MAT136H1 Lecture 5: Fundamental Theorem of Calculus Part 2

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

Document Summary

A(cid:374)d differe(cid:374)tia(cid:271)le o(cid:374) (cid:894)a, (cid:271)(cid:895) a(cid:374)d f"(cid:894)(cid:454)(cid:895) = f(cid:894)(cid:454)(cid:895) Last class - recall ftc (fundamental theorem of calculus part 1): If f is continuous on some interval [a, b] then the function (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1876) , what is the derivative of g(t): use the ftc. Mat136 lecture 5 fundamental theorem of calculus part 2. January 15, 2018 is continuous of [a, b] Consider: what if f has some jump discontinuities. ={(cid:1876) (cid:1858) (cid:1876)(cid:3410)(cid:882) (cid:882) (cid:1858) (cid:1876)(cid:3409)(cid:882) } discontinuities, what would happen to (cid:4666)(cid:1876)(cid:4667)= (cid:1858)(cid:4666)(cid:1872)(cid:4667) (cid:1872) (cid:4666)(cid:1876)(cid:4667)= (cid:4666)(cid:1876)(cid:4667) (cid:1876) (cid:3051)(cid:3028) (cid:3051) Part 1 assumes f(x) is continuous on [a, b]. If f(x) is not continuous, say, it has jump. So r(x) is not differentiable at x = 0 is continuous on [a, b] and if f is. Note: we are using ftc part 1 in greater generality. If f is integrable over [a, b], then (cid:4666)(cid:1876)(cid:4667)= (cid:1858)(cid:4666)(cid:1872)(cid:4667) (cid:1872) continuous at x0, then f is differentiable at x0 a(cid:374)d f"(cid:894)(cid:454)0) = f(x0) (cid:3051)(cid:3030)