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Chapter2

# Chapter2 - Section 2.1 The Tangent and Velocity Problems...

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1 Section 2.1 The Tangent and Velocity Problems • Goals – Use two problems to explain the need for the notion of limit : The tangent problem The velocity problem What would it mean in general for a line to be tangent to a curve? Tangent Problem touching once touching twice Example Let’s use a specific example to see how tangent lines in general could be defined: We want to find an equation of the tangent line to the parabola y = x 2 at the point P (1, 1) . Since we know the point P , the only question is that of the slope m of the line. Example (cont’d) Our idea is to use a nearby point Q ( x , x 2 ) on the parabola and find the slope m PQ of the secant line PQ . We choose x 1 so that Q P . Then 2 1 1 PQ x m x = Example (cont’d) Now we allow x to approach the value 1 without ever being 1 . – Meanwhile we monitor m PQ . The tables on the next slide give the values of m PQ as x approaches 1 both from above and below. In both cases it seems that m PQ approaches 2 ! Example (cont’d)

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