Find the taylor polynomial t3(x) for the function f centered at the number:
f (x) = 7 tan^-1(x), a = 1, t3(x)

Solution:

Given:

Function f(x) = tan^-1(x)

Taylor polynomial up to three terms at x = a is given by

f(x) = f(a) + [(x - a) / 1!] f¹(a) + [ (x - a)²/2! ]  f¹¹(a) + [ (x - a)³/3!] f¹¹¹(a) +....

If a = 1

Taylor polynomial up to three terms at x = 1 is given by 

f(x) = f(1) + [(x - 1) / 1! ] f¹(1) + [(x - 1)² / 2!] f¹¹(1) + [(x - 1)³/3!] f¹¹¹(1) +....

f(x) = tan^-1(x)

f(1) = tan^-1(1)

f(1) = 90°

First order derivative

f¹(x) = 1 / (1 + x²)

⇒ f¹(1) = 1/(1 + 1²) = 1 / 1 + 1

⇒ f¹(1) = 1/2

Second order derivative f¹¹(x)

Differentiating f¹ with respect to x gives

⇒ f¹¹(x) = - (2 × 2x) / (1 + x²)²

⇒ f¹¹(x) = - 4x / (1 + x²)²

⇒ f¹¹(1)= - 4 / (1 + 1²)²

= -(4)/2²

⇒ f¹¹(1) = - 4/4

= -1

Third order derivative f¹¹¹(x)

Differentiating f¹¹ With respect to x gives

⇒ f¹¹¹(x) = [(1 + x²)² (-4) - (-4x)(2(1 + x²)2x)] / (1 + x²)⁴

⇒ f¹¹¹ (x) = (6x² - 2)/(1 + x²)³

⇒ f¹¹¹(1) = (6(1)² - 2)/(1 + 1²)³

⇒ f¹¹¹(1) = (6 - 2)/(2)³

⇒ f¹¹¹(1) = 4/(2)³

⇒ f¹¹¹(1) = 4/8 = 1/2

Taylor polynomial up to three terms at x=a is given by

⇒f(x) = 90° + [(x - a)/1!] f¹(a) + [(x - a)²/2!] f¹¹(a) + [(x - a)³/3!] f¹¹¹(a) +....

⇒f(x) = 90° + [(x - 1)/1!] 1/2 + [(x - 1)²/2!]( -1) + 1/2 [(x - 1)³/3!] +….

⇒ f(x) = 90° + [(x - 1)/2] - (x - 1)²/2+ (x - 1)³/12+….

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Find the taylor polynomial t3(x) for the function f centered at the number:
f (x) = 7 tan - 1(x), a = 1 t3(x)

Summary:

The Taylor's series for the function f(x) up to the term containing of third degree is f(x) = tan^-1(x) = 90° + [(x - 1)/2] - (x - 1)²/2+ (x - 1)³/12 +…