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Browse High School Analysis

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Divide by 0 Undefined? [9/10/1996]
When something is divided by 0, why is the answer undefined?

Infinity over Infinity [4/10/1996]
My teacher says that infinity divided by infinity equals one. I say that infinity over infinity is indeterminate...

Infinity, Zero [1/4/1995]
You can't divide by zero, but no one can actually prove WHY. . . I wanted to see a real proof first. . . We learned in trig that you can't raise zero to the zero-th power because zero would equal one, obviously. I realize infinity is not so much a number as an endless amount, but if there are an infinite number of numbers between 1 and 2, and an infinite number of numbers between 1 and 50, wouldn't the second infinity be bigger than the first?

Line Segments and Size of Infinites [03/19/1997]
Divide a line segment into three parts, one half and one a quarter the length of the line segment. Choose a point at random along this line segment. What is that probability that this point lands in the 1/2 segment...?

Lines, Points, and Infinities [09/01/2001]
What is the cardinality of the set of real numbers between 0 and 1? Is this cardinality less than, greater than, or equal to the cardinality of real numbers between 0 and 2?

Patterns in Linear Systems [11/12/2001]
Given the equation 2x+5y=8, with the coefficients increasing by 3 each time, create linear systems that are similar using equations such as 3x+4y=5 or 4x+6y=8.

Point nine repeating equals one? [12/20/1994]
When I learned how to convert repeating decimals to fractions, we were given an example in which .9 repeating equalled one. The problem I have is that I can't logically believe this is true, and I don't see an error with the math, so what am I missing or forgetting to resolve this?

Proof of L'Hopital's Rule [12/23/1998]
Can you show me a proof of L'Hopital's Rule and say how it relates to the different versions of the Rule?

Rational and Irrational Numbers [11/12/1997]
Which set is bigger, the set of rational or irrational numbers?

Rational Numbers [11/24/1997]
Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?

0 Raised to a Power [04/25/2002]
What is the limit of 0^n as n approaches infinity? Can you explain why it equals zero or why it can't equal zero, or if it's undefined?

1 + 2 + 3 + 4 + ... Equals ... -1/12?! [09/18/2012]
Doctor Vogler explains how functions extended by analytic continuation can be evaluated to produce counterintuitive results.

Alternating Harmonic Series [11/18/1997]
I am trying to find the proof for the sum of the alternating harmonic series. I did find out that it is ln(2), but please tell me why?

Analysis and the Derivative [01/23/2001]
Suppose that f:RtoR is differentiable at c and that f(c)=0. Show that g(x):=|f(x)| is differentiable at c if and only if f'(c)=0.

Chaotic Functions [10/30/2000]
Can you give some mathematical examples of chaos theory?

Closure and Compactness in a Metric Space [10/08/2002]
Regard Q, the set of all rational numbers, as a metric space, with d(p,q)=|p-q|... Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q?

Closure and the Reals [03/26/1998]
Under what set of operations are the positive real integers closed?

Compact Sets and Hausdorff Spaces [03/19/2003]
How do you prove that every compact subset of a metric space is closed?

Complex Numbers in Quadratic Equations [11/09/1999]
How are imaginary numbers used in solving quadratic equations? How can solutions of this type be represented graphically?

Continuity [11/24/2001]
Suppose f and g are continuous on I = [a,b]... Prove there exists a k greater than 0 such that f(x) + k is less than or equal to g(x) for all x in I.

Convergence of Sums [05/07/1999]
Deduce that the following sum converges absolutely: Sum from 1 to infinity of (-1)^(n-1)/(n^2)!

Convergent and Divergent Series [07/14/1998]
I cannot seem to solve these problems...

Countability [09/09/1998]
How can I show that the rationals are countable and the irrationals are uncountable?

Creating a Smooth Acceleration in Animation [05/09/2001]
I am trying to figure out a way to set the value of X to achieve a specific difference between the last two sets in this problem...

Definition of the Limit [11/05/2002]
I need some help with the definition of the limit, particularly choosing delta for a given epsilon.

Delta-Epsilon Limit Proofs [09/05/1998]
Can you help me with some delta-epsilon proofs? How do you choose delta?

Dense and Nowhere Dense Sets [04/25/1999]
Can you define the mathematical terms 'dense' and 'nowhere dense'?

Derivation of the Power Series of Cosine [1/22/1996]
Would you please explain how the power series is derived from the cosine function and any other function?

Deriving the Gamma Function [12/15/2000]
How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional factorial like that equal to?

Descartes' Rule of Signs and Complex Roots [08/04/1998]
Prove that if p and q are real and q is not equal to 0, the equation x^3 + px +q = 0 has two imaginary roots.

Epsilon - Delta Proofs [1/23/1996]
I'm confused about the f(x) in the epsilon function and the x in the delta function...

Equations with Infinity [07/29/1998]
How can you use infinity in maths equations? Is it like other numbers?

Equations with Rational Expressions in Two Variables [12/07/2002]
Determine all positive integers a and b that satisfy the equation: 1/a + a/b + 1/ab = 1.

Eternity [12/30/1997]
If you must choose from two alternatives, eternity or half of eternity, which is the best?

Euler Formula: e^(pi*i) = -1 [6/5/1996]
Why does e^(pi*i) = -1?

Explaining the Intermediate Value Theorem [02/18/2004]
I want to know how to explain the Intermediate Value Theorem in its most simple form: If f is a continuous function on the closed interval [a,b] and N is any number between f(a) and f(b), there must be a number c in (a,b) such that f(c)= N.

Explain Supremum [02/02/1998]
Can you please explain, perhaps with an example, the concept of "supremum"?

Factoring Quadratics [03/05/1998]
Factoring quadratics that occur in fractions.

Fibonacci sequence [1/28/1996]
What is the explicit formula for the Fibonacci numbers?

Fibonacci Sequence - An Example [05/12/1999]
Glass plates and reflections.

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