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# 3.1.2-eg - Example Let f(x = 3x2 − 1 Use the deﬁnition...

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Unformatted text preview: Example Let f (x) = 3x2 − 1 . Use the deﬁnition of the derivative to ﬁnd f (2) = lim f (2 + h) − f (2) h h→0 , (1) and then write the equation of the tangent line to the curve y = f (x) at the point (2, f (2)). Solution: Making the substitution f (2 + h) = 3(2 + h)2 − 1 into (1), it follows that f (2) = = = = = = (3(2 + h)2 − 1) − (3 · 22 − 1) lim h→0 h [3 (4 + 4h + h2 ) − 1] − (3 · 4 − 1) lim h→0 h 12 + 12h + 3h2 − 1 − 11 lim h→0 h 12h + 3h2 lim h→0 h h(12 + 3h) lim h→0 h lim 12 + 3h h→0 = 12. Thus the slope of the tangent line to the curve y = f (x) at the point (2, f (2)) is f (2) = 12. Using the point-slope formula of the line through (2, f (2)) = (2, 11) with slope f (2) = 12, y − f (2) = f (2)(x − 2) or, y − 11 = 12(x − 2), so the equation of the tangent line to the curve through the point (2, f (2)) is y = 11 + 12(x − 2). ...
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