Vanderbilt University Department of Mathematics Math 1300 Fall 2015 Graded Homework 3 Due Monday, November 2, 2015 This is an open-note, open-book, take home assignment. You may work with other students in my class but you must write up your own work, on a separate sheet of paper to hand in to me. If you do work with other student(s) in my class, please list their name(s) at the bottom of that separate sheet of paper. What you turn in should include all your work on the assignment and should be done in a clear, logical and neat manner. No assignment will be accepted after the beginning of class Monday, November 2, 2015. Please see me if you have any questions. As usual, show all your work for full credit! (4 points) Imagine that someone has fallen out of a boat, and they are screaming for help in the water at point B . The line marked s is the shoreline. We are at point A on land, and we see the accident, and we can run and can also swim. But we can run faster than we can swim. What do we do? Do we go in a straight line? Yes, no doubt! However, by using a little more intelligence we would realize hat it would be advantageous to travel a little greater distance on land in order to decrease the distance in the water, because we go so much slower in the water. Following this line of reasoning out, we would say the right thing to do is to compute very careful what should be done!1 1. v1 s v2 b2 d2 C B x 2 1 ax d1 b1 A The setting of this problem has been taken, with minor modi\u001ccations, from The Feynman Lectures on Physics, see this webpage: http://www.feynmanlectures.caltech.edu/. 1 1 We are going to run, at velocity v1 in a straight line d1 up to a point C in the shoreline, and then we are going to swim from this point to point B , at velocity v2 also in a straight line d2 . We must chose the point C wisely! The distances from A and B to the shore are b1 and b2 , respectively. And measured over the shoreline, the displacement from C to B will be x, and thus from A to C will be a x, for a \u001cxed quantity a. (a) Taking into account that an object moving at velocity v travels a distance d in a certain time t = d/v , \u001cnd a function t(x) depending on x and the parameters a, b1 , and b2 , describing the time needed to go from point A to point B through point C as described above. (b) In order to simplify a little the problem, we are going to assume that we were by the shore at the time we saw the accident, that is to say, b1 = 0. (i) Simplify the function t(x), assuming that b1 = 0. (ii) Find the fastest path, that is, \u001cnd the x for which the time to go from A to B is minimum. (c) Finding the position of point C in the general case, that is, when b1 , b2 , and a are di\u001berent from 0, is more complicated. Instead of doing that, we will do something simpler, we are going to prove that the following relation between the angles 1 and 2 and the velocities v1 and v2 has to be satis\u001ced: v1 sin 1 = . (1) v2 sin 2 (i) Prove that, in the general case, the function t reaches its minimum at x if and only if (a x)v2 d2 = xv1 d1 . (ii) Prove equation (1). 2. (3 points) Consider the function f (x) = 1 1 + . 1 + |x| 1 + |x 2| (a) Remove the absolute values on f by rewriting it piece-wise. (b) Find the intervals where the function is increasing and decreasing. (c) Find the maximal value of f (x). (5 points) The objective of this exercise is understand the graph of any polynomial of degree 3, 1 p(x) = a3 x3 +a2 x2 +a1 x+a0 . Since we know how to obtain the graph of p(x) from the graph p(x) a3 (stretching/expanding with respect to the y -axis by a factor of 1/|a3 | and re\u001decting over x-axis if a3 < 0), we can assume that a3 = 1. We are going to restrict to the situation f (x) = x3 +ax2 +bx+c. 3. (a) In which intervals is f concave up? And concave down? (b) Show that f must have exactly one in\u001dection point and \u001cnd it. (c) When does the function have no critical numbers? And one? And two? Can it have more than two? (d) Prove that if f has only one critical number c, then (c, f (c)) is an in\u001dection point. (e) If the in\u001dection point is (0, 2) and f has one critical number, \u001cnd a, b and c. (f) Sketch the function for the above values of a, b and c. 2 4. (3 points) Compute (a) lim x x2 + 3x + 1 x (b) lim x x cos(1/x) x (c) lim n n i=1 i(3i 2) i2 (3i 2) + n3 n4 3 4