1. a. Show that a root of the equation x^{4} − 2x^{3} + 2x − 2 = 0 lies between x = 1 and x = 2.

b. Use the Newton-Rhapson method to find that root to 5 decimal places.

2. a. Show that a root of the equation log_{e}x = lies between 1 and 2.

b. Solve the equation log_{e}x - = 0 to 6 decimal places, using the Newton-Rhapson method and a starting value of x = 2.

3. A study suggests that a bird weighing x kg can maintain flight only if its average heat production is less than 700 − 2x calories. According to this theory the greatest weight , x kg, for a bird capable of flight is given by the equation

68x^{0.75} = 700 − 2x

Calculate the first iterate, x_{1}, of the Newton-Rhapson method to solve this equation for x, starting at x0 = 16.

Set out your working clearly.

4. Consider the function f(x) = 2e^{x} − 2x − 3

a. Differentiate f(x)

b. Starting with the value x_{0} = 1, calculate the first **two** iterates, x_{1} and x_{2},* *of the Newton-Raphson method to solve the equation f(*x*) = 0. Show all details of the calculations.

c. Starting with* *x_{0 }= 0, the Newton-Raphson method fails to solve the equation f(*x*) = 0 because f(0)/f '(0) is undefined and hence cannot x_{1} be evaluated. Explain **geometrically** why the Newton-Raphson method fails to solve the equation f(*x*) = 0 when starting with x_{0} = 0.