0
votes
1answer
23 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
3
votes
6answers
53 views
Determine the inverse function of $f(x)=3^{x-1}-2$
Determine the inverse function of $$f(x)=3^{x-1}-2.$$ I'm confused when you solve for the inverse you solve for $x$ instead of $y$
so would it be $x=3^{y-1}-2$?
3
votes
1answer
34 views
Using a definite integral, to create a specific recurrence relation.
Hello i have the integral:
$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$
I need to show that the integral can be represented by the recurrence relation below;
$$y_n= ...
0
votes
1answer
30 views
The sum of the integration of g and $g^{-1}$
Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto
$[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function.
Use geometric insight to visualize the ...
0
votes
2answers
37 views
$\sum \frac{ln(n)}{\sqrt{n^5}}$ test for convergence
Let $\sum a_{n}=\sum \frac{ln(n)}{\sqrt{n^5}}$. To find if the serie is convergence or not, I had some difficult on finding the proper serie to test the given one.
After some work around, I found ...
1
vote
1answer
52 views
Is the following differentiating under the integral sign correct?
Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
4
votes
4answers
74 views
Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
2
votes
2answers
24 views
Find the average temperature between $t=0$ and $t=24$ when $T(t) = 49+8t-(1/2)t^2$ degrees.
What was the average temperature during that period?
My initial thought was to take the derivative of the problem, plug in 24 for $t$ and solve. I was wrong. This is what I have
$T'=8-t=8-24=-16$ ...
-6
votes
1answer
49 views
need help on homework please [closed]
sam is $22$.If the amount that he invests is constant and if Sam wants his account to be worth $\$1,000,000$ when he is $69$ years old, how much should he invest each week? Assume that the account ...
0
votes
3answers
67 views
Calculate the angle between two curves $f(x)=x^2$ and $g(x)=\frac{1}{\sqrt{x}}$
I want to Calculate the angle between two curves on their intersect $f(x)=x^2$ and $g(x)=\frac{1}{\sqrt{x}}$, what I did so far is:
$$x^2=\frac{1}{\sqrt{x}} \rightarrow x=1$$then :
$$\tan(a)=\left ...
1
vote
5answers
41 views
For what values of a the function $y=x^6+ax^3-2x^3-2x^2+1$ is even
I want to know for what valuyes this function is even
I know that $f(x)=f(-x)$ to proove that function is even. how its helps me?$$y=x^6+ax^3-2x^3-2x^2+1$$
Thanks!
2
votes
2answers
60 views
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$.
(a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and ...
0
votes
2answers
28 views
Vector Field, Curl and Divergence. Simple computing and proving:
So I went through (a) and showed that curl( v x F) = 2v. (doing cross products twice, which comes out pretty simple.)
Now i am stuck at (b), it says use the results from part (a) to compute:
line ...
0
votes
0answers
22 views
Finding a volume of a solid by using integration. Calc III level.
Find the volume of the solid with the plane z = 0 as the bottom, the cylinder x^2+y^2=4 as the side, and the plane z = 3−x−y as the top.
So I set
0 < z < 3-x-y, 0 < x < 2, 0 < y < ...
0
votes
2answers
29 views
Using Lagrange multiplier to find maximum value.
The maximum value of the function $f(x, y) = xy$, and subject to condition $x^2+y^2=1$:
So do I apply Lagrange's Multiplier method to find the maximum value?
I tried to find the numbers just by ...
0
votes
1answer
18 views
Vector Field, Scalar Field. Which is meaningful and not?
Let a, b, c be vectors, f(x, y, z) be a scalar field, F(x,y,z) be a vector field. Which of the
following expressions are meaningful?
I. (a×b)×(c×b)
II. |a|(b· c) +|a|(b+c)
III. ∇ ×(f F)
IV. (∇ ...
0
votes
1answer
61 views
Prove that the derivative is unique
Can someone maybe elaborate on what this question wants me to do exactly and perhaps give me a hint?
Prove that $f:I\to \mathbb {R}$ has a derivative at $c\in I$ if and only if there exist a number
...
3
votes
2answers
56 views
+50
Monoton function proof of discontinuity
Let $f:[a,b] \to \mathbb {R}$ be nondecreasing.
(a) Recall from Thm. 2.1.9 (c) that $f(x-)$ and $f(x+)$ exist at every $x\in [a,b]$,
and conclude that $f$ is discontinuous at $x\in S $ iff $f(x-) ...
1
vote
1answer
15 views
continiouty of maping from set back into itself.
Let $f: [a,b] \to [a,b]$ be continuous. Show that the equation $f(x) = x$ has at least one solution in $[a,b]$.
Firstly im going to assume $x \in [a,b]$ thus a is the min and b is that max or vice ...
2
votes
4answers
70 views
$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$
I'm having trouble understanding how to apply the $\frac{d}{dx}$when taking the anti-derivative.
$$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$
In class it was mentioned we'll end up taking ...
1
vote
1answer
44 views
What about this $\lim_{x \to \infty}\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}$?
When I saw this limit, I didn't even try to solve it by an algebraic method. I thought about the assyntotic concept.
In the example,
$$\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}\sim ...
2
votes
4answers
60 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
1
vote
4answers
75 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
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 ...
1
vote
2answers
24 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?
2
votes
2answers
62 views
How do I solve for $dy/dx$ if $y=\ln (\sin x+\ln x)$?
Solve for $\frac{dy}{dx}$ if $y=\ln(\sin x+\ln x)$.
I know how to solve for integrals involving $du$ and $u$, but how do I do this type of problem (I think it's the opposite of the integral problem)?
...
1
vote
4answers
59 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
3
votes
3answers
102 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 $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
-1
votes
1answer
38 views
prove that $||d^2f(x)||\le M \Rightarrow ||df(x)||\le \sqrt{2Mf(x)}$
let E be a banach space , $f : E \to \mathbb R$ a function of $C^2$ / $f>0$ we suppose that $\exists M $ cte and :
$||d^2f(x)||\le M $
prove that :
$||df(x)||\le \sqrt{2Mf(x)}$
2
votes
0answers
33 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
31 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 ...
1
vote
1answer
26 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 ...
3
votes
3answers
41 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
2answers
56 views
Prove that $\lim _{x \to \infty} x\sin x$ doesn't exist (using delta epsilon).
Question:
Prove that $\lim _{x \to \infty} x\sin x$ doesn't exist (using delta epsilon).
What I did:
I've been struggling with this one for a long time. Really tried digging up the net for ...
1
vote
4answers
58 views
How do I evaluate a definite integral involving trigonometric functions?
Evaluate $$\int_0^{8\sqrt{2}} \dfrac{1}{\sqrt{256-s^2}}ds$$
I know that the antiderivative of $\dfrac{1}{\sqrt{1-x^2}}$ is $\sin^{-1}x + C$ but I am a little unsure how I would change the integrand ...
1
vote
1answer
39 views
Value of coefficients of the power series when radius of convergence is “less than 1” and “greater than or equal to 1”
Let $\sum_{n=1}^\infty c_n (x-a)^n $ be a power series. As "n" approaches infinity,the value of the coefficients "$ c_n $" may or may not be 0 when Radius of convergence R is such that 0< R ...
2
votes
2answers
39 views
power series and sequence
Let $\{ $a_n$ : n\geq 1\}$ be a sequence of real numbers such that radius of convergence of the power series : P(t) = $ \sum\limits_{n=0}^\infty a_n t^n $ satisfies $R > 0$.Then $ a_n \rightarrow ...
2
votes
2answers
30 views
Proving the coefficient of Power series is “0” always on given condition.
Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
1
vote
2answers
28 views
Proving a condition on a cont-differentiable function on positive real numbers.
Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$.
I am not getting how we will ...
1
vote
0answers
98 views
Help making a proof about a Lipschitz function 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
$|f(a)−f(b)|≤K|a−b$|
(a) Use the mean value theorem to show that the function $f(x)=2sin(x)$ is ...
1
vote
0answers
40 views
Approximation of (FEM) by (FDM)
[Ciarlet 3.4-6] Consider the functional
$$J_h : v = (v_i)\in\mathbb{R}^N\longrightarrow J_h(v)\ :=\ \frac{h}{2}\sum_{i=1}^N\left(\left[\frac{v_{i+1}-v_i}{h}\right]^2 + c_iv_i^2\right) - ...
1
vote
2answers
47 views
is $\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)<\infty$?
I don't know why but I'm having a hard time determining whether this series
$$
\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)
$$
converges to a real limit.
...
3
votes
2answers
63 views
Finding the Derivative without using Product or Quotient Rule
I have a math problem where I am required to find the derivative of a function with the limitations of not being allowed to use the Product or Quotient Rule of Differentiation.
The problem looks like ...
0
votes
1answer
139 views
making the domain of $z ↦\tan(z)$ injective
Given the following:
$\sin(z)$ = ($e^i$$^z$ - $e^-$$^i$$^z$)/$2i$
$\cos(z)$ = ($e^i$$^z$ + $e^-$$^i$$^z$)/$2$
$\sin(z)\cos(w) - \cos(z)\sin(w) = \sin(z-w)$
$\sin(z) = 0$ has solution $z = kπ$ for ...
2
votes
4answers
62 views
Finding $\lim_{x \to 0}\frac{\tan x-x}{x^3}$
Feeling like i did this wrong
$\displaystyle \lim_{x \to 0}\frac{\tan x-x}{x^3}$ $\to$ $\displaystyle \lim_{x \to 0}\frac{\sec^2x-1}{3x^2}$
$\displaystyle \lim_{x \to 0}\frac{2\tan x\sec^2x}{4x}$ ...
4
votes
3answers
98 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
1
vote
2answers
57 views
Heine-Borel Theorem implies compact proof
Let S be a nonempty subset of $\mathbb {R}$ such that H is any open covering of S, then S
has an open covering $H^{\sim }$ comprised of finitely many open sets from H. Show that
S is compact.
I got ...
1
vote
1answer
56 views
How to solve a tensor differential equation?
Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$
where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.
The original Problem
How does ...
1
vote
1answer
50 views
Laplace equation and integral
$$ \int_0^{2\pi} \frac{1+3 \sin{\phi}}{a^2-2ar \cos(\theta - \phi) + r^2 } d\phi$$
Help me plz ... I have tried to solve this. but I still don't know.
1
vote
2answers
70 views
Proof by counter example to false analysis statements.
(i)Disprove that the isolated points of a set form a closed set.
(ii)Disprove
every open set contains at least 2 points.
(iii) Disprove $\partial ( S \cup D) = \partial S \cup \partial D$
$S= ...
