# Let f: ℝ → (-π/2, π/2) be give

Let f: ℝ → (-π/2, π/2) be given by f(t) = tan-1(t)

a) What is Taylor’s polynomial P1(t) of order 1 f if t = 0? What is the residual E1(t) in Taylor’s formula f(t) = P1(t) + E1(t)?

b) Use Taylor’s residual formula to give an estimate of π/4 = tan-1(t) (without using π on the calculator). Show that the residual term E1(t) is between -1 and 0 and that π/4 is equal to P1(t) – 1/4 = 3/4 with an error less than 1/4

# Let f: ℝ → (-π/2, π/2) be give

Let f: ℝ → (-π/2, π/2) be given by f(t) = tan-1(t)

a) What is Taylor’s polynomial P1(t) of order 1 f if t = 0? What is the residual E1(t) in Taylor’s formula f(t) = P1(t) + E1(t)?

b) Use Taylor’s residual formula to give an estimate of π/4 = tan-1(t) (without using π on the calculator). Show that the residual term E1(t) is between -1 and 0 and that π/4 is equal to P1(t) – 1/4 = 3/4 with an error less than 1/4

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