PART I: CALCULUS PART A: KNOWLEDGE 1. a) Differentiate and factor fully if applicable. No negative exponents in solution. 3 y = 3'26 (1 mark) b) = % (3 marks) Evaluate the following limits. x +4 x g 2 . Given \"'0 = {-2306 x > 2 . find l''zx). (2 marks} \"ml"'111 (3 marks) For the curve with equation: xzl r 3_x2 1! 2(X26) Y = x3 Y 2 x' Y = T find the value of the intercepts. (1 mark) find the equations of any vertical or horizontal asymptotes. died: behowor dare to re Wrote; (4 marks) find all critical points and test them. Dreamer: any Mimi asymptote: in your amber\": (4 marks) find the points of inflection. Marmara any mm; wrote: in your mom (3 marks) sketch the graph being sure to label all aim} int: (1 mark) -A PART 8: APPLICATION 1. b) Determine the equation of the tangent to the g'aph f(x) 2 ea" sin(2x) atx = 0 . (4 marks) The curves with equations y : 33nd y = axz +2: +1: have the following properties: a common point where x = 2. a common tangent line where x = 2. both curves pass Through The point (1,4) Find The values of a, b and c. (4 marks) A drug is injected into the body in such a way that the concentration, C, in the blood at time t hours is given by the function C(t) =10(B_ZT 3'\"). At what time does the highest concentration occur within the first 5 hours? (4 marks) The position of an object moving along a straight line is given by s(T)=6T2 T3, tZO , where s is measured in metres and t is in seconds. Find The objects acceleration at T = 2 seconds. (2 marks) AT what Time(s) is The object at rest. (2 marks) When is The object moving in a positive direction? (2 marks) When is The object slowing down? (2 marks) What is the velocity of The object when it returns To its initial position? (2 marks) 5t+8 3t + 2 measwed in thousands and t is the time in years since 1950. The population of Narnia is modelled by P(t) = , where, the population is Determine the rate of change of The population in 1987. What does the value of This rate of change Tell you about what is happening in Narnia? (3 marks)\\ Is the population of Narnia ever expected to start increasing? If so when. Explain. (2 marks) A cylindrical storage tank with a capacity of 1000 m3 is to be constructed. The specifications call for the top and base to be made of sheet steel, which costs $10Cl/mz , and the wall to be made of sheet steel of a lower grade. which costs :liBO'iz . Determine the dimensions of the tank that would minimize the cost of construction. {V = 1'th SA = 2m: + anh)