Embed Size (px)
Citation preview
If k(x) =If k(x) =ff (g(x)), then (g(x)), thenk’(x)k’(x) = = f ’f ’ (( g(x)g(x) )) g’(x)g’(x)k(x) = k(x) = sinsin( x( x22 ) )
k’(x)k’(x) = = cos cos ( x( x2 2 ) ) 2x2x
If y = sec(3If y = sec(3t), find y’t), find y’
A.A. 33 sec(3 sec(3t) tan(3t) tan(3t)t)
B.B. 33 sec tan (3 sec tan (3t)t)
C.C. sec(3sec(3t) tan(3t) tan(3t)t)
If y = sec(3If y = sec(3t), find y’t), find y’
A.A. 33 sec(3 sec(3t) tan(3t) tan(3t)t)
B.B. 33 sec tan (3 sec tan (3t)t)
C.C. sec(3sec(3t) tan(3t) tan(3t)t)
If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’
A.A. -sec-sec22[sin(x)]cos(x)[sin(x)]cos(x)
B.B. secsec22[sin(x)]cos(x)[sin(x)]cos(x)
C.C. secsec22[cos(x)][cos(x)]
D.D. -csc-csc22[sin(x)]cos(x)[sin(x)]cos(x)
If y=tan(sin(x)), find y’If y=tan(sin(x)), find y’
A.A. -sec-sec22[sin(x)]cos(x)[sin(x)]cos(x)
B.B. secsec22[sin(x)]cos(x)[sin(x)]cos(x)
C.C. secsec22[cos(x)][cos(x)]
D.D. -csc-csc22[sin(x)]cos(x)[sin(x)]cos(x)
CorallaryCorallary
k(x) = gk(x) = gnn(x) = [g(x)](x) = [g(x)]nn
k’(x)k’(x) = n [g = n [g (x)] (x)] n-1n-1 g’(x)g’(x)
If y=(2x+1)If y=(2x+1)44, find y’, find y’
A.A. 4(2)4(2)33
B.B. 4(2x+1)4(2x+1)33
C.C. 8(2x+1)8(2x+1)
D.D. 8(2x+1)8(2x+1)33
If y=(2x+1)If y=(2x+1)44, find y’, find y’
A.A. 4(2)4(2)33
B.B. 4(2x+1)4(2x+1)33
C.C. 8(2x+1)8(2x+1)
D.D. 8(2x+1)8(2x+1)33
If y=x cos(xIf y=x cos(x22), find ), find dy/dxdy/dx
A.A. -x sin(x-x sin(x22) + cos(x) + cos(x22))
B.B. -2x sin(x-2x sin(x22) + cos(x) + cos(x22))
C.C. -2x-2x22 sin(x sin(x22) + cos(x) + cos(x22))
D.D. 2x2x22 sin(x sin(x22) + cos(x) + cos(x22))
If y=x cos(xIf y=x cos(x22), find ), find dy/dxdy/dx
A.A. -x sin(x-x sin(x22) + cos(x) + cos(x22))
B.B. -2x sin(x-2x sin(x22) + cos(x) + cos(x22))
C.C. -2x-2x22 sin(x sin(x22) + cos(x) + cos(x22))
D.D. 2x2x22 sin(x sin(x22) + cos(x) + cos(x22))
The chain ruleThe chain ruleIf y = If y = sin(sin(uu)) and u(x) = x and u(x) = x22
then dy/dx = dy/du du/dxthen dy/dx = dy/du du/dx
dy/du = cos(dy/du = cos(uu) d) duu/dx = /dx = 2x2x
dy/dx = dy/dx = cos(cos(uu) ) 2x2x
= = cos(cos(xx22) ) 2x2x
The chain ruleThe chain ruleIf y = If y = cos(cos(uu)) and u(x) = x and u(x) = x22 + 3x + 3x
then dy/dx = dy/du du/dxthen dy/dx = dy/du du/dx
dy/du = -sin(dy/du = -sin(uu) d) duu/dx = /dx = 2x + 32x + 3
dy/dx = dy/dx = -sin(-sin(uu) ) (2x+3)(2x+3)
= = -sin(-sin(xx22+2x+2x) ) (2x+3)(2x+3)
y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx
A.A. -10 csc-10 csc22(10x-5)(10x-5)
B.B. secsec22(10)(10)
C.C. -csc-csc22(10x-5)(10x-5)
D.D. 10 sec10 sec22(10x-5)(10x-5)
y=tan(u) u = 10x – 5y=tan(u) u = 10x – 5find dy/dxfind dy/dx
A.A. -10 csc-10 csc22(10x-5)(10x-5)
B.B. secsec22(10)(10)
C.C. -csc-csc22(10x-5)(10x-5)
D.D. 10 sec10 sec22(10x-5)(10x-5)
y= uy= u22+u u = 10x+u u = 10x22 – x – xfind dy/dxfind dy/dx
A.A. (20 x(20 x22 – 2x)(20x-1) – 2x)(20x-1)
B.B. (20 x(20 x22 – 2x +1)20x – 2x +1)20x
C.C. (20 x(20 x22 - 1)(20x-1) - 1)(20x-1)
D.D. (20 x(20 x22 – 2x +1)(20x-1) – 2x +1)(20x-1)
y= uy= u22+u u = 10x+u u = 10x22 – x – xfind dy/dxfind dy/dx
A.A. (20 x(20 x22 – 2x)(20x-1) – 2x)(20x-1)
B.B. (20 x(20 x22 – 2x +1)20x – 2x +1)20x
C.C. (20 x(20 x22 - 1)(20x-1) - 1)(20x-1)
D.D. (20 x(20 x22 – 2x +1)(20x-1) – 2x +1)(20x-1)
CorallaryCorallary
k(x) = [3xk(x) = [3x3 3 - x- x-2 -2 ]]2020
k’(x)k’(x) = 20 [3x = 20 [3x3 3 - x- x-2-2] ] 1919 ( (9x9x22+2x+2x--
33))
CorallaryCorallary
y = [3xy = [3x3 3 - x- x-2 -2 ]]2020
let u = [3xlet u = [3x3 3 - x- x-2 -2 ] ]
du/dx = (du/dx = (9x9x22+2x+2x-3-3)) y=uy=u2020 dy/dx=dy/du du/dx = dy/dx=dy/du du/dx = 20u20u1919 du/dx du/dx
= 20 [3x= 20 [3x3 3 - x- x-2-2] ] 1919 ( (9x9x22+2x+2x-3-3))
If y = (sec(x))If y = (sec(x))22=sec=sec22(x) (x) find dy/dxfind dy/dx
A.A. 2 sec(x) tan(x)2 sec(x) tan(x)
B.B. 2 sec2 sec22(x) tan(x)(x) tan(x)
C.C. 2 sec(x)2 sec(x) tantan22(x)(x)
D.D. secsec22(x)(x) tantan (x)(x)
If y = (sec(x))If y = (sec(x))22=sec=sec22(x) (x) find dy/dxfind dy/dx
A.A. 2 sec(x) tan(x)2 sec(x) tan(x)
B.B. 2 sec2 sec22(x) tan(x)(x) tan(x)
C.C. 2 sec(x)2 sec(x) tantan22(x)(x)
D.D. secsec22(x)(x) tantan (x)(x)
CorallaryCorallary
=[3x=[3x3 3 - x- x2 2 ]]1/21/2
k’(x)k’(x) = ½ [3x = ½ [3x3 3 - x- x22]]-1/2-1/2 ( (9x9x22-2x-2x))
3 2( ) 3k x x x
CorallaryCorallary
k’(x)k’(x) = =
( ) ( )k x g x
1'( )
2 ( )g x
g x'( )
2 ( )
g x
g x
CorallaryCorallary
k’(x)k’(x) = =
3 2( ) 3k x x x 29x -2x3 22 3x x
If y = find If y = find dy/dxdy/dx
A.A. csccsc3/23/2(x)(x)
B.B. ..
C.C. ..
D.D. ..
csc( )x
csc( )cot( )
2 csc( )
x x
x
csc( )cot( )
2 csc( )
x x
x
csc( )cot( )
csc( )
x x
x
If y = find If y = find dy/dxdy/dx
A.A. csccsc3/23/2(x)(x)
B.B. ..
C.C. ..
D.D. ..
csc( )x
csc( )cot( )
2 csc( )
x x
x
csc( )cot( )
2 csc( )
x x
x
csc( )cot( )
csc( )
x x
x
CorallaryCorallary
= [sin(2x)= [sin(2x) ]]1/21/2
k’(x)k’(x) = ½ [sin(2x)] = ½ [sin(2x)]-1/2-1/2 ( (cos(2x) cos(2x) 22))
( ) sin(2 )k x x
k(x) = sec(sin(2x)) k(x) = sec(sin(2x))
k’(x)k’(x) = =
secsec(sin(2x))(sin(2x))tantan(sin(2x))(sin(2x))((cos(2x) 2cos(2x) 2))
y = sec(sin(2x)) let u = y = sec(sin(2x)) let u = sin(2x) sin(2x)
dy/dx = dy/du du/dx dy/dx = dy/du du/dx
y = sec u y = sec u
y = sec(u) where u = sin(2x) y = sec(u) where u = sin(2x)
dy/dx = dy/du du/dx dy/dx = dy/du du/dx
= sec u tan u cos(2x) 2= sec u tan u cos(2x) 2
y = sec(u) where u = sin(2x) y = sec(u) where u = sin(2x)
dy/dx = dy/du du/dx dy/dx = dy/du du/dx
= sec u tan u cos(2x) 2= sec u tan u cos(2x) 2
secsec(sin(2x))(sin(2x))tantan(sin(2x))(sin(2x))((cos(2x) 2cos(2x) 2))
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
after the beginning of a race is after the beginning of a race is given bygiven by
a)a)Find and explain.Find and explain.b)b)Find R’(t).Find R’(t).c)c)Find R’(10) and explain.Find R’(10) and explain.d)d)Find R(10) and explain. Find R(10) and explain.
21400 3 204( )20
t tR t
t
lim ( )tR t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
a) Find and explain.a) Find and explain.
2 21400 3 20 /
4lim( 20) /t
t t t
t t
lim ( )tR t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
a) Find and explain.a) Find and explain.
2 21400 3 20 /
4lim( 20) /t
t t t
t t
lim ( )tR t
2
1 3 20400
4lim20
(1 )t
t t
t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
a)a) Find and explain.Find and explain.
2 21400 3 20 /
4lim( 20) /t
t t t
t t
lim ( )tR t
2
1 3 20400
4lim 20020
(1 )t
t t bpm
t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
a)a) Find and explain.Find and explain.
Mary’s maximum heart rate is Mary’s maximum heart rate is 200 bpm = 220 – age 200 bpm = 220 – age making her age close to 20.making her age close to 20.
2 21400 3 20 /
4lim( 20) /t
t t t
t t
lim ( )tR t
1 3 20400
4lim 20020
(1 )t
t t bpm
t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds after the beginning of a race is after the beginning of a race is given bygiven by
a)a)..b)b)Find R’(t)Find R’(t)c)c)Find R(10) = 115.47 bpmFind R(10) = 115.47 bpmd)d)Find R’(10) and explain. Find R’(10) and explain.
21400 3 204( )20
t tR t
t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
Find R’(t)Find R’(t)
2
2
2
400( 3) 12( 20) 400 3 2041
2 3 204'( )
( 20)
t
t t t
t tR t
t
21400 3 204( )20
t tR t
t
2
2 2
( 20)(200 1200) 800( 3 20)4
12 3 20( 20)4
tt t t
t t t
Number of heart Number of heart beats per minute, t beats per minute, t seconds seconds
Find R’(t)Find R’(t)R’(10) = 2.3094R’(10) = 2.3094bpm/minbpm/min
2
2 2
( 20)(200 1200) 800( 3 20)4
12 3 20( 20)4
tt t t
t t t
quizzquizz
1.Write the equation of the 1.Write the equation of the line tangent to the graph of line tangent to the graph of
y = x – cos(x) when x=0.y = x – cos(x) when x=0.
2. Diff. g(x)=cot x [sin x – cos 2. Diff. g(x)=cot x [sin x – cos x].x].
3. Find the x’s where the lines 3. Find the x’s where the lines tangent to y= are tangent to y= are horizontal.horizontal.
sin
1 cos
x
x
![g h k b f u l o g b q k d b Z f g l f h ` g g h ] h h x a Z « [ a h i Z k g h Z j n x … · 2017. 6. 2. · d b q _ k d h f m ] e Z f _ g l m L K ” a h i Z k g h k l b Z j n x](https://img.dokumen.tips/doc/110x75/6120b0b72cd12f00fb7f7d22/g-h-k-b-f-u-l-o-g-b-q-k-d-b-z-f-g-l-f-h-g-g-h-h-h-x-a-z-a-h-i-z-k-g-h-z.jpg)
![g g h ] [ x ^ ` l g h ] g Z m q g h ] m q j ` ^ g b y · СТЕНОГРАММА A : K ? > : G B Y > B K K ? J L : P B H G G H = K H < ? L : > 001.018.01 < N = ; G G](https://img.dokumen.tips/doc/110x75/5f026c837e708231d4043396/g-g-h-x-l-g-h-g-z-m-q-g-h-m-q-j-g-b-y-oeoe-a-.jpg)
![2D Convolution/Multiplication Application of Convolution Thm. · 2015. 10. 19. · Convolution F[g(x,y)**h(x,y)]=G(k x,k y)H(k x,k y) Multiplication F[g(x,y)h(x,y)]=G(k x,k y)**H(k](https://img.dokumen.tips/doc/110x75/6116b55ae7aa286d6958e024/2d-convolutionmultiplication-application-of-convolution-thm-2015-10-19-convolution.jpg)
![Y J ` g N h g a k R ^ a k x x x x x x x x x x x x x C x x ...€¦ · F Z j ] Z j y g ? = < _ q g u c f b j \ f b n _ b w i h k _ i b s Z ] _ j h _ \ d h k l b \ h e h k u [ h j h](https://img.dokumen.tips/doc/110x75/60615d7d1e06256081647567/y-j-g-n-h-g-a-k-r-a-k-x-x-x-x-x-x-x-x-x-x-x-x-x-c-x-x-f-z-j-z-j-y-g-.jpg)
![> 1 : = G H K L B D : 1 G > B < 1 > M : E V G H I K B O H ... · 17 : g Z e h ] q g h a Z ^ h i h f h ] h x k i _ p Z e v g b o d e x q \ a ^ c k g x } l v k y](https://img.dokumen.tips/doc/110x75/5f1050797e708231d4487f30/-1-g-h-k-l-b-d-1-g-b-1-m-e-v-g-h-i-k-b-o-h-17-g.jpg)
![Concurso 17 18 EDITAL COMPLETO2[1] · , a + * s i x i k s p b a b o d c j b " $ 1 " > $ # $ : b g i b a f x l x k x y g i b a h g f e k c b a b u h g e g p g h b a l 2 k j > i $ $](https://img.dokumen.tips/doc/110x75/5e3c521992f290383e2a93e6/concurso-17-18-edital-completo21-a-s-i-x-i-k-s-p-b-a-b-o-d-c-j-b-.jpg)
