Use newton's method to find the absolute minimum value of the function f(x)=x2+sinx correct to six decimal places.

The solution would be like this for this specific problem:

f(x) = x^2 + sin(x) 

f '(x) = 2x + cos(x) 

The minimum value is at f '(x) = 0, 

So, let g(x) = 2x + cos(x) 

Thus, g '(x) = 2 - sin(x) 

x(new) = x - g(x) / g '(x) 
or 
x(new) = x - [2x + cos(x)] / [2 - sin(x)] 

Calculation 

x1 = -0.5 - [2 * -0.5 + cos(-0.5)] / [2 - sin(-0.5)] 
= -0.4506266931 
x2 = -0.4501836476 
x3 = -0.4501836113 
x4 = -0.4501836113 

This value for x, f(x) = -0.2324655752. 

After converting to 6 decimal places: the minimum point is (-0.450184, -0.232466).

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.

answer: no, but if it does it's a obvious pattern.

step-by-step explanation: