Question

Find a linear approximation for the following functions at the indicated points.

(i) f(x) = x³ - 5x + 12, x₀ = 2

(ii) g(x) = √(x² + 9), x₀ = - 4

(iii) h(x) = x/(x + 1), x₀ = 1

Answer 1

(i) f(x) = x³ – 5x + 12 

f'(x) = 3x² – 5 

f(x₀) = f(2) = (2)³ – 5(2) + 12 = 8 – 10 + 12 = 10 

f'(x₀) = f'(2) = 3(2)² – 5 = 12 – 5 = 7 

The required linear approximation 

L(x) = f(x₀) + f'(x₀) (x – x₀) 

= 10 + 7(x – 2) 

= 10 + 7x – 14 

= 7x – 4

(ii) g(x) = √(x² + 9)

(iii) h(x) = x/(x + 1)