CALCULUS 1000A FW23 Homework Assignment #1 Due: October 6th at 11:59pm (EDT) on gradescope.ca For full credit, ensure you show all your work and explain your reasoning. Simply stating the nal answer for any of the following questions will not result in full credit. You should avoid using calculators and leave your answers in exact form, even in questions involving words like \"approximately\" 7 you won't increase your grade by showing us the decimal expansion of your results, so don't worry about that extra step. Make sure you read and follow the submission instructions for Gradescope. 1. (6 marks) Forg(:r;) = ln(1 m2) and f(:r;) = \\/m + 3, determine the subset of the domain of g on which the composition f o g is welldened. What is the domain of g o f? Find formulas for (f o g)(m) and (90f)($)- 2. (4 marks) Sketch a graph of f (3')) = % (3:- 2)2 3 by using a sequence of transformations of a wellknown function. For full points, make sure to describe the various transformations applied to the wellknown starting function. 3. (4 marks) Once activated, a population of selfreplicating nanobots is known to triple every four hours. Suppose that there are initially 50 nanobots. (a) (1 mark) What is the size of the population after 12 hours? (b) (2 marks) Use an exponential function C(t) = Coertto represent the population after 75 hours, and solve for Co and \"r. (c) (1 marks) Estimate the size of the population after 33 hours. Your answer should be given in the most exact representation (i.e., not a decimal). 4. (4 marks) Find the exact value of each a: E [0, 27F) for which cos2 (39) = %sin(2m). 5. (3 marks + 1 bonus mark) If gm) = #5:}: (a) (1 mark) What is the domain of 9(27)? (b) (1 mark) Verify (with a sketch and/ or short argument) that g is a onetoone function. ) ) (c (d (1 bonus mark) Find the range of g. (1 mark) Find a formula for the inverse function: that is, nd 9105). 6. (5 marks) Sketch the graph of a function f with the following properties: 0 f is not continuous when a; = 2 but limmng f (3:) exists. 0 f has a jump discontinuity when m = 1. o f is continuous at all points other than where :t; = 2 or x = 1