1. a. Show that a root of the equation x4 − 2x3 + 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 logex = lies between 1 and 2.
b. Solve the equation logex - = 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
68x0.75 = 700 − 2x
Calculate the first iterate, x1, 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) = 2ex − 2x − 3
a. Differentiate f(x)
b. Starting with the value x0 = 1, calculate the first two iterates, x1 and x2, of the Newton-Raphson method to solve the equation f(x) = 0. Show all details of the calculations.
c. Starting with x0 = 0, the Newton-Raphson method fails to solve the equation f(x) = 0 because f(0)/f '(0) is undefined and hence cannot x1 be evaluated. Explain geometrically why the Newton-Raphson method fails to solve the equation f(x) = 0 when starting with x0 = 0.