6 Introductory Proofs with Continuity Challenge Points: 14 A. The function / is given by: N(x) = if x > 2 Find a so that f is continuous. b. Assume the functions g, h satisfy g(x)' | h(x) = (asinx],x c R. Prove that: (a) g(0) = h(0) =0 (b) g, A are continuous at r = 0).Project Example (WW3.7 #9) This is a project paper if Webwork 3.7 89 if it wore a project prompt. Group Members: Sparty Spartan, Bucky Badger, and Brutus Buckeye Problem A woman is standing at the edge of a slow-moving river which is one mile wide, and she wishes to return to her campground on the opposite side of the river. Assume that the woman can walk at miles per hour and swim at 3 miles per hour, and that she will first swim to cross the river and then walk the remaining distance to the campground, which is 4 miles downstream from the point directly Ecross the river from the woman's starting point. What route will take the least amount of time? finish im start Mathematical Framing This Is an optimization problem that Involves finding the minimum time to complete the trip. According to the diagram above, we have the following variables: * is the distance between the point directly across the river from where the woman starts and the point she calls the river and starts walking. The domain off x is |04] because x-0 would represent swimming backward and we4 would represent swimming past the campground and chen walking backwards. & is the distance that the woman swims (at 3 miles per hour) "wis the distance that the woman walks [at 5 miles per hour) See the diagram for the following: The woman's starting location is marked with a red "%%, her ending location is marked with a yellow star, and one of the possible routes for her is marked with a red solid line. The variable s denotes the distance (In mij that she will swim, and the variable ww denotes the distance (in mi] that she will walk, Le. the distance between the red and blue arrows. Let joc denote the distance (in mij between the black and red arrows.Solution To solve this problem, we must do the optimization process from section 37, which involves calculating 3.1). a derivative [sections 2.3-2.5) to find the critical values and determine the absolute minimum fraction First we need to find the objective function. Then, we need to reduce the objective function to a single variable. We'll use the restrictions on the problem to write equations that relate the variables and substitute. W = 4- X 5 + Next, we need to find the absolute maximum of this function over the interval s in 10,4)- + LB) = 5 t(o)= #+$=1.1733 L(4) = 1. 06 66 7 *()=0. 4 =1.374 9/1+ * 7) = 257 2 ABSOLUTE MIN IS 1 0667 hr A IT accues ATFinally, weneed to mrtain our calculation in terms of the problem. In this case, that involves explaining how the x-value we found translates into "what route will take the least amount of time?" - 51- = 1.25 START The route that the woman should take to minimize her travel time is swimming to a point 3/4mi downstream from where the started, then she will have 3.25mi to walk to the campground. This route will take her 1 06657 hours to complete