Evaluate cube root of 64.04

Problem

√(3,64.04)

Solution

1. Identify the function and the point of approximation. Let ƒ(x)=√(3,x) We want to evaluate ƒ(64.04)

2. Choose a nearby value a where the cube root is an integer. Let a=64 so ƒ(64)=√(3,64)=4

3. Determine the change in x which is Δ(x)=64.04−64=0.04

4. Find the derivative of ƒ(x)=x(1/3)

d(x(1/3))/d(x)=1/3*x(−2/3)

5. Evaluate the derivative at a=64

ƒ(64)′=1/(3*(64)(2/3))

ƒ(64)′=1/(3*(4))

ƒ(64)′=1/48

6. Apply the linear approximation formula ƒ*(a+Δ(x))≈ƒ(a)+ƒ(a)′*Δ(x)

√(3,64.04)≈4+1/48*(0.04)

7. Simplify the calculation.

√(3,64.04)≈4+0.04/48

√(3,64.04)≈4+1/1200

√(3,64.04)≈4+0.0008333...

Final Answer

√(3,64.04)≈4.0008333

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