Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 2 - The Derivative - 2.2 The Derivative Function - Exercises Set 2.2: 11

Answer

$ f’(x) = 3x^2 $ The equation of the tangent line: $y = 0 $

Work Step by Step

$f(x) = x^3 $ Find the derivative: $ f’(x) = \lim\limits_{ Δx\to 0} \frac{f(x+Δx) - f(x)}{Δx} $ $ f’(x) = \lim\limits_{ Δx\to 0} \frac{(x+Δx)^3 - x^3}{Δx} $ $ f’(x) = \lim\limits_{ Δx\to 0} \frac{x^3 +3x^2Δx+3x(Δx)^2 + (Δx)^3-x^3}{Δx} $ $ f’(x) = \lim\limits_{ Δx\to 0} \frac{ Δx(3x^2+3xΔx + (Δx)^2)}{Δx} $ $f’(x) = \lim\limits_{ Δx\to 0} (3x^2+3xΔx + (Δx)^2) $ $ f’(x) = 3x^2+3x(0) + 0^2 $ $ f’(x) = 3x^2 $ When x = a, $f(0) = 0$ and $f’(0) = 0 $ Find the equation for the tangent line: $(y - y_1) = m(x-x_1) $ $(y - 0)=0(x-0) $ $y_{tangent} =0 $ $f’(x) = 3x^2 $ The equation of the tangent line: $y = 0 $
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