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Lecture

Trigonometric Derivatives.docx

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Department
Mathematics
Course
MATH113
Professor
Ian Blokland
Semester
Winter

Description
Part 1: The derivative as a rate of change (text: 3.4) Recall the falling basketball 2 d t =4.9t ∆d vavg ∆t v(t)=d'(t) More generally …. S(t) Let represent the position of an object s(t) Rate of change of : ' velocity→v t =s t ( ) speed→∨v(t)∨¿ Rate of change of velocity: ' acceleration→a t =v t =s''(t) ' ( ) ( ) s t = j t JERK s(4t SNAP s t CRACKLE (6) s t POP Example: Jumping on a trampoline 2 h t =7t−4.9t v t =h t =7∗1−4.9 2t =t−9.8t Attopof the jump,v=0→0=7−9.8t→t= 5 7 hmaxh 5 =2.5∧a(t)=v (t)=0−9.8=−9.8 (7  All the objects rise/fall with constant downward acceleration (Galileo) Application to economics: ¿ ¿ ¿ ¿ ¿ Marginal¿=“thederivativeof ”¿ C x (cost ¿of objects produce)→marginalcostc x ( ) Example: Explicitly, suppose: 1 c(x)=[100]+ 5x+ x ( ¿cost]+[variablecosts) [ 10 ] x=20 SUPPOSE: c = c(20)= 240 =12 Average cost avg 20 20 c (x)=0+5+ x →c (20)=9 Marginal cost 5 => cheaper than first 20 Part: 2 Derivatives of Trigonometric Functions (text 3.5) f (x)=sinx '( ) f (x+h)−f (x) f x =lih→0 h d sin(x+h)−sin(x) dx (sinx)=h→0 h Trigonometricidentity:sin(x+h)=sinx∗cos⁡(h)+cosx∗sin ⁡(h) d cos(h)−1 sinh ) dx (sinx)=lim sinx h +lim cosx h h→0 h→0 h+1 cos¿ ¿ ¿ h¿
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