MATH 2B Lecture Notes - Lecture 5: Antiderivative
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MATH 2B Full Course Notes
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Math2b lecture 5 5. 4 net change & indefinite integrals. Reminder from last lecture notes: ftc: (cid:1858) (cid:4666)(cid:4667) (cid:1856)=(cid:1858)(cid:4666)(cid:4667)|(cid:3028)(cid:3029) Fi(cid:374)di(cid:374)g the area u(cid:374)der the cur(cid:448)e (cid:449)ould ulti(cid:373)atel(cid:455) give us the anti-derivative. Once we add all the areas a+b+c together, we would get 0 as the answer. Anti-derivative of v(t) gives us the displacement over [a,b] (cid:4666)(cid:884)(cid:1872) (cid:883)(cid:884)(cid:1872)(cid:2870)(cid:4667)|(cid:2868)(cid:2872)=((cid:884)(cid:4666)(cid:886)(cid:4667) (cid:883)(cid:884)(cid:4666)(cid:886)(cid:4667)(cid:2870)) ((cid:884)(cid:4666)(cid:882)(cid:4667) (cid:883)(cid:884)(cid:4666)(cid:882)(cid:4667)(cid:2870)) The total distance traveled must be > 0. Example: a) if (cid:4666)(cid:1872)(cid:4667)=(cid:884) (cid:1872), what is the displacement over [0,4]? (cid:2872) (cid:2872) =(cid:4666)(cid:884)(cid:1872) (cid:883)(cid:884)(cid:1872)(cid:2870)(cid:4667)|(cid:2868)(cid:2870) (cid:4666)(cid:884)(cid:1872) (cid:883)(cid:884)(cid:1872)(cid:2870)(cid:4667)|(cid:2870)(cid:2872)=(cid:4666)(cid:886) (cid:884)(cid:4667) (cid:4666)(cid:882)(cid:4667) [(cid:4666)(cid:890) (cid:890)(cid:4667) (cid:4666)(cid:886) (cid:884)(cid:4667)]=(cid:886: we cannot take the anti-derivative of an absolute value. Therefore, we split the interval up at where the function is equal to 0: afterwards, we determine whether the function will be positive or negative. Well, we plugged in a number from the interval and checked to see if the solution we got was positive or negative. For instance, from [0,2], we can plug in 1 to |2-t| and get +1.