MAT136H1 Lecture Notes - Lecture 6: Antiderivative, Integral, Exponential Growth

• School: UTSG
• Department: Mathematics
• Course: MAT136H1
• Professor: Paul Milgram

Document Summary

Mat136 lecture 6 indefinite integrals & net change theorem. Last time: we discussed ftc-ii: i. e. if :[(cid:2183),(cid:2184)] is continuous and f is an antiderivative of f, then: =(cid:4666)(cid:4667): we also i(cid:374)troduced i(cid:374)defi(cid:374)ite i(cid:374)tegrals: i. e. if f is defi(cid:374)ed o(cid:374) so(cid:373)e i(cid:374)ter(cid:448)al, a(cid:374)d f" = f, the(cid:374): Today: we calculate some indefinite integrals with real world examples: (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:3029)(cid:3028) f [a,b) (continuous) area. (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:3028) f [a,b) (of t) (continuous) Constraint f needs to be continuous f needs to be continuous n/a to use ftc-ii. [where c is a constant: (cid:1858)(cid:4666)(cid:4667)+(cid:1859)(cid:4666)(cid:4667)(cid:1856)= (cid:1858)(cid:4666)(cid:4667)(cid:1856)+ (cid:1859)(cid:4666)(cid:4667)(cid:1856, (cid:1855) (cid:1858)(cid:4666)(cid:4667)(cid:1856)= (cid:1855) (cid:1858)(cid:4666)(cid:4667)(cid:1856, (cid:1872) (cid:4666)(cid:1872)(cid:2870)+(cid:885)(cid:1872)+(cid:884)(cid:4667)(cid:1856)(cid:1872, (cid:884)(cid:4666)(cid:883)+(cid:887)(cid:4667)(cid:1856)(cid:1872) Step 3: now find the antiderivatives for each part: Since [(cid:884)]=(cid:884)(cid:4666)(cid:884)(cid:4667), so (cid:2870)(cid:2870) is an antiderivative of (cid:884) Thus (cid:884)(cid:1872)(cid:1856)(cid:1872)+ (cid:883)(cid:882)(cid:1856)(cid:1872) (cid:884)(cid:884)+ (cid:883)(cid:882) (cid:883)(cid:882): where the co(cid:374)sta(cid:374)t (cid:862)c(cid:863) is fro(cid:373) all 3 parts of the i(cid:374)tegra(cid:374)d. Step 2: now use the properties of integers (#1 & #2) to split the integrand into 3 parts.