Higher order derivative patterns Polynomial function definition The degree is the highest exponent...

Higher order derivative patternsPolynomial function definition

f(x) =

Leading coefficient is

The degree is the highest exponent of “x”, in this case “n”

The exponents of base “x” are whole number values W={0,1,2,3,4,..}

The last term is a constant

Determine finite differences x y 1st

difference2nd

difference3rd

difference4th difference

-3 -104.5102.5

-2 -227.5

-7530

-1 25.5-17.5

-4530

0

0 8-32.5

-1530

0

1 -24.5-17.5

1530

0

2 -4227.5

4530

0

3 -14.5102.5

75

4 88

𝑦=5 𝑥3−7.5𝑥2−30 𝑥+8

Run=1Rise is not constant.Nonlinear

Not constant. Not quadratic

Third finite difference is the first constant; function was cubic, and the constant is 30 or 5(3)(2)(1) or 5(3!) MHF4U

All other finite differences will also be zero.

𝑑𝑦𝑑𝑥

=5(3)𝑥2−15 𝑥−30

𝑑2 𝑦𝑑𝑥2

=5 (3 ) (2 ) 𝑥−15

(1)

=5(3!), constant

For polynomial functions of degree “n”, both the finite differences and the higher order derivatives head towards “a(n!)

=a(n!) and the

and higher order derivatives for

finite difference=a(n!)

Predict with a formula, a) the derivative that first becomes constant and

the value of the constant.b) the value of the 12th derivative.

1¿ 𝑦=2−3 𝑥5−4 𝑥8 2) y = 2

3¿ 𝑦=14

𝑥3Thinking type question.

=(-4)(8!) or -4(4032) = -161 280

Polynomial function, degree 8The 8th derivative will be the first constant

= 0

Investigation required; generate data, seek patterns in the data using colour coding, make a formula prediction, verify formula, use formula to predict the 12th derivative.

Polynomial function, degree 6The 6th derivative will be the first constant

=(2(16))(6!) or 32(720) = 23 040

= 0

or

“She not be a polynomial type function”

For a polynomial function of degree “n”, =(a)(n!)