4. Newton’s method [23 marks]
a) Give a description of the Newton’s method to solve an equation.

b) i. Using curve transformation, sketch the curves
?? = ????+1 – 1 and ?? = 2??????( ?? – 2) for -10 < ?? < 2.
Explicitly describe the transformations used.
ii. Using your graph, give an approximate value of the root of
????+1 – 1 = 2 sin(?? – 2)
in the interval [-2,0].
c) Use Newton’s method to find the root of ????+1 – 1 = 2 sin(?? – 2), which is in the interval [-2,0], correct to 4 decimal places.

d) Explain why it’s impossible to implement Newton’s method to find a solution of . Illustrate your explanation with a sketch.
[23 marks]

5. Exponential decay [12 marks]
A common inhabitant of alien intestines is the bacterium GRE. A cell of this bacterium in special condition divides into 2 cells every (20 + ??) minutes. The initial population of the culture is (60 + ??) (where ?? and b are the 2 last digits of your student id.). Let ??(??) = ???????? be the population of the bacteria as a function of time ?? in minutes.
a) Find ?? and ??, give their exact values and an approximation, correct to 3 decimal places.
b) Find the value of the derivative of ?? two hours after the initial population is taken, correct to 3 decimal places.
c) Find the number of cells after 2 hours, give the exact value and an approximation, correct to 3 decimal places.
d) When will the population reach 20,000 cells?