Final answer:
Linear approximation is used to estimate the value of 3.001⁵ by expanding the function f(x) = x⁵ around the known point 3 and using the formula f(x) ≈ f(a) + f'(a)⋅(x - a).
Step-by-step explanation:
The subject of this question is Mathematics, and it pertains to the college level. The question asks us to use linear approximation to approximate the value of 3.0015. To achieve this, we can use the concept of a Taylor series expansion around a known point, in this case, 35. Since 3.001 is very close to 3, we can use the first order Taylor expansion which is essentially equivalent to finding the tangent line at x=3 for the function f(x) = x5.
To approximate 3.0015, let f(x) = x5, and find the derivative, f'(x) = 5x4. Evaluating this at x=3, we get f'(3) = 5(3)4 = 5(81) = 405. The linear approximation formula is f(x) ≈ f(a) + f'(a)∙(x - a), and setting a=3, x=3.001, we have f(3) = 35 = 243. So the linear approximation for 3.0015 is 243 + 405∙(3.001 - 3).
Performing this calculation, we get 243 + 405(0.001) = 243 + 0.405 = 243.405, which is the approximate value of 3.0015 using linear approximation.