‘Let L() be the linearization of the cube root function at X-64. When using L(x) to approximate the cube root of 65, how much is the answer greater than 4? 1/4 1/48 1/65 1/64 21/3’
The Correct Answer and Explanation is:
To solve this problem, we need to linearize the cube root function at x=64x = 64x=64. The general approach is to use the tangent line to the function f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x at x=64x = 64x=64, which will serve as the linear approximation, L(x)L(x)L(x).
Step 1: Find the cube root function and its derivative
We are given the cube root function:f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x
The derivative of f(x)f(x)f(x), using the power rule, is:f′(x)=ddx(x1/3)=13x−2/3=13x23f'(x) = \frac{d}{dx} \left( x^{1/3} \right) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}f′(x)=dxd(x1/3)=31x−2/3=33×21
Step 2: Linearize f(x)f(x)f(x) at x=64x = 64x=64
We need the equation of the tangent line to f(x)f(x)f(x) at x=64x = 64x=64. This tangent line is given by the linear approximation formula:L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x – a)L(x)=f(a)+f′(a)(x−a)
where a=64a = 64a=64.
- Find f(64)f(64)f(64): f(64)=643=4f(64) = \sqrt[3]{64} = 4f(64)=364=4
- Find f′(64)f'(64)f′(64): f′(64)=136423=13⋅16=148f'(64) = \frac{1}{3 \sqrt[3]{64^2}} = \frac{1}{3 \cdot 16} = \frac{1}{48}f′(64)=336421=3⋅161=481
Now the linear approximation L(x)L(x)L(x) is:L(x)=4+148(x−64)L(x) = 4 + \frac{1}{48} (x – 64)L(x)=4+481(x−64)
Step 3: Approximate the cube root of 65
We now use this linear approximation to approximate the cube root of 65. Substituting x=65x = 65x=65 into the linearization formula:L(65)=4+148(65−64)=4+148L(65) = 4 + \frac{1}{48} (65 – 64) = 4 + \frac{1}{48}L(65)=4+481(65−64)=4+481
Thus, the cube root of 65 is approximately:L(65)=4+148L(65) = 4 + \frac{1}{48}L(65)=4+481
Step 4: Find how much the answer is greater than 4
The difference between the linear approximation and 4 is:L(65)−4=148L(65) – 4 = \frac{1}{48}L(65)−4=481
Thus, the answer is 148\frac{1}{48}481, which means the linear approximation is greater than 4 by 148\frac{1}{48}481. The correct answer is:148\boxed{\frac{1}{48}}481
