1 esinx find where fx 0 fx 0 fx 0 f0xx

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Unformatted text preview: sin ( (1) ) = not defined (1 esin 1= not defined = f ( x) f (x) = e x! 1 (=) not defined lim f x = e 5. Intervals of Incr/Decr x! 1 esin(x) (find where f’(x) = 0, f’(x) > 0, f’(x) < 0) f(0xx) = cos(x)esin(x) 0 sin(x)= cos(x)esin(x) f 0 () f (x) = cos(x)e D = {x 2 R} 3 0 ⇡ ⇡3⇡ ⇡ , ⇡ 3⇡ ⇡ 3⇡ 3⇡ f 0cos(=)0==) ) x x) =),==) > x = ,2 , 2 (f )(x) = 00 = > cos(x 0 0,= x = xx cos( = , , ... =) = 22 22 22 2 (2 + 1)⇡ 0 (2k k + 1)⇡ , where k is an integer = 0 ( ) k 0 1)⇡ ff(xx(2= +=) xxwhere k2is an where k is an integer ) = 0 =) , = , integer x= 2 2 2 2 2 3⇡ ⇡ ( , ) 5⇡ , 3⇡ ) ⇡⇡ 5. Intervals of, Incr/ Decr ( ) 2 2( 2 22 sin(x) 2 f 5⇡ ) =⇡e ⇡ 3⇡ (x 3 0 3x)esin(x) ⇡ ⇡ ⇡⇡ ( , )( , f (x) = cos( , ) ( , ) ) ( 2 2 22 2 2 22 ⇡⇡ ⇡ 5⇡ 3⇡ 3⇡ , 3⇡ , 3⇡ ⇡ Intervx) = 0 ,=> ) (= , ) ( ⇡ , ) ( ⇡ , 32 ) ( 3⇡ , 5⇡ ) cos( ( x 2 2 2 2 22 22 22 22 al 3 ⇡ ⇡ 3⇡ 5⇡ 3⇡ 3⇡ ⇡ D⇡=5{x 2 R} (2k + 1)⇡ ) ( (, , )( , ) (2, 2) 2 +, where k is an integer 2 2-2 x =(x) 2+ 2 f‘ + 2 3⇡ 5⇡ ⇡ 3⇡ (,) (, ) ( 5⇡ , 3⇡ ) 22 22 f (x) Incr Decr I2ncr 2 Decr Incr 3⇡ 5⇡ (, ) ⇡ 3⇡ 22 (, ) 2 22 3⇡ 5⇡ (, ) 22 -5π/ 2 - 3π/2 - π/ 2 π/2 3 π/ 2 5 π/ 2 3⇡ 5⇡ (, ) 5. Local Max/Min 22 f ( x) = ⇡ ) = ) f (x + 2⇡ ) = esin(x+2esin(xesin(x) First Derivative Test: Local Max if f’(x) goes from + to – ⇡⇡ ( ,) sin(x+2⇡ ) sin(x) (x + 2⇡ ) = if f’(x) goes from – to + e =e 22 Local Min D ⇡ ⇡ x 3⇡ ⇡ ⇡ Neither Max or Min if f’(x) doesn’tmax at, x ) (2 3R}, ⇡ ) change ( = {= , Local 22 2 2 22 sign) 3⇡ ⇡ 5⇡ 3 ( , ) ⇡ , 3⇡⇡ ) ⇡⇡ 3⇡ ⇡ ( ,) 2 2( 2 Local min at x = ,2 Local max 2 2 = at x , 5⇡ 3⇡ ⇡ 3⇡ 22 2 3⇡ ⇡ ⇡⇡2 ( , )( , ) ( , )( ,) 2 2 22 2 2 2 23⇡ ⇡ ⇡ 3⇡ Local min5at x⇡= 3⇡ , ⇡ ⇡⇡ ⇡3 (, ) ( 3⇡ , 5⇡ ) 2 Interv ( 2 , 2 ) ( 22, 2 ) ( 2 , 2 ) 22 22 al f ‘ (x) f (x) 3⇡ 5⇡ ⇡ 3⇡ 5⇡ 3⇡ 3⇡ ⇡ (, ) ( , )( , ) (2, 2) 22 2 2 2+ 2 +⇡ -⇡ 3⇡ 5 ⇡3 (,) (, ) ( 5⇡ , 3⇡ ) 22 22 2 Incr Decr Incr2 Decr 3⇡ 5⇡ + Incr 22 ( x) = e = not defined 3⇡ 00 sin(x) ⇡ 2 sin(x)) = esin(x) f (x) minsin( 6. Concav...
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This document was uploaded on 03/04/2014 for the course MATH 1A03 at McMaster University.

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