Explanation:
f(x) = sin(x), a = π/6, n = 4
Find the derivatives.
f⁽⁰⁾(π/6) = sin(π/6) = ½
f⁽¹⁾(π/6) = cos(π/6) = ½√3
f⁽²⁾(π/6) = -sin(π/6) = -½
f⁽³⁾(π/6) = -cos(π/6) = -½√3
f⁽⁴⁾(π/6) = sin(π/6) = ½
T₄(x) = ½ (x − π/6)⁰ / 0! + ½√3 (x − π/6)¹ / 1! − ½ (x − π/6)² / 2! − ½√3 (x − π/6)³ / 3! + ½ (x − π/6)⁴ / 4!
T₄(x) = ½ + ½√3 (x − π/6) − ¼ (x − π/6)² − ¹/₁₂√3 (x − π/6)³ + ¹/₄₈ (x − π/6)⁴
Find the fifth derivative.
f⁽⁵⁾(x) = cos(x)
So the next term of the series would be:
cos(z) (x − π/6)⁵ / 5!
For 0 ≤ z ≤ π/3,│f⁽⁵⁾(x)│is a maximum at z = 0. Therefore:
│R₄(x)│≤ │cos(0) (0 − π/6)⁵ / 5!│
│R₄(x)│≤ 0.000328
│R₄(x)│is always positive, so we can ignore the bottom two graphs. The top right graph has a y-intercept of approximately 0.0003, so that is the correct graph.