3. [5 marks] Find all the values of m such that 4. [5 marks] Let f(x) = >. Use a definition of the derivative to find f'(x). No credit will be given for solutions using differentiation rules, but you can use those to check your answer. f (x ) = 623 - 2m ifx 5 -1 2x2 + 5m if x> -1 is continuous. Answer: Answer: Student number: 6 Student number: 75. [5 marks] Suppose the function f(x) is defined and continuous on all real numbers, and 6. [5 marks] Pictured below is the graph of a function. On the blank set of axes, sketch that the graph of its derivative. lim f( 5+ h) - 3 h :-5. y Given this information, we can find the equation of the tangent line to the curve y = f(x) at a particular point. What is the point, and what is the slope of the tangent line to the curve y = f(x) at that point? Answer: Student number: 8 Student number:11. For any positive integer n, the Hassell model of exponent n (c) [3 marks] Let n, and n; be distinct positive integers. Find all points of intersection RI between the graphs of the Hassell models of exponent m and n2. f(z)=( M>U,R>1 HM)\" describes the size f(I) of a population given the size I 2 0 of the population in the previr ous generation. (The special case 1L = 1 gives the BcvcrtonrHolt model from Assignment 1.) (a) [3 marks] Let n 2 2. Solve the equation f(z) : 1'. (d) [3 marks] Pictured below is a graph of the Beverton-Holt model On the same set of axes, sketch the graph of a Hassell model of exponent n 2 2. (You may assume that the parameters M and R are the same in both models.) (b) [2 marks] Let n 2 2. Find the horizontal asymptote of f( ) y Student number: Student number: (e) [3 marks] The exponent n is sometimes interpreted to indicate the degree to which 12. The cumulative exponential probability distribution describes the probability P(t) that a scarce resources are distributed equally in a population. Based on your answers to particular radioactive atom will decay within t minutes. It is given by previous parts of this question, do you think an exponent of n 2 2 corresponds to more equal or less equal distribution of scarce resources, compared to an exponent P(t) = 1-e-dt of n = 1 (i.e. the Beverton-Holt model)? Justify your answer in a few sentences. where A is a positive constant. The domain is restricted to t 2 0. (a) [2 marks] Find the horizontal asymptote of P(t) Answer: (b) [2 marks] Find the intercepts of P(t). Answer: Student number: 16 Student number: 17(c) [3 marks] Draw a large sketch of the graph of P(t) on the axes below, using the information determined in the previous parts of this question. y (d) [3 marks] The half-life of a radioactive isotope is the time taken for a sample of the isotope to decay to half of its original mass. It is equal to the time by which the probability of decay of an individual atom is :7. What is the halfrlife of the isotope whose atoms have the cumulative exponential probability distribution described above? Student number: 13. Let f(:c) I . (a) [3 marks] Find the equation of the tangent line to f(z) at the point (10,13). Your answer should be in the form y = mx + b, and may include the quantity In. (b) [3 marks] Imagine you were given a numerical value for 1'). Explain in 173 sentences how you would determine whether or not the tangent line described in part (a) passes through the point (2, 3). (c) [1 mark] Write down the equations of all tangent lines to f (I) that pass through the point (0, 0). Your answer(s) should be in the form 1/ : ms + 12. Your answer(s) do not have to be justified. Answer: Student number: (d) [3 marks] Let q > 0. How many tangent lines to f (x) pass through the point (0, q)? 14. Hyperbolic functions are analogues to trigonometric functions, defined with respect to (Remember to justify your answer.) the unit hyperbola instead of the unit circle. They are particularly useful in the study of differential equations. They can also be defined in terms of exponential functions. One example is the hyperbolic cotangent function coth(x) = - tex ex - e-z' defined on the restricted domain x # 0. (a) [3 marks] Find all horizontal asymptotes of coth(x). Answer: (e) [4 marks] Let q > 0. Find the equations of all tangent lines to f(x) that pass through the point (0, -q). Your answer(s) should be in the form y = mx + b. Answer: (b) [2 marks] Find all vertical asymptotes of coth(x). Answer: Answer: Student number: 20 Student number: 21(c) [4 marks] Determine where coth(x) is increasing and where it is decreasing. (d) [3 marks] Draw at large sketch of the graph of coth(z) on the axes below, using the information determined in the previous parts of this question Student number: Student number