Given the function y= (2x-3)(3x-1), which finite differences will be constant?

a) first differences
b) second differences
c) third differences
d) fourth differences

linear: 1st

quadratic: 2nd
cubic: 3rd
and so on

actually, "... and higher" will also apply, since after you reach a constant difference, all the rest are zero, which is also constant.

Well, let's see if we can find out which finite differences will be constant.

To find the first differences, we need to subtract each consecutive term in the sequence.

The function y = (2x-3)(3x-1) can be written as y = 6x^2 - 5x - 3.

Now, let's find the first differences:

(2(1) - 3)(3(1) - 1) - (2(0) - 3)(3(0) - 1) = 10 - (-3) = 13
(2(2) - 3)(3(2) - 1) - (2(1) - 3)(3(1) - 1) = 36 - 13 = 23
(2(3) - 3)(3(3) - 1) - (2(2) - 3)(3(2) - 1) = 70 - 23 = 47

Hmm, it seems that the first differences are not constant.

Now let's find the second differences by subtracting the first differences:

23 - 13 = 10
47 - 23 = 24

The second differences are not constant either.

Finally, let's find the third differences by subtracting the second differences:

24 - 10 = 14

And the third differences are also not constant.

Therefore, none of the finite differences for this function are constant. So the answer is none of the above! Keep in mind that this is not always the case with functions, but it happens to be true for this one. I hope that gave you a good laugh!

To determine which finite differences will be constant, we need to find the differences between consecutive terms of the function.

First, let's expand the given function:

y = (2x - 3)(3x - 1)
= 6x^2 - 2x - 9x + 3

Simplifying, we get:

y = 6x^2 - 11x + 3

Now, let's find the first differences by subtracting consecutive terms:

First differences = (y[1] - y[0]), (y[2] - y[1]), (y[3] - y[2])...

For our function, the first differences will be:

(y[1] - y[0]) = (6(x[1])^2 - 11(x[1]) + 3) - (6(x[0])^2 - 11(x[0]) + 3)

Simplifying, we get:

First differences = 6(x[1]^2 - x[0]^2) - 11(x[1] - x[0])

Since x[1] and x[0] are consecutive terms, their difference is constant (let's say it's 'h').

First differences = 6(x[1]^2 - x[0]^2) - 11h

The term x[1]^2 - x[0]^2 is also a constant difference because the square terms cancel out.

Therefore, the first differences will be constant.

To answer the question:
a) The first differences will be constant.

The second differences can be found by subtracting consecutive first differences.

Second differences = (first differences[1] - first differences[0]), (first differences[2] - first differences[1]), (first differences[3] - first differences[2])...

Similarly, the second differences can be shown to be constant as well.

Therefore, the answer is:
b) The second differences will be constant.

To find out which finite differences will be constant for the function y = (2x-3)(3x-1), we need to calculate the differences for consecutive values of y.

1) First Differences:
First, we calculate the differences between consecutive y-values. To do this, we evaluate the function for two consecutive x-values and find the difference between their corresponding y-values.

Let's calculate the first differences using x-values from 1 to, say, 5:

x = 1: y = (2(1)-3)(3(1)-1) = (-1)(2) = -2
x = 2: y = (2(2)-3)(3(2)-1) = (1)(5) = 5
x = 3: y = (2(3)-3)(3(3)-1) = (3)(8) = 24
x = 4: y = (2(4)-3)(3(4)-1) = (5)(11) = 55
x = 5: y = (2(5)-3)(3(5)-1) = (7)(14) = 98

Now, let's calculate the differences between consecutive y-values:
First Difference = y(n) - y(n-1)
Difference between y(2) and y(1) = 5 - (-2) = 7
Difference between y(3) and y(2) = 24 - 5 = 19
Difference between y(4) and y(3) = 55 - 24 = 31
Difference between y(5) and y(4) = 98 - 55 = 43

The first differences are not constant because they differ from one another. Thus, option (a) is not correct.

2) Second Differences:
To find the second differences, we calculate the differences between consecutive first differences.

Second Difference = First Difference(n) - First Difference(n-1)

Difference between first differences:
Difference between the second difference and the first difference of 7 is 19 - 7 = 12
Difference between the third difference and the second difference of 31 is 43 - 31 = 12

The second differences are constant at 12. Therefore, option (b) is correct.

3) Third Differences:
To find the third differences, we calculate the differences between consecutive second differences. However, since we only found one value for the second differences, we cannot determine if it will be constant. Therefore, option (c) cannot be determined.

4) Fourth Differences:
Since we only have one value for the second differences, we cannot calculate the fourth differences.

In conclusion, for the given function y = (2x-3)(3x-1), the second differences will be constant, which means option (b) is correct.