Derivatives, Integration

    1. [15 marks] The size of a population, P of toads t years after it is introduced into a wetland is given by P(t) = 1000 1 + A(1/2)t where A is a constant. (a) Find the value of A if at time t = 0, twenty (20) toads were introduced into the wetland ? (Assume that there were no toads present in the wetland prior to their introduction into the wetland). (b) How many toads are present at t = 5 and t = 10? (Give the answer after rounding off to the appropriate integer ). (c) How long does it take the toad population to reach 500? (d) As time increases without bound, does the toad population increase without bound or does it reach a point of saturation. Support your answer with numerical proof. 2. [3 marks] Differentiate the following y = e x ln x 3. [4 marks] Find the rate of change of f(x) = e 3x 1 + e x at x = 1. 1 4. [16 marks] Evaluate the following integrals (a) Z 2 5 x 4 − x 2 + 2x + 1 dx (b) Z 2x 3 − x 2 x 3 4 dx (c) Z 1 0 √ t √ t + 2t 2 dt (d) Z 2 0 (1 − z)(z − 1) dz. 5. [ 12 marks] Find the equation of the tangent to the curve y = (x 2 + 4)√ 3x + 10 at x = 2. 6. [10 marks] Find (a) d dz " −1 (z + 3z 2) 1 2 # (b) Hence evaluate Z 2 1 1 + 6z 2(z + 3z 2) 3/2 dx