Question

3x^2+x-5=0 to 2 decimal place 3x^3+2x-4=0 to 2 decimal place

Answer 1

Answer:

1. 3/5

2. 1.219841

3. 2 - f(2)/7

4. -1.9682

5. 1.502446

Step-by-step explanation:

If we call

then the second approximation to the root of the equation f(x)=0 would be

hence

2.

Here we have

Let's start with  

then

Since the first 6 decimals of  and  are equal, the desired approximation is 1.219841

3.

If the line y = 4x − 1 is tangent to the curve y = f(x) when x = 2, then f'(2) = 4*2 - 1 = 7, so

4.

Here

5.

We want to find all the values x such that

or what is the same, the x such that

so, let f(x) be

and let's use Newton's method to find the roots of f(x).

Since

f is strictly increasing, and since

f(1) = e+3-9 = e - 6 < 0

and

f has only one root in [1,2]

By using Newton's iterations starting with  

Since  then

x=1.50244564 is the desired root.

The answers as a comma-separated list would be

1,  1.573899431,  1.503982961,  1.502446348, 1.50244564

The answer rounded to six decimal places would be

1.502446