Final answer:
To approximate √65 - √63 using the tangent line approximation, we find the equation of the tangent line at x = 64 and use it to approximate the values at x = 65 and x = 63. Then, we calculate the difference between these approximations to get the final result of 1/8.
Step-by-step explanation:
To approximate √65 - √63 using the tangent line approximation for f(x) = √x at x = 64, we first find the equation of the tangent line at x = 64. The equation of the tangent line is y = f'(64)(x-64) + f(64).
Taking the derivative of f(x) = √x, we get f'(x) = 1 / (2√x). Plugging in x = 64, we find f'(64) = 1 / (2√64) = 1 / 16.
So, the equation of the tangent line is y = (1/16)(x-64) + 8.
Now, we can approximate √65 by plugging in x = 65 into the equation of the tangent line: f(65) ≈ (1/16)(65-64) + 8 = 8 + 1/16 = 129/16.
Similarly, we can approximate √63 by plugging in x = 63 into the equation of the tangent line: f(63) ≈ (1/16)(63-64) + 8 = 8 - 1/16 = 127/16. Finally, we calculate the approximation √65 - √63:
√65 - √63 ≈ (129/16) - (127/16) = 2/16 = 1/8