One example of a polynomial sequence that shows that the terms of the second difference are constant is the sequence of perfect squares. The terms of this sequence increase by adding consecutive odd numbers (1, 3, 5, 7, ...).
For example, the sequence of perfect squares is:
1, 4, 9, 16, 25, 36, 49, ...
The first differences of this sequence are:
3, 5, 7, 9, 11, 13, ...
The second differences of this sequence are:
2, 2, 2, 2, 2, ...
As we can see, the second differences are constant at 2.
Which polynomial sequence shows that the terms of the second difference are constant
5 answers
if y = x^2
first derivative = slope = dy/dx = 2 x (linear)
second derivative = curvature = d^2y/dx^2 = 2, (constant)
first derivative = slope = dy/dx = 2 x (linear)
second derivative = curvature = d^2y/dx^2 = 2, (constant)
Yes, you are correct. The polynomial sequence y = x^2 demonstrates that the terms of the second difference (second derivative) are constant. The first derivative, which represents the slope of the curve at any given point, is 2x, a linear function. The second derivative, which represents the curvature of the curve at any given point, is 2, a constant value. This indicates that the terms of the second difference are constant.
any quadratic function will generate constant second differences.
Yes, you are right. Any quadratic function will generate constant second differences. The second difference of a quadratic function is always constant and equal to twice the leading coefficient. This means that the terms of the second difference will always be the same value, regardless of the specific quadratic function being considered.
1 answer