interval of convergence of the series ...
1
vote
1answer
21 views

General technique of proving limit exists as x approaches infinity

I'm looking for a way of proving that $f(x)$ will have some limit (without specifying what it is) as $x \to \infty$. To make this more concrete, I'm asked to (1) prove the limit exists and (2) find ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
1
vote
1answer
36 views

Interchange of the expected value and infinite summation $E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$

Let $Y_t$ be a random variable (Not positive necesarily). Can I make the next assumption? $$E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$$ Thanks! I think it is correct ...
1
vote
1answer
29 views

implicit multivariable derivative

I didn't really understand how implicit multi-variable functions are derived; I thought of another method which may fit and may not; suppose we have $xy^2z^3=8$ and we want to derive it; it is the ...
1
vote
4answers
52 views

Prove that $\lim_{n\to \infty} n\cdot r^n=0$ where $(0\leq r <1)$ without using ratio test

$\lim_{n\to \infty} n\cdot r^n=0$, where $0\leq r <1$, can be obtained by vanishing condition (considering $\sum^{\infty}_{n=1}n\cdot r^n$, which converges, using ratio test). Is there a direct ...
1
vote
0answers
15 views

Volumes of solids (of rotation). Any real world applications?

Washer method, disc method, etc. In what areas or fields would someone make these calculations? For example, I think 3D printers do some sort of "slicing" algorithm in their CPU in order to print ...
1
vote
2answers
42 views

$\sum_{n=2}^\infty {(-2)^n \over n} $ How does this converge or diverge using the alternate series test?

$$\sum_{n=2}^\infty {(-2)^n \over n} $$ When I took the limit I got -2, I also tried using ratio and root test and got the same answer. The answer is supposed to be divergent I think but I thought if ...
0
votes
1answer
20 views

Solving Recurrence Relations with Geometric Series

If given the following problem... $$4T \left(\frac n2\right) + c$$ after getting the pattern down you see the following $$4^k T\left(\frac {n}{2^k}\right) + 3^{k-1}c + 3^{k-2} c + \cdots + 3c + c$$ ...
0
votes
1answer
22 views

limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
0
votes
2answers
41 views

How do I go about proving that an even function about x=0 has a derivative of 0?

I do not have mean value theorem and can't use it. All I have are IVT and that the derivative of an even function is odd and the definition of a derivative. With those tools, what can I do?
2
votes
3answers
58 views

How can I compute the following integrals

What is the best way to compute the following integrals $$\int_{0}^{1}\int_{0}^{1}\frac{x^2-y^2}{(x^2+y^2)^2}dydx$$ And $$\int_{0}^{1}\int_{0}^{1}\frac{x^2-y^2}{(x^2+y^2)^2}dxdy$$ I know the ...
-1
votes
0answers
29 views

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a$ is of the form $\xi(f) = \sum_{i} c_i \frac{\partial f}{\partial x_i}(a)$? [on hold]

How to show that each tangent vector to $\mathbb{R}^n$ at a point $a = (a_1, \ldots, a_n) $ is of the form $\xi(f) = \sum_{i=1}^n c_i \frac{\partial f}{\partial x_i}(a)$? Thank you very much. Edit: ...
1
vote
1answer
17 views

Differentiation and Integral

Let $\phi:[0,\infty)\to[0,\infty)$ be increasing and left-continuous, with $\phi(0)=0$. Suppose on $(0,\infty)$ that $\phi$ neither identically zero nor identically infinite. Then the function $\Phi$ ...
1
vote
0answers
12 views

r(tau) within of H´s level curves

Im currently working on a math project and have one question regarding level curves. I´ve been given a function $$H(\theta,\omega)=\frac{1}{2}\omega^2-C\theta-\cos{\theta}$$ And I´ve shown that if ...
5
votes
2answers
1k views

What is cosine to the power of zero?

I was doing a question relates to substitution rule under integration. The question is as follow: Evaluate $\int{1\over{(1+x^2)^n}}dx, n\in \mathbb{Z}^+$ We have seen that ...
0
votes
1answer
33 views

Volume of paraboloid

A paraboloid is formed by revolving a parabola, $y=kx^2$, about its axis of symmetry. The paraboloid is bounded by a plane cutting the axis of symmetry perpendicularly at the point (0,20). The ...
1
vote
1answer
24 views

Linearizing an expression involving exponentials

How can I linearize $f(x) = A(1-\text{exp}(Bx))$? I tried to take the natural logaritm, but could not find something that looks like linear. I am trying to find a fitting curve for this by hand. $A$ ...
0
votes
1answer
35 views

Prove limit exists if and only if left and right limits exist and are equal

Prove $\lim_{x\to a}f(x)=L \iff \lim_{x\to a^+}f(x)=L=\lim_{x\to a^-}f(x)$ I have no problem with the $(\Leftarrow)$ direction but I can't do it for the other direction. Proofs for both directions ...
0
votes
1answer
18 views

Nonnegative Dini derivative implies nondecreasing function

This was posed as one of the proposition in my lecture note: If $f$ is continuous on $[a,b]$ and one of its Dini derivative is everywhere nonnegative on $(a,b)$, then $f$ is nondecreasing on ...
1
vote
2answers
50 views

Area inside loop of polar equation, unsolvable problem?

Is this problem solvable? "Please find the area inside the first loop of the following equation (using polar coordinates): r = cos$(\theta)$ - sec$(\theta)$." From what I can tell, this function ...
0
votes
1answer
25 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
1
vote
2answers
56 views

Method to integrate $\cos^4(x)$

Here my attempts for integrating $\cos^4(x)$ in few methods. 1st method. $(\cos^2x)^2=(\frac{1}{2})^2(1+\cos2x)^2$ $=\frac{1}{4}(1+2\cos2x+\cos^22x)=\frac{1}{4}(1+2\cos2x)+\frac{1}{4}(\cos^22x)$ ...
0
votes
0answers
23 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
0
votes
0answers
12 views

Show that convex function crosses linear function

Let $u(x)$ be a real positive function and define $$ g(x)=\frac{1}{u'(x)} $$ Assume that $u'(x)$ can be either linear (L) or concave (C), such that $g(x)$ is either linear or convex. Is it then ...