Find the approximate value of (17/81)^(1/4) using differentiation.

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question is will you evaluate (97/81)^(1/4) in the same manner?
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Let y = f(x) = x^(1/4) To find: Approximate value of f(17/81) using differentiation We know that, f (x + delta x) = f(x) + .....using first principle 1st degree approximation. Here 'delta x' is the increment in x. Let us choose x such that f(x) is a rational number. If, x = 16/81 f(16/81)...
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Let y = f(x) = x^(1/4) To find: Approximate value of f(17/81) using differentiation We know that, f (x + delta x) = f(x) + [f'(x)* (delta x)].....using first principle 1st degree approximation. Here 'delta x' is the increment in x. Let us choose x such that f(x) is a rational number. If, x = 16/81 f(16/81) = 2/3 Therefore, delta x = (17/81)-(16/81) = 1/81 And f'(x) = d(x^(1/4))/dx = (1/4)* [x^ (-3/4)] So, approx. f(17/81) = f(16/81) + [(1/4)* {(16/81)^(-3/4)}] * (1/81) = (2/3) + (1/4) * (27/8) * (1/81) = (2/3) + (1/96) = 65/96 = 0.677 This is the approx value of (17/81)^(1/4) read less
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Question is will you evaluate (97/81)^(1/4) in the same manner?
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question is will you evaluate (97/81)^(1/4) in the same manner
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65/96 is the answer. (16/81)^(1/4) = 2/3 which is useful to check your answer
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Let y = f (x) = x^(1/4). Now differentiate to find delta y in terms of delta x and then find y at the given point. Realise that (16/81)^(1/4) = 2/3. On solving you will get 65/96.
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(17/81)^1/4 = (17)^1/4/(81)^1/4 =(17)^1/4/3 Let f(x)= x^1/4 f'(x)=1/4x^3/4 {f(x+∆x)-f(x)}= f'(x)*∆x {f(x+∆x)-f(x)}=1/4x^3/4*∆x Write 17=16+1 then put x=16 abd ∆x=1 {f(16+1)-f(16)}=1/4(16)^3/4*1 f(17)-f(16)=1/4*2^4 f(17)=1/32 + f(16) f(17) = 1/32 + (16)^1/4 f(17) = 0.03125 + 2 (17)^1/4 = 2.03125 Approximate...
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(17/81)^1/4 = (17)^1/4/(81)^1/4 =(17)^1/4/3 Let f(x)= x^1/4 f'(x)=1/4x^3/4 {f(x+∆x)-f(x)}= f'(x)*∆x {f(x+∆x)-f(x)}=1/4x^3/4*∆x Write 17=16+1 then put x=16 abd ∆x=1 {f(16+1)-f(16)}=1/4(16)^3/4*1 f(17)-f(16)=1/4*2^4 f(17)=1/32 + f(16) f(17) = 1/32 + (16)^1/4 f(17) = 0.03125 + 2 (17)^1/4 = 2.03125 Approximate value of (17/81)^1/4 = 2.03125/3 = 0.677 read less
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