Thread: Using numerical methods to solve problems

Question One.
Use an appropriate method to find the root of x³ + x + 1 = 0 correct to 2 decimal places. Start with initial interval -0.8 and -0.6. Show all working.

Question Two.
Human blood pressure P (in mm of mercury) varies periodically with time (t in seconds). The first time during the first second of a cycle when the pressure P is 90 mm can be expressed by:
80 + 300t - 2700t³ = 90
or:
f(t) = 2700t³ - 300t + 10 = 0

Take an initial value of t = 0.05 seconds and use a method different to Question One to find a solution to f(t) = 0 accurate to 4 decimal places.

Originally Posted by Kiwigirl

Question One.
Use an appropriate method to find the root of x³ + x + 1 = 0 correct to 2 decimal places. Start with initial interval -0.8 and -0.6. Show all working.

Let:

.

We wish to find a zero of in the interval . We know it has a zero
in this interval as it changes sign between and .


Here we are going to use binary chop, In the following tableau we have
in each row a lower limit of the interval containing the zeros, the
value of the function at the lower limit, the upper limit of the interval
containing the zeros, the value of the function at the upper limit,
the mid-point of the interval and the value of the function at the mid=point.

To obtain the following row from a row the value of the upper or lower limit
of the interval is replaced by the mid-point according to the sign of the
function at the mid-point. This process is continued until the upper and
lower limits of the interval are equal to two significant digits.

So the zero we seek is .

RonL

Originally Posted by Kiwigirl

Question Two.
Human blood pressure P (in mm of mercury) varies periodically with time (t in seconds). The first time during the first second of a cycle when the pressure P is 90 mm can be expressed by:
80 + 300t - 2700t³ = 90
or:
f(t) = 2700t³ - 300t + 10 = 0

Take an initial value of t = 0.05 seconds and use a method different to Question One to find a solution to f(t) = 0 accurate to 4 decimal places.

Here we will use Newton-Raphson itteration. This generates a new estimate
for the root from an old estimate using the relation:

So to four significant digits is the required root.

RonL