What is f'(1) if f(x) = 3x^3 - 5x^2 + 2 and f'(x) = 9x^2 - 10x?

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Multiple Choice

What is f'(1) if f(x) = 3x^3 - 5x^2 + 2 and f'(x) = 9x^2 - 10x?

Explanation:
To find the slope of the tangent at x = 1, plug that value into the derivative f'(x). With f'(x) = 9x^2 - 10x, evaluate at x = 1: f'(1) = 9(1)^2 - 10(1) = 9 - 10 = -1. The instantaneous rate of change at x = 1 is -1, meaning the tangent line there slopes downward. This result would be the same if you differentiated f to get f'(x) and then substituted x = 1.

To find the slope of the tangent at x = 1, plug that value into the derivative f'(x). With f'(x) = 9x^2 - 10x, evaluate at x = 1: f'(1) = 9(1)^2 - 10(1) = 9 - 10 = -1. The instantaneous rate of change at x = 1 is -1, meaning the tangent line there slopes downward. This result would be the same if you differentiated f to get f'(x) and then substituted x = 1.

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