M AT H 1 2 0 Differential and Integral Calculus Winter 2017 - Problem Set #16 Due Wednesday, February 15. Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your final write-up is clean and clear and effectively communicates your reasoning to the grader. 1) Consider the parabola y = x2 . Find the point on the parabola at which the curvature is the largest. What is the value of the curvature at this point? 2) Consider the parametric curve given by \u001a x (t) = t cos(t) y(t) = t sin(t) 0 t 2. Find all the values of t such that the curvature of the curve at the point ( x (t), y(t)) is 85 . 3) If we think of an electron as a particle at a certain distance X from the centre of an atom, then the probability that the electron is within distance r is 1 (2r2 + 2r + 1)e2r . In other words, X is a random variable such that P(0 X r ) = 1 (2r2 + 2r + 1)e2r . The distance is measured in Bohr radii; 1 Bohr radius = 5.29 1011 m. For instance, P(0 X 1) = 1 5e2 0.32 means that the electron is within 1 Bohr radius form the centre of the atom 32% of the time. (a) Find a formula for the density of the random variable X. Sketch the graph of this density. (b) Find the mean distance. (c) Find an approximate value for the median distance. (A median is defined as a number m such that the probability that X is below m equals the probability that X is above m)