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,..}

Determine finite differences x 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 =5 3 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 =5 3 2 (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!)

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) = 23 040 𝑑 8 𝑦 𝑑 𝑥 8 =(-4)(8!) or -4(4032) = -161 280 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.