What is Absolute value: Definition and 367 Discussions
In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
this is the question,
Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.)
I know you have to use the upper and lower bounds to prove this statement but i don't know where to start...
this is the question,
Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.)
I know you have to use the upper and lower bounds to prove this statement but i don't know where to start?
Thanks
1. Evaluate
\int_{-1}^{3} \left|x^2 -4\right| dx
3. The Attempt at a Solution
This is the first time I'm trying this type of question & I think I need to use the following theorem for such questions;
f is integrable on a closed interval a to b.
\int_{a}^{b}f(x)dx =...
I'm taking an algebra & triginometry class at my college and my professor is kind of slow and unclear. I think I'm a fast learner and a good understander which is why I came here to get this info down. We're up to complex fractions or radical equations right now I think, forgot which. Something...
Prove the following: if |a| \leq b then -b \leq a \leq b (where b \geq 0 ).
So a \leq b and -a \leq b . Then -b \leq a so that -b \leq a \leq b .
Suppose that -b \leq a \leq b . Then a \leq b and -a \leq b so that |a| \leq b .
Is this a correct proof? You don't have to...
Homework Statement
Rewrite the following expressions without absolute value signs, treating various cases separately where neccesary
Homework Equations
a-Abs[(a-(abs)a)]
the question is do i have 2 answers to this ?
Homework Statement
F(x) = (8-12ln|x|)/(x^4) > 0
(a) For what values of x is the expression F(x) defined?
Write your answer in interval notation.
(b) At what value(s) of x is the expression F(x) equal to zero?
If there is more than one answer separate them by commas.
(c) The set of...
Homework Statement
Question is: how can you tell if there are any places you can't take the derivative of an equation that has an absolute value (using logic, not just graphing it)
example equations
1. \left|x-5\right|
2. \left| x3+4x2+9x+17 \right|
x2+1
3...
Here, it says that for the limit f(x) = |x| / x,
|x| = { x, x > 0
-x, x < 0 }
What I don't undestand is why is |x| = -x for values under zero? Isn't the absolute value for negative values just x and not -x?
thanks.
EDIT: I don't want to start a new thread, but I got stuck on this...
Ok the problem is:
lim x->-1 |x+1| / x2-1
(sorry i don't really know how to type the equation out)
I think that you have to find the limit as x->-1 from both the left and right sides
from right:
so I got lim x->-1 = (x+1)/x2-1 = (x+1)/(x+1)(x-1) = 1/(x-1) =1/-1-1 =-1/2
How would I...
While reading my text, I came across an inequality that I couldn't convince myself of...
For real numbers a,b: \left|a+b|<= |a|+|b|. Is this something proven? Or is it an axiom or something?
Rigorous proof of basic "absolute value" theorem?
Hey :) I'm working through a Real Analysis text, and I came across this theorem and "proof":
http://img352.imageshack.us/img352/6725/proofbx2.png
It kind of took me by surprise, because the author of the text is usually very careful about...
Just wanted to say hi before I start my post! :smile:
As you may know there is a property of the absolute value that states; for a, b \in R;
|ab| = |a||b|
Well, my friend asked me if I knew a proof for this... but I don't know...
How can we prove this statement/property? I know there...
Ok this is something i learned few years ago and I am a bit rusty.
So i have to find the absolute value of:
\frac{1 - 2i}{3 + 4i} + \frac{i - 4}{6i - 8}
So first i add the two fractions and i get:
\frac{(1 - 2i)(6i - 8) + (i - 4)(3 + 4i)}{(3 + 4i)(6i - 8)}
Next i simplify and then...
Hey folks,
I'm looking for a little guidance in solving the derivative y'(x)of the following function containing an absolute in the exponent:
y(x)=e^{a|x|}
I'm pretty sure its not as simple as
y'(x)=a e^{a|x|}
Any suggestions??
Homework Statement
a) Find the average value of the function f(x)=|x-1| over [0,2].
b)Verify the mean value theorum for integrals for the function and interval in part (a).
Homework Equations
mean value theorum for integrals: \int_a^{b} f(x)dx=f(c)(b-a)
mean value theorum for...
I found a study page which lists the absolute value of x for x<0 as -x. I think this has to be a typo. The study area is real analysis. Does anyone have better information? Maybe it is some special notation?
Homework Statement
Resolve these equations and these inequations with the absolute values. Give the solutions in the form of interval :
|2-x|< 4
|6-2x| = 3
|x+2| > 3
|4x²-12x+9| = 4
|3x+1|+|1-x|>3
|1-x²|=2x
|x+2|<|x+3|
|x^3-1|+pi[tex]>[tex]\sqrt{3}
3<|x+2|<4...
Homework Statement
\left|x-5\right|\leq\left|x+3\right|
i know how to slove abs inequalities when there is one. like \left|x-5\right|\leq5. put x-5 between 5 and -5. then solve for x.
i don't know how to start this.
can someone provide some insight.
Homework Statement
a and b are real numbers.
Show l a-b l < l a l + l b l
Homework Equations
Well, I know la+bl < lal + lbl by the triangle inequality.
The Attempt at a Solution
If I can prove that la-bl < la+bl, then I'm done, but that most recent inequality almost seems...
Homework Statement
Prove that abs(a+b+c) is less than or equal to abs(a)+abs(b)+abs(c)
Homework Equations
None
The Attempt at a Solution
This makes sense to me that this would always be true, but i just can't seem to figure out how to write it out
Hello,
abs(x^2 - 4) < 1
implies that:
x^2 - 4 < 1
and
4 - x^2 < 1
solving first equation for x gives:
-sqrt(5) < x < sqrt(5)
solving second equation for x gives:
-sqrt(3) < x < sqrt(3)
Now, my question is, what does that mean??
How do I give the solution set, without a...
Hello everyone,
For the following absolute value equations, I have no trouble solving them and finding the valid x solutions by plugging all the x solutions into the original equation.
However, I am just wondering if could someone please show me how to set restrictions for the following...
Homework Statement
25|x| = x^2 + 144
Homework Equations
none
The Attempt at a Solution
okay well, I'm not quite sure what to do, do i try to isolate the |x|? and then break it up into a postive and negative?
|x| = (x^2 + 144)/25 ?
but from here i become lost...
[SOLVED] Absolute Value in a double integral
Homework Statement
If \Omega = [-1,1] x [0,2], evaluate the double integral \int\int_{\Omega} \sqrt{|y-x^{2}|} dA given that it exists.
Homework Equations
None
The Attempt at a Solution
I know that in order to integrate with the...
Hello,
My question is as follows: Show that the function f(x,y) = sqrt(abs(xy)) is not differentiable at (0,0).
I was going to go with trying to show that the directional derivatives don't all exist here, but that would require finding the gradient, and I always get confused when trying to...
Homework Statement
Find the real/imaginary parts of sinh(x+yi) and its abs value.Homework Equations
The Attempt at a Solution
I am able to decompose sinh(x+yi) = cosy*sinh(x) + isin(y)cosh(x) - (which is correct according to my book)
Now finding the absolute value is kind of causing some...
Homework Statement
If |x| = -x + 2x*H(x) what is |x - a|? This isn't the actual question, just something I need to know to solve the question.
Homework Equations
H(x) is the Heaviside function which is:
y = 1 if x >= 0
y = 0 if x < 0
The Attempt at a Solution
Well, I'm...
Homework Statement
The problem is: Draw the graph of the following function:
f(x)=|x+|x+|x-1|||
Homework Equations
|x|=\left\{\begin{array}{cc}x,&\mbox{ if } x \geq 0\\-x,&\mbox{ if }x<0\end{array}\right
The Attempt at a Solution
If the function were, for instance...
Homework Statement
Prove the following statement:
For all real numbers x and y, |x| times |y| = |xy|
Homework Equations
I really don't know how to start this as a formal proof.
The Attempt at a Solution
I was thinking I'd have to break it down into four cases and logically prove...
Okay, we don't understand this and we need much help
the largest pyramid included in the first wonder of the world is Khufu. IT stands 450 feet tall and its base is 755 feet long. Imagine that a coordinate plane is placed over side of the pyramid. In the coordinate plane, each unit...
Please help me out here with this problem:
37. Express the function f(x)=l x l + l x-2 l
without using absolute value signs.
Okay, I've took a sneak peak at the answer key and it is given in kinda of like piece-wise notation where they have 3 equations and different intervals of x for...
Hi. For an integral like this, for example:
\int {\sqrt {1 - \cos ^2 x} dx}
The most obvious way of solving would be to make use of the pythagorean identity, to get:
\int {\sqrt {\sin ^2 x} dx}
Now, I've been taught to simply evaluate it like this:
\int {\sqrt {\sin ^2...
Hey i have 2 questions here that i have finished and i have no answers to my solutions so i would like someone to check it over and see where i went wrong , I am not 100% sure on this stuff and need some help , thanks a lot !
Question 1: Using the definition of a derivative, show that ƒ (x) =...
The equation |2x+a|=|b-x| has exactly 2 solutions, at x=-4 and x=2/3. Find the value(s) of a and b.
Ok so the questions is asking me to find possible values of a and b which make the equation true for ONLY x=-4 and x=2/3.
So for:
|a-8|=|b+4|
|a+4/3|=b-2/3|
I need to find the values...
Given the function:
f(x) = \left| {x + 1} \right| + \left| {x + 2} \right|
How can i write that as a piecewise function? If i was given something in the form of f(x) = \left| {g(x)} \right|, i know to write it as:
f(x) = \left\{ {\begin{array}{*{20}c}
{g(x),} & {g(x) \ge 0}...
What I have is this:
Let P_n(x)=(x-x_0)(x-x_1)...(x-x_n), _i are subscripts.
Prove that the maximum value of |P_1(x)| for x in [x_0,x_1] is h^2/4, where h =x_1 - x_0.
All the x_i terms are evenly spaced. That is, x_(i+1)-x_i is the same for all i.
What I noticed is that P_1(x_0)=P_1(x_1)=0...
I need some help graphing an Absolute value. I know what it look like from a graphing calculator but I don't understand how to get to the answer.
http://media.twango.com/m1/original/0042/5fda15431b96400eb32e66afa3627624.jpg
Thanks
I'm confused by why my work is wrong:
3x^{2}+12x>0
3x(x+4)>0
3x>0 , x+4>0
x>0 , x>-4
However, the correct answers are: x>0 and x<-4? Why is the sign reversed on -4? I thought you were only suppose to reverse it if you multiply or divide the equation by a...
Homework Statement
Question straight off the book "In this lesson, you have explored the absolute value of a function. How is it defined?"
Homework Equations
In this lesson, I did questions like xE[0,9] and was told to write each intervals in absolute value notation.
The Attempt at a...
Homework Statement
Consider two vectors by A=5i-3j and B=-i - 2j
Calculate A+B
Calculate A-B
Calculate absolute value of A+B
Calculate absolute value of A-B
Homework Equations
R= (Ax + Bx)i + (Ay + By)j
R= (Ax + Bx)i - (Ay + By)j
The Attempt at a Solution
I'm not sure...
what is the absolute value of imaginary numbers, why not "queer" numbers?
the square root of -1 is "i".
the absolute value of an interger is itself, and of a negative number, it is a positive interger.
|-5| = 5
|5| = 5
what is
|5i| = ?
|-5i| = ?
why not invent a queer number...
Hi. I need help computing a range.
The question is : Find the domain and range of y=\frac{|x+2|}{x}.
The domain is obvious, x can't be 0, (-inf,0,) U (0,inf). But how do I find the range?? Can someone help me out? I have tried messing around with the definition of absolute value... if...
I am trying to understand the meaning of the s-domain absolute function derived from taking the laplace transform of a t-domain function. I know for sure that the real part of the complex frequency in the time domain is the sinusoidal frequency and the imaginary part of the complex frequency in...
How does a norm differ from an absolute value? For example, is
\|\mathbf{x}\| = \sqrt{x_1^2 + \cdots + x_n^2}
any different than
|\mathbf{x}| = \sqrt{x_1^2 + \cdots + x_n^2}
??