Homework Statement
Find all tangent lines of the graph f(x)=x+3/x that have a y intercept of 4.
Homework Equations
The Attempt at a Solution
Assume a is the x coordinate of a point of tangency. Thus the point of tangency is (a, a+3/a). We know the tangent line must pass...
Bah. Never mind, I was looking at something wrong.
So would the set of all n=infinity locations be the whole argand plane? It seems that varying values the angle theta and the step function could eventually allow the sum of values to "converge" to any point on the plane.
Thank you guys, that takes care of the angle problem.
So would the general nth position be Zn=1*e^(i0)+f*e^(i*theta)+...+f^n*e^(n*i*theta)? Somehow I doubt this is correct - shouldn't the radius continually be decreasing?
Oops - I meant to the *horizontal*. After the initial step theta...
Homework Statement
An man starts on 0+0i of the complex plane. During the first walk step he moves a distance 1 to the right and lands on 1+0i. At each walk phase after this initial one the walk length decreases by a factor f<1 and he changes direction by an angle theta (measured from the...
I may be wrong about the holder condition, but it looks to me like f(x) is holder continuous as long as the exponent in the condition is equal to or less than 1/2.
Homework Statement
Given each of the functions f below, describe the set of points at which the Fourier
series converges to f.
b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x)
Homework Equations
Theorem: If f(x) is absolutely integrable, then its fourier series converges to f...
Homework Statement
Let (a,b)=XUY, X,Y arbitrary sets where (a,b) is an arbitrary interval. Prove that either X or Y has the same cardinality as that of (a,b).
Homework Equations
The Attempt at a Solution
Really lost.
Homework Statement
I reduced another problem to the following problem:
If I is an interval and A is a subset of I, then A is either an interval, a set of discreet points, a union of the two.
Homework Equations
The Attempt at a Solution
Is this trivial?
I suppose by z as the right endpoint if z is greater than or equal to all y in I and likewise for x. So if we define x to be the smallest element in I and z to be the largest, by the betweenness property we have that I contains every point in between x and z, i.e. for any x<y<z, y is in I...
Ah forgot about the other direction. Let some subset of R be I such that I contains each point between any two of its points. Thus if x,z are in I and x<y<z, then y is in I. Let x,z be the endpoints of I. Thus by the betweenness property any y such that x<y<z implies y is in I. So I is an...
Homework Statement
a) Let I be a subset of the real line. Prove I is an interval if and only if it contains each point between any two of its points.
b) Let Ia be a collection of intervals on the real line such that the intersection of the collection is nonempty. Show the union of the...
Homework Statement
Prove that a(s) is a straight line if and only if its tangent lines are all parallel.
Homework Equations
Frenet serret theorem
The Attempt at a Solution
I'm confused on the direction "if the tangent lines are parallel then a(s) is a straight line".
Assume all the...
so how would I start the proof that the diameter is equal to the diagonal of the rectangle formed by the k-cell? it seems so simple but im having trouble seeing where to begin
Generally in this problem set if not specified, you couldn't assume a particular metric. But in any case I think its safe to assume its euclidean. Otherwise the diameter would just be infinite.
Ah thanks. Thats precisely what I confused on. I thought diam(K) meant the sup of all possible distance functions so it would have to be infinite, which made no sense.
so I can take it that the question means the euclidean metric?
I think my main concern is how the euclidean metric yields the largest distance when there are an infinite amount of other possible metrics. How would that account for it?
Yes, thats what I mean.
Isn't it true that they are as far apart as possible only in terms of the euclidean distance metric? In finding the diameter, aren't I supposed to find the least upper bound for all possible distance metrics on all possible points in this 2 cell, not just the euclidean...
Ah sorry. A 2-cell is a subset of R^2 of the form [a,b] x [c,d], or {(x,y) in R^2: a <= x <= b and c <= y <= d}.
By diameter, since no distance function is specified, how do we know we can't define one that will give a value larger than the straight line path between the two farthest points in...
Homework Statement
Define diam(X) of a set X, X a nonempty subset of arbitrary metric space, as diamX=supremum{distancefunction(x,y):x,y in X}.
Let K be the two cell formed by two points a=(2,3) and b=(4,6) in the xy plane. Find diameter of K and show this is the diameter.
Homework...