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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!)