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 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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.4: Logarithmic Functions
Problem 46E
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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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