MAT 21A Lecture Notes - Lecture 8: Farad, Tangent, Adverb
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MAT 21A Full Course Notes
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Mat21a - lecture 8 - 3. 1: tangents and the derivative at a point & 3. 2: the derivative as a. Slope of the graph y=f(x) at any point (a,b) (cid:1864)(cid:1865) . This is the same as the tangent line to y=f(x) at (a,b) This is called the derivative of f(x) at x=a. Tells us how fast y is changing with respect to x (cid:1006) (cid:1007) (cid:1007) (cid:1013) (cid:1006) (cid:1006) at (3, (cid:1007) Example 1: find the tangent line to = (cid:1006) (cid:4666) + (cid:4667) (cid:4666) (cid:4667) (cid:1007)+ (cid:1007)=(cid:1864)(cid:1865) (cid:1004) (cid:1007)+ (cid:1006) (cid:1007)+ (cid:1006) (cid:1864)(cid:1865) (cid:1004) (cid:1006)(cid:4666)(cid:1007)(cid:4667) (cid:1006)(cid:4666)(cid:1007)+ (cid:4667) Find a common denominator: (cid:1864)(cid:1865) (cid:1004) (cid:1007)(cid:4666)(cid:1007)+ (cid:4667) (cid:1007)(cid:4666)(cid:1007)+ (cid:4667) = (cid:1010) (cid:1010) (cid:1006) . Multiply by the reciprocal: (cid:1864)(cid:1865) (cid:1004) (cid:1006) (cid:1007)(cid:4666)(cid:1007)+ (cid:4667) (cid:1007)(cid:4666)(cid:1007)+(cid:1004)(cid:4667)= (cid:1006) Plug in h=0: (cid:1864)(cid:1865) (cid:1004) (cid:1006) Example 2: say the population of a bacterial colony is (cid:1006)+ (cid:1013) + (cid:1005), in millions, after t hours. Derivative of (cid:4666) (cid:4667)= (cid:1006)+ (cid:1013) + (cid:1005)at t=4 (cid:1864)(cid:1865) (cid:1004)