12. prove that sin x and cos x are orthogonal functions in the vector space co, of exercise 10, with the inner product defined there
10. let cab be the vector space of all continuous real-valued functions with domain a s xs b prove that defined in cj by (f, g) xg(x) dx, is an inner product in c

Monica build a remote control toy airplane for science project to test the plane she launched it from the top of a building the plane travel a horizontal distance of 50 feet before landing on the ground a quadratic function which models the height of the plane in feet relative to the ground at a horizontal distance of x feet from the building is shown

An annual marathon covers a route that has a distance of approximately 26 miles. winning times for this marathon are all over 2 hours. the following data are the minutes over 2 hours for the winning male runners over two periods of 20 years each. earlier period 14 12 15 22 13 10 19 13 9 14 20 18 16 20 23 12 18 17 6 13 recent period 7 11 7 14 8 9 11 14 8 7 9 8 7 9 9 9 9 8 10 8 (a) make a stem-and-leaf display for the minutes over 2 hours of the winning times for the earlier period. use two lines per stem. (use the tens digit as the stem and the ones digit as the leaf. enter none in any unused answer blanks. for more details, view how to split a stem.) minutes beyond 2 hours earlier period 0 1 2 (b) make a stem-and-leaf display for the minutes over 2 hours of the winning times for the recent period. use two lines per stem. (use the tens digit as the stem and the ones digit as the leaf. enter none in any unused answer blanks.) minutes beyond 2 hours recent period (c) compare the two distributions. how many times under 15 minutes are in each distribution

Use sigma notation to represent the following series for 9 terms.