Chapter 3 test, Question # 12

a) Use linear approximation techniques to estimate 256.8^(1/4). Leave your answer as a number plus or minus a fraction. Show your work.

b) Determine the calculator-generated value of 256.8^(1/4).

c) To how many decimal places is your estimate similar to the calculator-generated value?

Solutions
First make a general equation: f(x) = x^(1/4)
Find the derivative of that function: f’(x) = (1/4)x^(-3/4)
Choose an a close to the original x-value that gives a nice answer: a = 256
Substitute the a-value into f(x): f(a) = 4
Substitute the a-value into f’(x): f’(a) = (1/4)256^(-3/4) = 1/256
Use the equation for linear approximation: L(x) = f(a) + f’(a) (x-a)
L(x) = 4 + (1/256)(.8) = 4 + (1/320)


a) Answer: 4 + (1/320)

Enter in calculator: 256.8^(1/4) = 4.00312…
b) Answer: 4.00312…

Compare the two answers
c) Answer: 5 decimals