2. Given 𝑓(𝑥) = 𝑥^1.857

Question

a. Find the linear approximation to the function at a = 1.

b. Use the linear approximating function to estimate the value of the function at x = 1.1, 1.01, and 1.001.

c. Use a calculator to get the actual values at those stated in (b) and compare the approximations to the actual values using the relative error.

Answer 1

a:

f(1) = 1 --> (x,y) = (1,1)

f '(x) = 1.857x^0.857

m = f '(1) = 1.857

y -1 = 1.857(x-1) --> y = 1.857x -0.857

b:

y(1.1) = 1.1857

y(1.01) = 1

y(1.001) = 1.001857

c:

f(1.1) = 1.19362 --> 0.66% error

f(1.01) = 1.0186 --> 1.8% error

f(1.001) = 1.0018578 --> 0.00008% error