Math Answers

Show whether the following functions are uniformly continuous on the given domain.

1. F(x)=x^3 on [-1,1]

2. F(x)= 2x/2x-1 on [1, infinity]

3. F(x)= sinx/x on (0,1)

4. F(x)= 1/x on (0,1)

A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found!

83 purchased a dog product

101 purchased a cat product

22 purchased a fish product

31 purchased a dog and a cat product

8 purchased a dog and a fish product

10 purchased a cat and a fish product

6 purchased a dog, a cat, and a fish product

34 purchased a product for a pet other than a dog, cat, or fish.

Draw a Venn diagram to find that:

(1) How many purchases were for a dog product only?

(ii) How many purchases were for a cat product only?

How many purchases were for a dog or a fish product?

(iv) How many purchases were there in total?

A peanut butter seller sells an average of 250 bottles of 1 kg week with a standard deviation of 42 kg. The sales are normally distributed.

a) If a sample of 16 weeks was selected for a study, find the probability that the sample mean is greater than 300 bottles

b) In one year how many weeks would record sales of less than 150 bottles

A random variable X ~ bino(5, 0.3). g(x) is a continuous and differentiable function on the interval (0, 5).

a. Write down the cumulative distribution function F(x) for the random variable X.

b. g(x) is an approximation of F(x). Construct g(x).

L + x be a nœemal randome variable with mean 4 and variouse dot alpha Find the probability 8li * e_{s}

1) Between 4 and 6

2) Between 5 and 7.5

3) ・2 and 3.

4) greater them d * 75

5) Less than 4.025.

Solve the following Bernoulli's Differential Equations. Show your solutions.

a. dy/dx + (1/3) y = e^x y²

b. x (dy/dx) + y = xy³

c. dy/dx + (2/x) y = -x² cos x y²

d. x²y-x³ (dy/dx) = y² cos x

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c, then a ≤ c. 4. Do you think that a ≤ b and b ≤ a imply

a = b? Explain your reasoning. (Hint: This is not as trivial as it might look.)

Determine whether -104 is a quadratic residue or non residue of the prime 997.

Given a die it has 6 faces in which gas either dot of x = 1, 2,3, 4, 5,6. Consider the sample of size n= 3. Illustrate the sampling distrubution and the central limit theorem.

Find an approximate value of the double integral below where 𝑅 is the rectangular region having

vertices (−1, 1) and (2, 3). Take a partition of 𝑅 formed by the lines 𝑥 = 0, 𝑥 = 1, and 𝑦 = 2, and take (𝑢𝑖

, 𝑣𝑖) at the

center of the 𝑖th sub region.

∬(3𝑦 − 2𝑥

2)𝑑𝐴

𝑅