1. Higher order derivative patterns
Polynomial function definition
The degree is the highest exponent of “x”, in this case “n”
f(x) = 𝑎 1 𝑥 𝑛 + 𝑎 2 𝑥 𝑛−1 + 𝑎 3 𝑥 𝑛−2 +…+ 𝑎 𝑛 𝑥 1 + 𝑎 𝑛+1 𝑥 0
The last term is a constant
Leading coefficient is "𝑎 1 "
The exponents of base “x” are whole number values W={0,1,2,3,4,..}

2. Determine finite differencesx y
1st difference
2nd difference
3rd difference
4th difference -3
-104.5
102.5 -2
27.5
-75 30 -1
25.5
-17.5
-45 8
-32.5
-15 1
-24.5 15 2
-42 45 3
-14.5 75 4 88
and higher order derivatives for
𝑦=5 𝑥 3 −7.5𝑥 2 −30𝑥+8
𝑑𝑦 𝑑𝑥 =5(3) 𝑥 2 −15𝑥−30
𝑑 2 𝑦 𝑑 𝑥 2 = 𝑥−15
For polynomial functions of degree “n”, both the finite differences and the higher order derivatives head towards “a(n!)
Not constant. Not quadratic
Run=1
𝑑 3 𝑦 𝑑 𝑥 3 = (1) 𝑥 0
Rise is not constant.
Nonlinear
All other finite differences will also be zero.
𝑑 3 𝑦 𝑑 𝑥 3 =5(3!), constant
Third finite difference is the first constant; function was cubic, and the constant is 30 or 5(3)(2)(1) or 5(3!) MHF4U
𝑑 𝑛 𝑦 𝑑 𝑥 𝑛 =a(n!) and the
𝑑 4 𝑦 𝑑 𝑥 4 =0 as well as all other
higher order derivatives
𝑛 𝑡ℎ finite difference=a(n!)

3. Predict with a formula, a) the derivative that first becomes constant and the value of the constant. b) the value of the 12th derivative.
For a polynomial function of degree “n”, 𝑑 𝑛 𝑦 𝑑 𝑥 𝑛 =(a)(n!)
2) y = 2 4 𝑥 3 −5 2 or
𝑦=2(4 𝑥 3 −5)(4 𝑥 3 −5)
1) 𝑦=2−3 𝑥 5 −4 𝑥 8
𝑦=2(16 𝑥 6 −40 𝑥 3 +25)
Polynomial function, degree 8
The 8th derivative will be the first constant
Polynomial function, degree 6
The 6th derivative will be the first constant
𝑑 6 𝑦 𝑑 𝑥 6 =(2(16))(6!) or 32(720) =
𝑑 8 𝑦 𝑑 𝑥 8 =(-4)(8!) or -4(4032) =
𝑑 12 𝑦 𝑑 𝑥 12 = 0
𝑑 12 𝑦 𝑑 𝑥 12 = 0
3) 𝑦= 14 𝑥 3
Thinking type question. 
“She not be a polynomial type function”
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.