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2.1.4-eg-1

# 2.1.4-eg-1 - h 6 = 0(b To ﬁnd the slope of the tangent to...

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Example Consider the curve y = f ( x ) = ( x + 1) 2 which goes through the point P (2 , f (2)) = P (2 , 9). (a) Find the slope of the secant line joining P (2 , f (2)) and Q (2 + h, f (2 + h )), for some h 6 = 0. (b) Using your answer in (a), what is the slope of the tangent line to the curve at the point P ? Solution: (a) The slope of the secant line joining P (2 , f (2)) and Q (2 + h, f (2 + h )) for h 6 = 0 is given by f (2 + h ) - f (2) (2 + h ) - 2 = ((2 + h ) + 1) 2 - (2 + 1) 2 h = (2 + h ) 2 + 2(2 + h ) + 1 - 9 h = 4 + 4 h + h 2 + 4 + 2 h + 1 - 9 h = h 2 + 6 h h = h + 6 where the cancellation of h
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Unformatted text preview: h 6 = 0. (b) To ﬁnd the slope of the tangent to the curve at the point P (2 ,f (2)), we must consider the slope of the secant line PQ as the point Q approaches P . That is, as h approaches 0. But as h gets closer and closer to zero, h + 6 gets closer and closer to 6. The slope of the tangent line to the curve at the point P (2 ,f (2)) must be 6....
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