Find the approximate value of the expression (1.99)^7 by using linear approximation.
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Given : 1.99⁷ and Linear approximation method
To find : Value of 1.99⁷ using Linear approximation method
Solution:
f(x) = x⁷
at x = 2
=> f(2) = 2⁷
=> f(2) = 128
f'(x) = 7x⁶
f'(2) = 7(2)⁶
=> f'(2) = 448
L(x) = f(x) + f'(x) (x - a) at x = a
=> L(x) = 128 + 448(x - 2)
putting x = 1.99
=> L(1.99) = 128 + 448(1.99 - 2)
=> L(1.99) = 128 - 4.48
=> L(1.99) = 123.52
1.99⁷ = 123.52 using linear approximation.
1.99⁷ = 123.5866 ( using calculator)
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