Evaluate cube root of 1.1

Problem

√(3,1.1)

Solution

1. Identify the function and the point of approximation. To evaluate √(3,1.1) we use the function ƒ(x)=√(3,x) and the linear approximation at a=1

2. Calculate the value of the function at a=1

ƒ(1)=√(3,1)=1

3. Find the derivative of ƒ(x)=x(1/3)

d(x(1/3))/d(x)=1/3*x(−2/3)

4. Evaluate the derivative at a=1 to find the slope of the tangent line.

ƒ(1)′=1/3*(1)(−2/3)=1/3

5. Apply the linear approximation formula L(x)=ƒ(a)+ƒ(a)′*(x−a) where x=1.1

L(1.1)=1+1/3*(1.1−1)

6. Simplify the expression to find the approximate value.

L(1.1)=1+1/3*(0.1)

L(1.1)=1+0.0333...

L(1.1)≈1.0333

Final Answer

√(3,1.1)≈1.0333

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