Calculus 1000A/B Lecture Notes - Lecture 5: Piecewise

• School: Western University
• Department: Calculus
• Course: Calculus 1000A/B
• Professor: Allen O'Hara

Document Summary

Ideal form is point-slope form: (cid:1877) (cid:1877)(cid:2868)=(cid:4666)(cid:1876) (cid:1876)(cid:2868)(cid:4667, need to find slope of the tangent. Ne(cid:449)to(cid:374)"s solutio(cid:374: approximate slope carefully, using a second point (x, f(x)) and secants based on given curve, as x gets closer to a, the secant slopes get closer to the tangent. Slope = (cid:3045)(cid:3046)(cid:3045) = (cid:3052)(cid:3118) (cid:3052)(cid:3117) (cid:3051)(cid:3118) (cid:3051)(cid:3117) lim(cid:3051) (cid:4666)(cid:1876)(cid:4667) = l (cid:862)the li(cid:373)it of f(cid:894)x(cid:895) as x te(cid:374)ds to a is the (cid:448)alue l(cid:863) Considers behaviour of one thing depending on the second as it approaches the third. Ex. (cid:1812)(cid:1813) (cid:2777)(cid:1814) (cid:4666)(cid:4667) lim(cid:3051) (cid:2868)sin (cid:4666)(cid:3051)(cid:4667)(cid:3051) = 1 lim(cid:3051) (cid:2868)sin (cid:4666)(cid:2869)(cid:3051)(cid:4667) = dne (does not exist) Cannot be written as a single equation. Defined by one expression on some range of values and others expressions on other ranges. Turns out, lim(cid:3051) (cid:4666)(cid:1876)(cid:4667) exists lim(cid:3051) +(cid:4666)(cid:1876)(cid:4667), lim(cid:3051) (cid:4666)(cid:1876)(cid:4667) exists and lim(cid:3051) (cid:4666)(cid:1876)(cid:4667) = lim(cid:3051) +(cid:4666)(cid:1876)(cid:4667) (cid:2779)(cid:3409) lim(cid:3051) (cid:2870)+(cid:4666)(cid:1876)(cid:4667) = -4 lim(cid:3051) (cid:2869)+(cid:4666)(cid:1876)(cid:4667) = 2 lim(cid:3051) (cid:2869) (cid:4666)(cid:1876)(cid:4667) = 4 lim(cid:3051) (cid:2869)(cid:4666)(cid:1876)(cid:4667) = dne.