Real Analysis Problem

The inverse cosine function has domain [-1,1]and range [0, pi]. Prove that (cos^-1)'(x) = -1/ sqrt(1-x^2). This needs to be proved from a real analysis point of view not a calculus.

Real Analysis: Proof by induction that a polynomial of degree n>0 has at most n roots.

A polynomial of degree n>0 has at most n roots. (A root of a function is a point at which the function has value 0.) I need a proof by induction to show this.

MacLaurin Series And Laplace Transforms : Absolute Convergence

Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f ...continues

Combinations : Problem Solving With Dice

Please see the attached file for the fully formatted problems. Question1: How many dots at the outer sides of the dice. Given view is from the top of these four dice? Question2: In the newly formed shaped below do you think answer is the same or different from above? What does your intuition say without investigating? Q ...continues

Real Anaylsis - Mean Value Theorem

Define f : [ -2,2]---> R by f(x) = x^3 - 3x + 3. Find all numbers p in [-2,2] that satisfy the conclusion of the Mean Value Theorem. How do we know we have them all?

Probability: Variance, Mean and Standard Deviation

1) Suppose we have an aisle with storage racks on both sides of the aisle. The aisle is 100 feet long. A worker is stationed at one end of the aisle. The worker needs to retrieve an item from storage. Assume that the items are divided into two groups: high turnover and low turnover. The high turnover items are stored in the loc ...continues

Probability: Joint Probability Mass Function, Covariance and Variance

Let X and Y have joint probability mass function Pr{X = i, Y = j}= c(i + 1)(j + 2) for i >= 0, j >= 0, and i + j < 4. Determine a) the marginal probability mass function of X b) the probability mass function of Y c) the conditional probability mass function of X given Y = 0 d) the probability mass function of Z = X + Y ...continues

Real Analysis Functions with Equal Derivatives

Let f be a function given by x + 2 if x < 0 f(x) = x if x >= 0 Is there a function g: R ---> R such that g'=f? *be careful applying definition of the derivative

Finding the Area of a Circle using the Unit Circle

Please see the attached file for the fully formatted problems. Given square is 11*11, unit square is 1*1 without using any algebraic equation by using scissors or folding it, we can find the area of 11*11 square in terms of unit square. With a similar idea we can do triangles (Main triangle and unit triangle). ...continues

Serial Problem

At Very Long Hotel in Florida, there are n rooms located along a very long corridor and numbered consecutively from 1 to n. One night after a party, n people, who have been likewise numbered from 1 to n, arrived at this hotel and proceeded as follows: Guest 1 opened all the doors. Then Guest 2 closed every second door beginnin ...continues