Stewart6e3_3 - MATH 191 Sections 1 and 3 Calculus I Fall...

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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.3: Differentiating trig functions The goal of today’s class is to nail down one particularly difficult derivative, and to derive (no pun intended!) a few others from this one: d dx (sin( x )) = cos( x ) . The first computations we perform are quite straightforward: we simply apply the definition of the derivative to f ( x ) = sin( x ) and see how far we can get using trig identities: f ( x ) = lim h → ... Notice that we used the law sin( A + B ) = sin( A ) cos( B ) + sin( B ) cos( A ) in the above computations. Everything therefore comes down to computing the following two limits: lim h → sin( h ) h and lim h → cos( h )- 1 h . Let’s use Mathematica to get a reasonable guess as to what this first limit might be. Put your guess in the space below: To prove this limit we argue geometrically. The following diagram will help us out immensely : Step 1. By the definition of radian measure, the length of the arc AB in the above diagram is...
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Stewart6e3_3 - MATH 191 Sections 1 and 3 Calculus I Fall...

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