MATH100 Quiz: Theorem & derivative test.docx

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MATH100 Full Course Notes
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MATH100 Full Course Notes
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Rolle"s theorem f " (c)=0 for some cbetweena b( assume f ( x)is differentiable that fa)=f (b) Show that: f ( x)=x3+5x+2 has exactly one root (f ( x)=0) Intermediate value theorem: f ( x)is continuous f (0)=2 f ( 1)= 4( ) Suppose f (a)=0 f (b)=0for 2different x values. f " ( x)=3 x2+5. No 2 solutions can exist, therefore there is exactly one root. Mean value theorem (rolle"s theorem on a slant ) f ( x)is differentiable ,f "(c)= f ( b) f (a) b a. A car is located at xa at t a. If vavg is over the speed limit, the mean value theorem guarantees that the instantaneous speed v must have matched vavg at least once -> you get a speeding ticket. Prove that : f " (x)=0 onan interval[a,b]then f (x)=c. M. v. t f " (c )= f (b ) f (a ) b a.

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