Math 1A Worksheet 25November 7th, 20071. Story problem time! Sophie Snell, a lifeguard, is sitting at her lifeguardpost, 20 meters from the water’s edge. She spots a swimmer in trouble;he’s floundering 20 meters out into the water, and the point on thebeach nearest to him is 40 meters away from the point in the waternearest to Sophie.a) Draw a picture to illustrate the above, or ask me very nicely todraw a picture.b) On land, Sophie can run at 6 meters per second, while in the water,she moves at only 2 meters per second. Given this, discuss whereon your picture Sophie should enter the water to reach the swimmerfastest. No equations yet!c) Now, let x be the distance between the point on the water closest toSophie’s lifeguard post and the point where Sophie enters the water.Add x to your picture. Set up an equation for the time it takes Sophieto reach the swimmer in terms of x. Don’t solve this yet!d) Take a derivative of the equation you’ve found and set it equal to 0.Note that there will only be one solution. Now, use the equation youget to find a relationship between the cosines of the angles betweenthe shoreline and each of the two legs of Sophie’s path. Use this tomake a rough sketch of the shortest path.e) Now solve for x to an accuracy of 1 meter (by method of your choice;calculator allowed).2. Consider the functionf(x) =√x, x ≥ 0−√−x x < 0.Show that no matter what initial approximation x16= 0 you pick,Newton’s method will fail to find the root of f .13. Find all functions f(x) with f00(x) = ex.4. (Berkeley Prelim Spring 2001) Suppose f(x) is a continuous function,defined for all real numbers, and suppose moreover that f is perio dicwith period 1, i.e. f (x+1) = f(x) for all x. Show that f (x) is boundedfrom above and below (i.e. there exist constants m and M such thatm ≤ f(x) ≤ M for all x), and show that f has an absolute maximumand
View Full Document
Unlocking...