10 Derivatives ( )
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1 Instructor: Micael Medvinsky 0 Derivatives (.6-.8) Te tangent line to te curve yf() at te point (a,f(a)) is te line l m + b troug tis point wit slope Alternatively one can epress te slope as f f a m lim a a provided tat tis limit eists. ( + ) f a f a m lim Instantaneous vs. Average Velocity: If you drive a car from SLC to Las Vegas, your f ( t+ ) f ( t) average velocity is given by (note tat velocity mean speed displacement time and direction, i.e. it can be eiter negative or positive). However wen you ride te car your speedometer sows you different values of te speed\velocity during te trip, because it represents an instantaneous velocity, wic can be epressed by v a ( + ). 0 f a lim f a Rate of cange: denote cange in as and cange in y as y y y f ( ) f ( ), ten te rate of cange is given cange in y cange in y f f and instantaneous rate of cange is lim f f 0. Te derivative (.7) df Def: Te Derivative of f() at a is denoted as f '( a). f '( a ) is an d instantaneous rate of cange if yf() wit respect to wen a, i.e. f f ( a) f '( a) lim provided te limit eists. It can be of course alternatively defined a f a+ f a f a f a f ' a lim lim as and even f '( a) a ( + ) ( ) f a f a lim. E. Find an angle between f 8, g at intersection point. We first find te intersection ( , terefore 0,3, but since 8 3 < 0 te only intersection is at 0. Te angle between a line y m + band X-ais is given by
2 Instructor: Micael Medvinsky arctan m arctan y', i.e. te angle doesn t depend on b (wy?). Te angle between function f(), tat is not necessary a line, at a and X-ais is te angle of te tangent line of f() at a. Te formula of te tangent line is given by y f '( a) + b. Te angle between lines is te difference between teir angles to X-ais. Terefore we first derive f() and g(): ( a) ( 8 ) ( 8 ) f f a a a f '( a) lim lim lim lim( ) a a a ( + 4 a+ ( + 4+ a+ a a ( a)( + 4+ a+ g g a a g' lim lim 4 lim + 4 a+ 4 a 4lim 4lim 4lim ( a)( + 4+ a+ ( a)( + 4+ a+ ( + 4+ a+ 4 4 ( a+ 4 + a+ 4 ) a+ 4 a+ 4 We net evaluate it at 0: f, g. And finally π π π θ arctan arctan ( ) 4 4 Def.:If we allow te number to vary, we can redefine te derivative as a function f ' lim ( + ) f f Def.: A function f is differentiable at a if f (a) eists. It is differentiable on an open interval (a,b) (can also be alf- inf or inf) if it is differentiable at every ( ab, ) Tm: If f is differentiable at a ten f is continuous at a.. Proof: ( ) lim f lim f a + f f a lim f a + lim f f a f a + f f a f f a + lim a f a + lim lim a f a + f ' a 0 f a a a E. f > 0 0 is differentiable on R including 0 since f f 0 lim lim lim lim 0 ( + ) ( ) ( 0+ ) ( 0 ) f
3 Instructor: Micael Medvinsky 0.. Not differentiable functions: In general a function fails to be differentiable wen it discontinuous or ave a corner or kink, or ave a vertical tangent line at a i.e. lim f ' (like in, ' at 0). f f Important note, tat despite tat every differentiable is continuous, not every continuous function is differentiable E 3. f it is continuous at 0 since lim 0 f ( 0), but lim lim lim lim 0 > 0 + ( + ) lim lim lim lim( ) < 0 terefore te derivative doesn t eists. E 4. f 3 is differentiable at any but ( ) lim lim lim lim lim + + E 5. Find a necessary and sufficient condition for g a f to be differentiable at a. g g( a) a f a a f ( a) a lim lim f ( a) lim ± f ( a) L a a wrong ± a means L f a f a f a 0 only for eplanation onesided derivative Def.: A Necessary Condition for some statement S is a condition tat must be satisfied in order for S to obtain. Def.: A Sufficient Condition for some statement S is a condition tat, if satisfied, guarantees tat S obtains. E 4. For differentiability of function f(), continuity is necessary condition; owever it is not sufficient condition, since some continuous functions aren t differentiable.
4 Instructor: Micael Medvinsky E 5. For continuity of function f(), differentiability is a sufficient condition, owever it is not necessary condition, and since tere are a non-differentiable continues functions. 0.. Higer Derivatives Since te derivative of a function f() is a function f () we can derive it again (given it is differentiable). We denote following derivation as f f ' f f f '' f f f ' n n n ( n f ) f f n f n f E 6. ( ) ( n ) ( ( + ) + ( + ) ) ( + ) + ' lim ( ) ( + ) lim lim lim ( ) ( + ) ( 4( + ) + ) ( 4+ ) f ' f ' 4 + '' lim lim lim Wat does f, f say about f? (.8) Derivatives are very important ting in calculus. One can learn a lot about function f() using information about its derivatives. For eample te sign of f provides us information about direction of te function: If f '( ) > 0on some interval, ten f is increasing tere. If f '( ) < 0 on some interval, ten is decreasing tere. Furtermore, te second derivative can provide even more information:
5 Instructor: Micael Medvinsky If f ''( ) > 0 on some interval, ten f is concave upward (appy smile) tere. If f ''( ) < 0 on some interval, ten f is concave downward (conve, sad smile) tere. Def (Even function): f()f(-), i.e. it is symmetric wit respect to te y-ais, tus te grap remains uncanged after reflection about te y-ais. E 7. cos,, ^ Def (Odd function): f() -f(-) i.e. it as rotational symmetry wit respect to te origin, tus te grap remains uncanged after rotation of 80 degrees about te origin. E 8. E 9., ^3, sin Te derivative of even function is odd function and vice verse ( + ) ( ) ( ) ( + ) f f f f f t f f ( ) lim lim f t t 0 t g( + ) g( ) g( ) + g g( + t) g g ( ) lim lim g t t 0 t Antiderivatives: Some time we know f() and we need to find anoter function F() suc tat F ()f(). If suc F() eists we call it antiderivative. Since f() is derivative of F() we ave lot information about F() to work wit. We ll learn it in future.
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