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University of Toronto St. George

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MAT137Y1

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MAT137Y1b.doc Lecture #12 Thursday, October 16, 2003 D IFFERENTIATION (c, fc )) P f Q (c + h, (c+ h)) c c + h What is the slope of the secant line PQ ?f ( + ) f( ) = f(c+h )f c( ). (c + c h Idea: Get the slope of the tangent line as a limit of slopes of secant lines. The slope of the tangent line at x = c ought to be limf(c + ) f c ). h0 h Example f(x)= x at x = 0 This DOESNT have a well defined tangent line Definition f(c +h ) f( ) f is differentiable at x = c if the lih0 lim h exists. If it does, we call it the derivative of f at c and we denote it by fc ). Geometrically f (c) is the slope of the tangent line going throu(c, fc ). What is the equation for tangent line? f(c)= f (c)x c ) Page 1 of 38 www.notesolution.com MAT137Y1b.doc Lecture #13 Tuesday, October 21, 2003 Example 2 2 2 (2+ h ) 2 For function f ( ) x , the derivative of f atc =2 is f(2 = h0 h = h0 4+ h = )4 . 2 The derivative of f is itself a function (x) = x repeat the same calculation for any value of c. 2 2 2 2 2 f c )= lim f c + h) f c)= lim (c+ h) ()c = lim c + 2ch+ h c = lim(2c + h =)2c h0 h h0 h h0 h h0 At any fixed value of x, f(x) = 2x . Definition f x+ h) f x ) The derivative of fs a function, denoted f , and f x )= lim , if it exists. h0 h Terminology To differentiate a function is to find the derivative. Notice: The function f has to be defined in the interval, x+ ) in order for f(x) to be defined. Example Actually, even if f is continuous o(x , x+ ), it doesnt mean f(x) f (x) is defined. Consider f(x)= x ,c =0 x Theorem If f is differentiable at x, the f is continuous. Being differentiable is better than being continuous. Proof: f x+ h) f x ) Because f is differentiable ath0 h = f (x ) lim ( (x+ h )f x = lim) f(x+ h) f ( )h = f(x 0 = 0 h0 h0 h So f is continuous. D IFFERENTIATION R ULES Building Blocks If f(x)= c (a constant function), then f(x)= 0 for all x. If f (x)= x , then f(x)= 1 for all x. Page 2 of 38 www.notesolution.com MAT137Y1b.doc Theorem: Sums and Scalar Multiples Let f, g be differentiable at x and a constant. Then (f + g) and f are differentiable, then (f + g )x = ) x + g( ) ( ) (f )x(= f x ( ) Proof: (f +g x( +h )( f+ g x ) (f + g )x(= lim h0 h f x +h +)g x+ h f x) g x( ) ( ) = h0 h f(x + h )f x (g)x + h )g x ( ) = h0 h + h = f (x ) g x ( ) Example If f (x)= 10x , f x )=10 Theorem: Differences and Linear Combinations From the Sums and Scalar Multiples rule, (f g )x = ) x g( ) ( ) (1 1+ 2 +2..+ f n n x = 1) 1 x + 2f 2 x(+...+ n f n x ( ) ( ) Theorem: Product Rule If f and g are differentiable at x, then f g differentiable and(f g x = )f x (x + f ( g x ( ) ( ) Proof: (f )x(=)lim (fg x( + h)( fg x( ) h0 h = lim f x + h x +(h f x x ( ) ( ) h0 h f x + h x +(h f x x + h + f x x)+ h (f x x ) ( ) ( ) = lim h0 h f x + h )f x ( ) g x + h )g x ( ) = h0 h g x + h +)h0 f x ( ) h Because f , g aredifferentiable, they arecontinuous. = f (x g x(+)f x g (x) ( ) Theorem: Power Rule Using the Product Rule, we derive the Power Rule. For n > 0,nZ , if f( )= x , then f x( )nx n1. Proof (by induction): True for k =1 Assume (x) = kxk1. Prove for k +1: Page 3 of 38 www.notesolution.com

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