How do you estimate the quantity using the Linear Approximation and find the error using a calculator #(15.8)^(1/4)#?

Question

Charles Anctil · Accepted Answer

The linear approximation (about #x=16#) is:

# (15.8)^(1/4) ~~ 1.975 #

This is within #1%# of the actual value.

Linear Approximation The linear approximation formula for #f(x)# about #x=a# is; # f(x) ~~ f(a) + f'(a)(x-a) # Let #f(x) = x^(1/4) => f'(x)=1/4x^(-1/4) # And so the Linear approximation for #f(x)# is: # f(x) ~~ f(a) + (x-a)/(4a^(1/4)) # If we choose #a=16# (close to #15.8#); then # f(x) ~~ 16^(1/4) + (x-16)/(4*16^(1/4)) # # \ \ \ \ \ \ \ = 2 + (x-16)/(4*2) # # \ \ \ \ \ \ \ = 2 + (x-16)/8 # So a linear approximation of #15.8# is given by; # f(15.8) ~~ 2 + (15.8-16)/8 # # \ \ \ \ \ \ \ \ \ \ \ = 2 - 0.2/8 # # \ \ \ \ \ \ \ \ \ \ \ = 2 - 0.025 # # \ \ \ \ \ \ \ \ \ \ \ = 1.975 # Calculator Result Using a calculator we find: # f(15.8) = 1.99372048 ... # Error Analysis The %age error is calculated using: # %"age error" = |("estimate-actual")/"actual "* 100| # # " " = | (1.975-1.99372048 ...)/(1.99372048 ... ) * 100| # # " " = 0.9389725 ... # So our linear approximation estimate was within #1%# of the actual value.

Ezekiel Alexander · Answer

undefined To estimate the quantity using linear approximation, we use the formula:

$$ f(x) \approx f(a) + f'(a)(x-a) $$

where $ f(x) $ is the function we want to approximate, $ f'(a) $ is the derivative of $ f(x) $ evaluated at $ x = a $, and $ a $ is the value near which we want to approximate.

In this case, we want to estimate $ (15.8)^{\frac{1}{4}} $. Let's choose $ a = 16 $ because it's a convenient value near $ 15.8 $. Now, we find the derivative of $ f(x) = x^{\frac{1}{4}} $ and evaluate it at $ x = 16 $.

$$ f'(x) = \frac{1}{4}x^{-\frac{3}{4}} $$
$$ f'(16) = \frac{1}{4}(16)^{-\frac{3}{4}} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} $$

Now, we plug these values into the linear approximation formula:

$$ f(15.8) \approx f(16) + f'(16)(15.8 - 16) $$
$$ (15.8)^{\frac{1}{4}} \approx (16)^{\frac{1}{4}} + \frac{1}{8}(15.8 - 16) $$
$$ \approx 2 + \frac{1}{8}(-0.2) $$
$$ \approx 2 - 0.025 $$
$$ \approx 1.975 $$

Now, to find the error, we calculate the actual value of $ (15.8)^{\frac{1}{4}} $ using a calculator:

$$ (15.8)^{\frac{1}{4}} \approx 1.976 $$

The error is:

$$ |1.975 - 1.976| = 0.001 $$