Calculus 1 Question Estimating Derivatives

Question

The following functions have derivatives at the given points. Estimate the derivative to two decimal places using a numerical technique.

f (x) = x^1/3, x = 4

Answer 1

f(x) = x^1/3

To find the slope of the curve at 4, use the formula:

m = (f(x+Δx) - f(x-Δx))/(2Δx) , Δx = 0.01

m = ((4.01)^1/3 - (3.99)^1/3)/0.02

= (1.5887 - 1.586)/.02 = 0.0027/.02 = 0.135 ≈ 0.14

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Compare to f '(x) = (1/3)x^-2/3, f '(4) = (1/3)/4^2/3 = (1/3)(0.39685) = 0.13228 --> 2% error