2. Let f(x) = ??x?. i. Use the limit definition of derivative to find the exact value f'(4). ii. By EQUATION 1, f'(4) ? (f(4+h)-f(4))/h = (??4?+?h?-??4?)/h. Therefore we estimate f'(4) by calculating (??4?+?h?-2)/h for values of h near 0. Complete Columns 2 and 5 of TABLE 1 below and describe how (??4?+?h?-2)/h behaves as h approaches 0. | h | (??4?+?h?-2)/h | Error* | h | (??4?+?h?-2)/h | Error* | |---|---|---|---|---|---| | 0.1 | | | - 0.1 | | | | 0.01 | | | - 0.01 | | | | 0.001 | | | - 0.001 | | | | 0.0001 | | | - 0.0001 | | | TABLE 1
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First, we need to find the derivative of the function f(x) = βx using the limit definition of the derivative: f'(x) = lim (h->0) [(f(x+h) - f(x))/h] In this case, f(x) = βx, so we have: f'(x) = lim (h->0) [(β(x+h) - βx)/h] Now, we want to find f'(4), so we will Show moreβ¦
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