Question

Use the Intermediate value theorem to find an interval of length one that contains a root of the equation. a) x^3=9 b) 3x^3+x^2=x+5 2. Still consider equations 1a and 1b A)How many iterations are required to find the roots by the Bisection Method within six decimal places B) Use the interval you find in problem 1 that contains a root to compute the first 4 iterations and table the results

Answer 1

Follow the steps for better understanding.

Step 1.) Understand the Intermediate value theorem.

Intermediate Value Theorem states that,

If f(x) is a polynomial function, if f(x₁) and f(x₂) have opposite sign, for (x₁ < x₂). Then there must be at least one zero (or one root for the function f(x)) on the interval [x₁, x₂].

Step 1.1) Use Intermediate Value theorem to find the interval at which the roots lies for the given function.

above equation can be written as,

Substitute x = 2, in above equation.

Substitute x = - 2, in above equation.

(In the above case, you have to choose the value of 'x' such that value of f(x) is negative and choose the second choice of 'x' at which the value of the function is positive (This can only be done with 'hit and trial' method))

Now we are with an interval at which the root of the above function must lie.

Above interval satisfy Intermediate value theorem.

If f(x) is a polynomial function, if f(2) and f(2.5) have opposite sign, for (2 < 2.5). Then there must be at least one zero (or one root for the function f(x)) on the interval [2, 2.5].

Step.2.) How many iterations required to find the roots by a Bisection method.

The no. of iteration can be determined by below formula (with the desired accuracy).

Step 3.) Use the interval you find in problem 1 that contains a root to compute the first 4 iterations and table the results

**Please comment below if you have any doubt regarding this answer**

and for 1.b the function is a little bit confusing whether it is

b) 3x^3+x^2=x+5

or

b) 3x^3+x^2=(x+5)^2

or

b) 3x^3+x^2=x+5^2

**Please follow the above procedure and solve the second part, you will face no problem, if you face please comment below**