10. At time t (in days), the size S of a population of butterflies is given by the formula 600 S = 1+49(0.6)' Use the graph below and your knowledge of derivatives to answer the following: (6 marks) a) What is the growth rate after 7 days? b) When does the maximum population occur? c) Explain why the shape of the graph looks this way (ie. what is happening to this population of butterflies).11. Carbon-14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a half-life (the time it takes for half the amount of a sample to decay) of approximately 5730 years. Using this known piece of information, scientists can date objects such as the Dead Sea Scrolls. (5 marks) The function N(t) = Noe-At represents the exponential decay of a radioactive substance. / is the amount remaining after time t in years, No is the initial amount of the substance and A is the decay constant. (1 = 1.21 x 10*) a) If the percentage of carbon-14 atoms remaining in a sample is 75%, how old is the sample? b) Find the rate of change of an initial amount of 1 gm of Carbon-14 found in the scrolls, if the decay constant is given as 1 = 1.21 x 104.Provide complete solutions for each of the following questions. 9. Find the derivative of each of the following functions. (18 marks) a) y = 4x5 - 3x2 + - 4x b) f(x) = 5e-5x c) g(x) = Vx5 d) h(x) = COS X x2 +1 e) s(x) = In(3x+2) 3x+2 f) t(x) = (3x2 + 1)(2x3 - x2 + 3) g) y = (2x2 + 3x + 1)-4 h) y = cos3x sinx12. The volume of water, V litres, in a tank t minutes after the tank has started to drain is given by the formula- V(t) = 1000(25 - t2) , Osts5 How fast is the water draining after 2 minutes ? (3 marks) 13. An object starts at t = 0 and moves in a straight line. It's distance, d in metres, from it's starting point at time t seconds is given by the equation - 4t d(t) = 1+ t2 How far is the object from the starting point when it's velocity is 0 m/s. (4 marks)