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

𝑦 = 𝑒𝑥+1 − 1 and 𝑦 = 2𝑠𝑖𝑛( 𝑥 − 2) for -10 < 𝑥 < 2.

Explicitly describe the transformations used.

ii. Using your graph, give an approximate value of the root of

𝑒𝑥+1 − 1 = 2 sin(𝑥 − 2)

in the interval [-2,0].

c) Use Newton’s method to find the root of 𝑒𝑥+1 − 1 = 2 sin(𝑥 − 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 + 𝑏) minutes. The initial population of the culture is (60

+ 𝑎) (where 𝑎 and b are the 2 last digits of your student id.). Let 𝑦(𝑡) = 𝛼𝑒𝛽𝑡 be the population of the bacteria as a function of time 𝑡 in minutes.

a) Find 𝛼 and 𝛽, give their exact values and an approximation, correct to 3 decimal places.

b) Find the value of the derivative of 𝑦 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?