2
votes
3answers
95 views

Understanding of the formal and intuitive definition of a limit

The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...
-3
votes
1answer
34 views

using the definition to find derivatives [closed]

Calculate the derivate of the given function directly from the definition of derivative, and express the result using differentials when $f(x) = \sqrt{x^2 + 3}.$
1
vote
4answers
182 views

Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?

For writing a (german) article about the power with natural degree I have the following question: In school one defines the power with natural degree via $$n^k = \underbrace{n\cdot n\cdot \ldots ...
0
votes
2answers
33 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
3
votes
1answer
89 views

Has anyone succeeded in formalizing Leibniz notation in such a way that the chain rule and inversion rule “work”?

The notation $\frac{\partial}{\partial x}$ is ubiquitous and totally useful, but also kind of weird. It seems to be doing the following: Bind $x$ Compute the derivative Evaluate at $x$ To ...
4
votes
4answers
128 views

Why is the derivative at a jump undefined even if the slope remains the same?

I've searched online and found almost nothing. What in the mathematical definition of a derivative makes it so that the derivative of the following is undefined at ...
0
votes
2answers
36 views

A question about the boundedness theorem

I have a question about the boundedness theorem: http://en.wikipedia.org/wiki/Extreme_value_theorem The boundedness theorem which states that a continuous function $f$ in the closed interval ...
1
vote
2answers
67 views

The sequence in the definition of the integral

In my high school Calculus class, we learned this definition of the definite integral: $$\int_a^b f(x)dx = \lim_{n\to \infty} \sum_{i=1}^n f(x_i) \frac{b-a}{n}$$ Now that I know more about sequences ...
1
vote
1answer
39 views

Limit of vector-valued function is equal to the limit of its components

Let $f: \Bbb R^m \to \Bbb R^n$. Express $f(x)$ in terms of components: $$f(x)=(f_1(x), f_2(x), ... , f_n(x))$$ I need to prove that $f$ is continuous at $a$ if and only if each $f_i$ is continuous ...
1
vote
3answers
91 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
3
votes
1answer
115 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
5
votes
1answer
60 views

Halmos on Definability and Luzin on Division by 0

For a successful introduction of a new symbol (e.g. '$\emptyset$') into a mathematical discourse it is necessary and sufficient that the symbol refer to something (e.g. Existence + Specification in ...
0
votes
1answer
19 views

Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
6
votes
1answer
147 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
1
vote
1answer
58 views

Precise definition of oscillation behavior of functions like $\sin(\frac1{x})$

I tried today precisely defining the oscillation behavior present in functions like $\sin(\frac1{x})$ i.e: To do this, I started with the domain of function as in limits, and let $f(x)-L$ have ...
3
votes
1answer
94 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
0
votes
0answers
46 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If $\sum_{n=1}^\infty a_n$ ...
2
votes
2answers
50 views

A problem with the domain of function in the defintion of limits

My Stewart's Calculus gives the following definition of limit: $f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for ...
0
votes
1answer
76 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
11
votes
3answers
396 views

A doubt in the rigorous definition of limits.

I studied the definition of limits today, and I think I mostly understood it, but I have a little doubt. In the definition: $f(x)$ is defined on some open interval containing $a$, except at possibly ...
0
votes
2answers
72 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that ...
1
vote
2answers
113 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
1
vote
4answers
92 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
0
votes
1answer
47 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
0
votes
2answers
92 views

I Need Help Understanding the Formal Definition of A Limit

I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that: $$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ ...
0
votes
0answers
27 views

Definition of $\partial^2{u}/ \partial {n^2}$

I want to solve a boundary value problem on a square, that one of the boundary condition is $$\frac{\partial^2{u}}{ \partial {n^2}}=0.$$ The test example that I want to solve is: $$\Delta^2u=f,$$ ...
19
votes
5answers
595 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
3
votes
2answers
126 views

Radius of Convergence and its application to a Power Series including $x^{2n}$ rather than $x^n$

(Radius of Convergence) Consider the Power Series $f(x)=\sum_{n=0}^{+ \infty}a_n x^n$, the radius of convergence $\rho$ can be found using $$\rho = \displaystyle \lim_{n \to + \infty} \left| ...
0
votes
1answer
33 views

Question on definition of little o

I would like to generalize the definition of little o. The definition from Wikipedia is as such: Let $f$ and $g$ be two real valued functions. We write $f(x) = o(g(x))$ as $x \to \infty$ if for all ...
0
votes
1answer
65 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
0
votes
1answer
65 views

Definition of continuity without limit

Without limits, I define $f(x)$ is continuous at $x=a$, when it: $f(a)$ exists; For every $d>0$, in the close interval $[a-d,a+d]$, there exist a maximum $M$ and a minimum $m$; For every ...
1
vote
2answers
81 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
0
votes
3answers
148 views

Solution of a differential equation having a singularity (not everywhere defined) [closed]

Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity). I want to remember the exact definition of a solution with singularity, which I ...
0
votes
1answer
144 views

Limit of Identity Function vs. limit of Squaring Function

$$\lim_{x\rightarrow a} x = a$$ and $$\lim_{x\rightarrow a} x^2 = a^2$$ $f(x)=x^2=x \times x$, i.e.: two identity functions. I'm a bit confused on how $x^2$ can be interpreted as being ...
6
votes
5answers
183 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
1
vote
2answers
92 views

What *really* are the local maxima and local minima

In math is the local max and local min just any peak ... point where slope of the function changes from positive to negative or vice-versa... Or are the LOCAL max and min just the highest point of the ...
0
votes
1answer
398 views

Why is the direct substitution property so specific

Mt text book states the Direct Substitution Property as If f is a polynomial or a rational function and a is in the domain, then $$\begin{align*} \lim_{x\to a} f(x)=f(a) \end{align*}$$ Why does ...
5
votes
2answers
124 views

Differential — Mathematically conform?

In calculus, I know that one defined the differential quotient $$\frac{dy}{dx} := \lim\limits_{h \rightarrow 0}{\frac{y(x+h)-y(x)}{h}}$$ I learned that it is not a quotient, but can be treated as one ...
2
votes
2answers
78 views

Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $?

Is the equality below true? $$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
4
votes
4answers
243 views

Using the definition of the derivative prove that if $f(x)=x^\frac{4}{3}$ then $f'(x)=\frac{4x^\frac{1}{3}}{3}$

So I have that $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ and know that applying $f(x)=x^\frac{4}{3}=f\frac{(x+h)^\frac{4}{3} -x^\frac{4}{3}}{h}$ but am at a loss when trying to expand ...
3
votes
4answers
2k views

Continuity on open interval

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval. But an open interval $(a,b)$ doesn't contain $a$ and $b$, so we never ...
2
votes
0answers
1k views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
3
votes
3answers
129 views

Calculating derivative by definition vs not by definition

I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. The following two examples confused me: $$ g(x) = \begin{cases} x^2\cdot ...
2
votes
2answers
183 views

Defining infinitesimals

Can such definition of infinitesimals hold? $$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$ And, if the above definiton works, then obviously ...
0
votes
1answer
83 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
1
vote
3answers
141 views

Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
2
votes
1answer
926 views

Definition of local maxima, local minima

Wikipedia says that: A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when |x − ...
2
votes
3answers
2k views

What's the difference between direction, sense, and orientation?

I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...
1
vote
1answer
31 views

Change Along A Tangent Line

I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$