MAT135H1 Lecture 1: Derivative Review Notes

• School: UTSG
• Department: Mathematics
• Course: MAT135H1
• Professor: N.Garcia- Fritz

Document Summary

Rationalizing denominators: recall: (cid:3028) (cid:3029)= (cid:3028) (cid:3029) (cid:3029) (cid:3029)= (cid:3028)(cid:3029)(cid:3029, (cid:1876)+ (cid:1877) (cid:1853)(cid:1871) (cid:1853) (cid:1855)(cid:1867)(cid:1866)(cid:1862)(cid:1873)(cid:1859)(cid:1853)(cid:1872)(cid:1857) (cid:1867)(cid:1858) (cid:1876) (cid:1877) If (cid:1876) (cid:882) we make (cid:1858)(cid:4666)(cid:1876)(cid:4667) as large as we like, therefore = . If (cid:1876) we make (cid:1858)(cid:4666)(cid:1876)(cid:4667) as small as we like, therefore = 0. Test by substitution, then divide by the highest degree. When we talk about a function being continuous at a point, we mean that the graph passes through the point without a break. A graph that is not continuous has a break at the point: continuous for all values of the domain: 2. Discontinuous at (cid:1876)=(cid:883) (point discontinuity: discontinuous at x = 1 (jump discontinuity) 4. Discontinuous at x = 1 (infinite discontinuity: continuity at a point: the function (cid:1858)(cid:4666)(cid:1876)(cid:4667) is continuous at (cid:1876)=(cid:1853) if (cid:1858)(cid:4666)(cid:1853)(cid:4667) is defined and if im(cid:3051) (cid:3028)(cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1858)(cid:4666)(cid:1853)(cid:4667). Otherwise, (cid:1858)(cid:4666)(cid:1876)(cid:4667) is discontinuous at (cid:1876)=(cid:1853). im(cid:3051) (cid:3028)(cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1857)(cid:1876)(cid:1861)(cid:1871)(cid:1872)(cid:1871) (cid:1861)(cid:1858) (cid:1853)(cid:1866)(cid:1856) (cid:1867)(cid:1866)(cid:1877) (cid:1861)(cid:1858) im(cid:3051) (cid:3028) = im(cid:3051) (cid:3028)+(cid:1858)(cid:4666)(cid:1876)(cid:4667) (left hand limit = right hand.