Math 1A Worksheet 25 November 7th 2007 1 Story problem time Sophie Snell a lifeguard is sitting at her lifeguard post 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 the beach nearest to him is 40 meters away from the point in the water nearest to Sophie a Draw a picture to illustrate the above or ask me very nicely to draw 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 where on your picture Sophie should enter the water to reach the swimmer fastest No equations yet c Now let x be the distance between the point on the water closest to Sophie 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 Sophie to 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 you get to find a relationship between the cosines of the angles between the shoreline and each of the two legs of Sophie s path Use this to make 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 function x x 0 f x x x 0 Show that no matter what initial approximation x1 6 0 you pick Newton s method will fail to find the root of f 1 3 Find all functions f x with f 00 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 periodic with period 1 i e f x 1 f x for all x Show that f x is bounded from above and below i e there exist constants m and M such that m f x M for all x and show that f has an absolute maximum and minimum 2
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