MATH 1680 Lecture Notes - Lecture 1: Differentiable Function
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Today the following will be explained: average rate of change. This is also called derivative of (cid:1858)(cid:4666)(cid:1876)(cid:4667) at (cid:1876)=: when do we say (cid:1858)(cid:4666)(cid:1876)(cid:4667) is differentiable function, tangent line of (cid:1858)(cid:4666)(cid:1876)(cid:4667) at (cid:1876)=. (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1858)(cid:4666)+ (cid:4667) (cid:1858)(cid:4666)(cid:4667) (cid:1871)(cid:1864)(cid:1867)(cid:1868)(cid:1857)=lim 0(cid:1858)(cid:4666)+ (cid:4667) (cid:1858)(cid:4666)(cid:4667) Because slope is the same as tangent line, it is the same. Finding the derivative of (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1876). (cid:1858)(cid:4666)(cid:1876)(cid:4667)=sin(cid:1876). (cid:1858)(cid:4666)(cid:1876)(cid:4667)=cos(cid:1876). (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1876). (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1876). (cid:1858)(cid:4666)(cid:1876)(cid:4667)=. We say that (cid:1858)(cid:4666)(cid:1876)(cid:4667) is differentiable function if derivative of (cid:1858)(cid:4666)(cid:1876)(cid:4667) exists for each x in the domain. (cid:1876),sin(cid:1876), make a table of derivatives special functions. Notation (cid:1858) (cid:4666)(cid:4667) is the derivative (slope of tangent line (visually)) at (cid:1876)=. Example: (cid:1858) (cid:4666)(cid:882)(cid:4667)=(cid:882) (cid:1858) (cid:4666)(cid:883)(cid:4667)=? no tangent line at (cid:883) (cid:883) * not differentiable because there is a place where tangent line doesn"t e(cid:454)ist. (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1876)2. + (cid:1876) (cid:1856)(cid:1876) is again a small change in (cid:1876). (cid:1856)(cid:1877) is an implied change on the tangent line. (cid:1856)(cid:1877)
