image due to macros in Overleaf
ok I think (a) could just be done by observation by just adding up obvious areas
but (b) and (c) are a litte ?
sorry had to post this before the lab closes
This is an exercise given in my real analysis course. We have definitions for continuity, uniform continuity, l.s.c and u.s.c., but I don't see how to apply them.
I guess I want to construct $φ$ piece by piece of the interval to eventually cover all of [a,b]?
Exercise:
A continuous function...
Hi!
I'm looking at some piece-wise function right now and I can't help but wonder what all these parts are called. I'm learning to use and write this type of functions now and I think I have a pretty good understanding of how they work. I even took the extra step of learning some "set builder...
Hello,
Me and my friend were talking about differentiability of some piece-wise functions, but we thought of a problem that we could were not able to come to an agreement on. If the function is:
y=sin(x) for x≠0
and
y=x^2 for x=0,
Is this function differentiable? The graph looks like a normal...
I recently plotted a piecewise function:
Plot[Piecewise[{{1 - Exp[-.002*t],
0 <= t < 120}, {-Exp[-.002*t] + Exp[-.002*(t - 120)],
120 <= t}}], {t, 0, 5000}, PlotRange -> {0, 0.25}]
I then defined the function which I am calling q[t_] as follows:
q[t_] := Piecewise[{{1 -...
Homework Statement
I've entered the following piecewise equation into Mathematica:
Plot[Piecewise[{{sin (t), 0 <= t < \[Pi]}, {5 + 5 cos (t) + sin (t), \[Pi] <= t < 4*\[Pi]}, {10 cos (t) + sin (t), 4*\[Pi] <= t}}], {t, 0, 20*\[Pi]}]
But I am getting a blank graph in return. I've proofread my...
Homework Statement
##f:\mathbb{Z} \to \mathbb{N}##
## f(x) = \begin{cases} 2x+2 & \text{if } x \geq 0 \\ -2x-1 & \text{if } x < 0 \end{cases} ##
Prove that f is a bijection from ##\mathbb{Z}## to ##\mathbb{N}##
Homework EquationsThe Attempt at a Solution
Proving that it is an injection if quite...
Homework Statement
http://s23.postimg.org/wsj9e91wb/IMG_1334.jpg[/B]
photo of the problem
g(x)=∫ƒ(t) dt from -5 to x
ƒ(t) = (0 if x < -5
5 if -5≤x<-1
-3 if -1≤≤x<3
0 if x≥3)
(a) g(-8) = 0
(b) g(-4) = 5
(c) g(0) = ?
(d) g(4) = ?
Homework Equations
∫ƒ(x) from...
The function is g(x) = (x^2 - a^2)/(x - a) if x doesn't equal a; and the second part is g(x) = 8 when x = a. The question asks for me to find a specific value for a so that the function might be continuous on the entire real line.
I know that each part of the piece-wise function needs to...
f: Z -> Z defined by f(x) = x/2 if x is even, (x-1)/2 if x is odd.
Proof: If x is even:
x1 = 2k1
x2 = 2k2
Suppose f(x1) = f(x2), then
2k1/2 = 2k2/2
k1 = k2
So if x is even, the function is one to one? Is this an okay proof for the first half of if x is even, then I just do the...
1. Suppose f(x)=0 if x is irrational, and f(x)=x if x is rational. Is f differentiable at x=0?
2. the derivative= lim[h->0] [f(a+h)-f(a)]/h
3. I don't really know how to start, but I do know that between any two real numbers, there exists a rational and irrational number. So I'm...
Homework Statement
Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold.
f(x)= {x^3 if x < or = -2
{2 if x > -2
Homework Equations
The conditions are that a function is said to be...
Homework Statement
Function f and g are defined as follows :
f(R)=R , f(x)=x^2 , g(R)=R , g(x)=x+1,x>=0 , -x , x<0 (its a piecewise function) . Find fg(x) and gf(x) .
Homework Equations
The Attempt at a Solution
fg(x)=
(x+1)^2 , x>=0
x^2 , x<0
gf(x)=
x^2+1 , x>=0...
I hate a piece-wise function
f(x)= [ 1 if 0 </eq x </1
[ 0 if 1 < x < 2
[ -2x+4 if x >/eq 2
If I let F(x)= integral (from 0->x) f(t) dt, how can i find a formula for F(x) without using integrals? I think it may have something to do with the 2nd definition of the...
Homework Statement
Please help me sketch the graph of the given piecewiswe-defined function and its transformations:
The function f(t)={t+4 ;-4<=t<0
={-3/2(t)+10 ; 0<=t<2
={1/2(t) ; 2<=t<=4
Homework Equations
The transformations of the...
I am NOT trying to solve this as a homework problem, I'm just trying to dig out basic concepts that I'm missing when using this battery+resistance diode model.I'm having trouble seeing why, when solving for current, KCL equation does NOT work out as:
1) I= (0-Va)/rd
2) I=(Va-(-5))/R...
[SOLVED] Continuity on a piece-wise function
Problem:
Suppose:
f(x)=\left\{\begin{array}{cc}x^2, &
x\in\mathbb{Q} \\ -x^2, & x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right
At what points is f continuous?
Relevant Questions:
This is in a classical analysis course, not a...
I am doing my calculus homework and two problems are holding me up.
The first says:
Using one-sided derivatives, show that the function f(x) =
x^3, x_<_1
3x, x>1
does not have a derivative at x=1
Now it is painfully obvious that the function is not continuous at x=1. however, i...
I'm doing a review of fuctions, and a nagging question popped up in my mind after completing this problem.
After graphing y = |x| + x, express this equation as a piece-wise function with no absolute values.
I did graph it; it was simple (following is a sketch without values)...
Let D[a,b] be the set of piece-wise continuous functions on the close interval [a,b]. Show that D[a,b] is a subspace of the vector space P[a,b] of all functions defined on the interval [a,b].
Can someone get me started? Do I just need to show that they are closed under addition and...