5. (7 marks) Suppose HI) = 2a$3 l bgr 40 where (Lb and c are real constants. Suppose that f has a horizontal tangent line at :0 : 2, and that the tangent line to f at 12 : 0 is given by y : 8$ + 4. Use the given information to find all possible values for (Lb. and 0. Explain your reasoning. 3. (7 marks) Let f(x) = =2 x and let g be the function defined by (2 - 3) (2 - 5) if a 3 (a) Evaluate lim (f o g) (ac). (b) Is fog continuous at a = 3? If not, does fog have a removable discontinuity, a jump discontinuity and / or an infinite discontinuity at x = 3?6. (7 marks) (a) Let f be defined on [3.45] with f(3) = .30 and f(45) = 2. If f is continuous on the interval (3,45), is it true that there must be some number 0 6 (3,45) such that f(c) : 10? If you believe it is true, justify your reasoning. If you believe it is false, show it with an example. (b) Let g be some function such that 91(2) : 1 and 11m 9 : 6. Find the value of hm q(1:) Please . ' .540 h. T>2L justify all your reasoning. | At birth Now | | Height 20 inches 69 inches | Weight under 9 pounds over 180 pounds Fully justify your answers to the questions below, using facts and theorems from the course. Note: Ivan has had a quiet and simple life and nothing sudden or unexpected has happened in terms of his height or weight. (a) Show that at least once during his life, lvan's weight was 50 pounds. (b) Show that at least once during his life, lvan's height in inches equalled his weight in pounds. 1 \\/3:+1 Be sure to justify your computations. Specifically, if at any point you evaluate a limit by inserting a value. explain WHY it is possible to do that. 2. (6 marks) Find the derivative of f[:c) : at :3 : 0 using the definition of the derivative as a limit. 1. (7 marks) Let f be a function that is defined and differentiable on all of R. Here is some information about f and its derivative f': 2 3 4 f (x 8 3 - 3 V2 0 f' (ac) 1 7 4 -6 2 (a) Suppose h(x) = (23 - 4x-1+ 5) . f (ac). Find h'(2). (b) Suppose R(x) = , _572 . Find R'(-1). f (ac)