7_ (an) (a) (5 marks) Using the substitution ;1'.' = tan(t), show that 1 f (11: = tan'1 3: + (2'. 1 + .13 (h) (7 marks) Using a suitable substitution show that a J: j , , d1? = tanl () + C, a2 + 2 a (c) (8 marks) The number (in thousands), N , of tadpoles in a pond after t days may be approximated by the solution to the equation where a is some constant. dN _ 20 E 17000 + t2)' Given that there are initially 0 tadpoles in the pond, nd N as a function of it. What happens to N in the limit t > ac? \\Vhat may we conclude about the long-time population of tadpoles