The function f ( x ) = cos( πx ) for x ∈ [0 , 1] is approximated by the following piecewise polynomial p of degree ≤ 3. The interval [0 , 1] is divided into k subintervals [ a j − 1 ,a j ] with a j = j/k for j = 0 , 1 ,...,k , and in each subinterval p is the polynomial interpolating f at 4 equally spaced points: a j − 1 , a j − 1 + 1 / (3 k ), a j − 1 + 2 / (3 k ), and a j . How should k be chosen so that the error max 0 ≤ x ≤ 1 | f ( x ) − p ( x ) |≤ 10 − 8 ? That is, derive an error upper bound, find the magnitude of | f (4) ( x ) | , and next get the desired value of k .

According to statistics a person will devote 32 years to sleeping and watching tv. the number of years sleeping will exceed the number of years watching tv by 18. over a lifetime, how many years will the person spend on each of these activities

On the way home from visiting his family, vincent’s plane cruised at an altitude of 3.2 × 10^4 feet. he had heard that a man named felix baumgartner skydived from an altitude of 1.28 × 10^5 feet. vincent wants to know how much higher felix went on his skydiving trip. question 1 you have already seen how adding numbers works in either scientific notation or standard notation. but can you subtract numbers in scientific notation and get the same results as subtracting in standard notation? to find out, first solve vincent’s problem in standard notation. part a write the cruising altitude of vincent’s plane in standard notation.

Astrid spent $36 on new shoes. this was $12 less than twice what she spent on a new skirt. part a: which equation would solve for how much she spent on the skirt? part b: solve the equation in the workspace provided below. how much did astrid spend on her skirt?