Sample Calculus problems

7/24/2019 Sample Calculus problems

1/141

Last revision: September 3, 2015


2/141


3/141

limx5

4

3x + 1

x2 7x + 10

limx5

4

3x + 1

x2 7x + 10 =lim

x5

(4 3x + 1)(4 +3x + 1)

(x2 7x + 10

)(4 +

3x + 1

)=limx516

(3x + 1

)(x2 7x + 10)(4 +3x + 1)=lim

x5

15 3x(x 5)(x 2)(4 +3x + 1)=lim

x5

3(5 x)(x 5)(x 2)(4 +3x + 1)=lim

x5

3(x 2)(4 +3x + 1)=

3

(5 2

)(4 +

3 5 + 1

)= 33 8=

1

8

=

=

limx5

4

3x + 1

x2 7x + 10 lim

x5

(4 3x + 1)(4 +3x + 1)(x2 7x + 10)(4 +3x + 1)limx5

15 3x(x 5)(x 2)(4 +3x + 1)limx5

3(5 x)(

x 5

)(x 2

)(4 +

3x + 1

)


4/141

=

limx5

4

3x + 1

x2 7x + 10 lim

x5

(4 3x + 1)(4 +3x + 1)(x2 7x + 10)(4 +3x + 1) lim

x5

15 3x

(x 5)(x 2)(4 +3x + 1) lim

x5

3(5 x)(x 5)(x 2)(4 +3x + 1)

=limx5

(4

3x + 1

)(4 +

3x + 1

)(x2 7x + 10

)(4 +

3x + 1

)

limx5

(4 3x + 1)(4 +3x + 1)(x2 7x + 10)(4 +3x + 1)

x2 1

x 1 =x + 1 x=1

x

x2 =4

limx5

4

3x + 1

x2 7x + 10 =(4 3x + 1)(4 +3x + 1)(x2 7x + 10)(4 +3x + 1)

=3

(x 2

)(4 +

3x + 1

)=3

(5 2)(4 +3 5 + 1) =

3

3 8

= 1

8

3(x 2)(4 +3x + 1)

1

8

x = 1

3

(x 2

)(4 +

3x + 1

) =

3

(1 2)(4 +3 1 + 1) =1

2= 1

8


5/141

m

y =x2

x + 2

(3,9)

m

m

m

y =f(x)

(x0, f

(x0

))

m = limxx0

f(x) f(x0)x x0

m = limh0

f(x0 + h) f(x0)h

.

m = limx3

x2

x+

2

(9

)x (3) =limh0 (3 + h

)2

3+

h+

2

(9

)h .

limxc

f(x) =0 limxc

g(x) = 0 limxc

f(x)g(x)

limxc

f

(x

)g

(x

)

L =limxc

f

(x

)g

(x

)

limxc

f(x) = limxcf(x)

g(x) g(x) = limxc f(x)g(x) limxc g(x) = L 0 = 0.

limxc

f(x) =0

limxc

f(x)g(x)

limxc

f(x) = 0 K g(x) K x=c c limxc (f(x)g(x)) = 0

f(x)g(x) =f(x) g(x) f(x) K

Kf(x) f(x)g(x) Kf(x) .

limxcf(x) =0


6/141

limx0+

f(x) =A limx0

f(x) = B

limx0

f(x2 x) limx0

f(x2) f(x) limx0+

f(x3 x)

limx0

f

(x3

) f

(x

)

limx1

f

(x2 x

) x 0 x >0 x2 x >0 x 0

x < 0

x2

x

limx0

f(x2) f(x) = limx0

f(x2) limx0

f(x) =A B

0


7/141

x2 + y2 =r2 (x 1)2 + y2 =1 x =r22

x2 + y2 =r2

Q(r22,r2 r44)

R(a, 0)

R

S

Q

x

RSQ

ROP

a

r

2

2r2 r44 = ar

a =r32

r

r2 r44 .

limr0+

a = limr0+

r32r

r2 r4

4

= limr0+ r32r2 (r2 r44) (r +r2 r44)

=2 limr0+

(1 +1 r24)=2 (1 +1 024) = 4 .

R

(4, 0)

r 0+

lim

x12

1

x

=2

> 0 >0

0


8/141

2 = 0

=2

2 2

2 x .


9/141

x

0


10/141

(x4 + 7x 17) 43 = x4 + 7x 60 =(x + 3)(x3 3x2 + 9x 20)

0


11/141

c >0 m 1m

c

c

1n

n

=c 1m >0

> 0

0 0

0


12/141

1

x2

4 =cos x

f(x) =1 x24 cos x

1

x2

4 =cos x

f

f(0) = 0

x = 0

f(2) = 1 216 >0 f() =2 24 >3 f

f

[

2,

]

c

f(c) = 0

f

f(c) = f(c) = 0 x = c

45

= 4

= 74

= 154 +

T()

T( + 2) = T() T

c

T(c + ) =T(c)

f() =T( + ) T() T f

c

T

(c +

)=T

(c

)

c

f

(c

)=0

f

(0

)=0

c =0

f(0) =0 f(0) = T()T(0) = T()T(2) = f() f(0)

f()

f

[0, ]

c [0, ]

f(c) = 0


13/141

y =x3 (2, 4)

dydx = d(x3)dx = 3x2

(x0, x30) y x30 = 3x20(x x0) (2, 4)

4 x3

0 = 3x2

0(2 x0) x30 3x20 +2 = 0

x0 =1

x303x2

0+2 =

(x01)(

x202x02)

x0 =1

3

y =x3

(1, 1)

(1 +3, 10 + 63) (1 3, 10 63)

(2, 4)

y =3x 2

y =(12+ 63)x (20+ 123) y =(12 63)x (20 123)

limx0

1 + sin2 x2 cos3 x2

x3 tan x

limx01 + sin2 x2 cos3 x2x3 tan x =limx01 + sin2 x2 1x4 + 1 cos3 x2x4 xtan x

=limx0sin x2

x22 1

1 + sin2 x2 + 1+

1 cos3 x2

x4 x

tan x

limx0

sin x2

x2 =1

limx0

1

1 + sin2 x2 + 1

=1

1 + sin2 02 + 1

=1

2

limx0

xtan x

=1


14/141

limx0

1 cos3 x2

x4 =lim

x01 cos x2

x4 (1 + cos x2 + cos2 x2)

=limx0

2sin2(x22)

x4

(1 + cos x2 + cos2 x2

)= 12 limx0 sin(x22)x222 limx0(1 + cos x2 + cos2 x2)=

1

2 12 3

=3

2 .

limx0

1 + sin2 x2 cos3 x2

x3 tan x =

1

1

2+

3

2

1 = 2 .

y = sin2(x36)

x = 1

y = sin2(x36) dy

dx = 2sin(x36) cos(x36) 3x26

dy

dx

x=1

=2 sin

(

6

) cos

(

6

)

2 =

3

4

yx=1 =14

y 1

4 =

3

4(x 1)

y =

3

4 x +

1

3

4 .

f(x) =

2x + x2 sin1x

x=0,

0

x = 0.

f(x)

x

f

0


15/141

x=0

f(x) = ddx2x + x2 sin(1x) =2 + 2x sin(1x) + x2 cos(1x) (1x2)

x

=0

x =0

f(0) =limh0

f(0 + h) f(0)h

=limh0

2h + h2 sin(1h)h

=limh0

2 + limh0

h sin1h = 2 + 0 = 2

limh0

hsin(1h) = 0

sin(1h) 1

h=0

h sin

(1

h

)=

h

sin

(1

h

)

h

h

=0

h h sin1h h

h=0 .

limh0h =0 = lim

h0(h)

limh0

hsin(1h) = 0

f

(x

)=

2 + 2x sin

1

x

cos

1

x

x

=0,

2

x = 0.

limx0

f(x)

limx0

2 =2

limx0

2xsin(1x) = 0

limx0

cos(1x)

limx0

f(x)

f

0


16/141

d2y

dx2(x,y)=(2,1)

y

x

x3+2y3 =5xy

x3 + 2y3 =5xyddx

3x2 + 6y2dy

dx =5y + 5x

dy

dx ()

x = 2, y = 1

12 + 6dy

dx

=5 + 10dy

dx

4dy

dx =7

dy

dx =

7

4

(x, y) =(2, 1)

(

)

x


17/141

3x2 + 6y2dy

dx =5y + 5x

dy

dxddx

6x + 12y dy

dx2

+ 6y2d2y

dx2 =5

dy

dx+ 5

dy

dx+ 5x

d2y

dx2x =2, y =1,

dy

dx=

7

4

12 + 12742 + 6d2y

dx2 =10

7

4+ 10

d2y

dx2

d2y

dx2 =

125

16

(x, y) =(2, 1)

y

() y = 5y 3x2

6y2 5x

x

y

P(x, y) Q(a, 0)

xy

P

Q

x

OP

x

P(x, y)

Q(a,0)

x

y

x2 + y2 = 25

14

55

a =11 da

dt = 1200

d

dt =?


18/141

x2 + y2 =52

(x a)2 + y2 =142 .

a =11

x2 + y2 =52 (x 11)2 + y2 =142

22x112 =52142

x

x = 2511

y =20611

t

xdx

dt + y

dy

dt =0

(x a

)

dx

dt

da

dt + y

dy

dt

=0 .

a =11

da

dt =v = 1200 x = 2511

y =20

611

5dx

dt + 4

6dy

dt =0

146dx

dt + 20

6

dy

dt = 146v .

121dxdt

= 146 v dxdt

=146121

v

dy

dt =

365

242

6v

d

dt

tan =yx

sec2 d

dt =

xdy

dt y

dx

dtx2

sec2 =1 + tan2 =1 + (yx)2

d

dt =

xdy

dt y

dx

dtx2 + y2

.

x =

25

11

y =20

6

11

dx

dt =

146

121v

dy

dt =

365

242

6v

d

dt =

73

1106v =

1460

6

11


19/141

1460

6

11

1460

6

11

60

2

x2 +52

2 5 x cos =142 t

ddt

= 5cos

115sin

v .

x = 11

cos = 511

sin = 4

611

3

a

V

V = a3

t

dV

dt =3a2

da

dt

dV

dt = 100 3 a = 5

da

dt =

4

3

4

3

r

h

V

V =

3r2h dV =

2

3 rhdr +

3r2dh

dV

V =2

dr

r +

dh

h .

r

dr

r 1% dh

h 2%

dV

V =2dr

r +

dh

h 2 dr

r + dh

h 2 1% + 2% =4% .


20/141

r

h

L

y

V0

V0 = 72 L

3r2h = 72 L

32

5h2h =49L 4

75h3

rh =25

t

0 =d

dtV0 =49

dL

dt

4

25h2

dh

dt .

h = 5

49dL

dt =4

dh

dt .

h =L y dh

dt =

dL

dt

dy

dt

dh

dt =

dL

dt + 3

dydt = 3

dL

dt =4

15


21/141

3

3 4

13

4

5

4

r h

V

rh =25

V =

3r2h =

32

5h2h = 4

75h3

dV

dt =

4

25h2

dh

dt

dV

dt = 3

3

h = 4

dhdt dhdt = 75(64) 75(64)

dV

dt = k

h

k

kh = 425h2dhdt


22/141

dhdt = 13 h = 5

k

5 =4

2552 1

3

k4 = 42542dhdt

h = 4

dhdt

dhdt =5

524

5

524

4

25h32 dh = k dt

8

125h52 = kt + C

C

h52 = at + b

a

b

h =5

t =0

b = 552

h = 4

t =3

452 = a 3 + 552

a =(552 452)3

h =0

t =ba =3 552(552 452)

3(1 (45)52)


23/141

f(x) = x43xx13

[1, 6]

f

[a, b]

f

f (a, b)

a

b

f

f

[a, b]

f

(x

)=

4

3x131

1

3x23

x = 0

x =0

f(x) = 0

4

3x131

1

3x23 =0

z=x13

4z33z2

1 =0 z=1 4z33z21 =(z1)(4z2+z+1)

z = 1

x =z3 =13 =1

[1, 6]

f(0) = 0

f(1) = 1

f(1) = 3

f(6) = 5 613 6

5 613 6 >3 5 613 >9 53 6 > 93 725 > 721

f(x) = x43xx13 [1, 6] 5 613 6 1

f(x) = x + 1x2 + x + 9

[ 0,)

f(x) = x2 + 2x 8(x2 + x + 9)2 f(x) =0 x = 4

x = 2

x = 2

[ 0,)

0 2

f(0) =1

9

f(2) =1

5

limxf(x) = 0 1

5 >

1

9 >0

1

5


24/141

f

f(x) = f(x2) x f(0) = 1

f(x0) = 0 x0 >0 x1 0


25/141

y =5x23 2x53

y =

10

3x13

10

3x23 =

10

3x13(1 x) y > 0 (0, 1)

y 0 (,12) y 1

3 3

2 >3

x = 0

limx0+

y = limx0+

103

x13(1 x) =

limx0

y = limx0

103

x13(1 x) = . (

0, 0

)

x

y =0 5x23 2x53 =0 2x23(5 2x) = 0 x =0

x =52


26/141


27/141

L

x

y

L

3x = y2

L =(x2 + 9)12 + (4 + y2)12 =(x2 + 9)12 + (4 + (6x)2)12 .

L =(x2 + 9)12 1 + 2x 0 < x < .

dL

dx =(x2 + 9)12 x 1 + 2

x + (x2 + 9)12 2

x2 =0

x + 2 2 + 18x2 =0 x = 1813

L =(223 + 323)32

limx0+

L = limx

L =

(223 + 323)32

(223 + 323)32

7.02


28/141

S

S=2r2+2rh

h =

(1 r2

)12

S=

2r

2+

2r(1 r2

)1

2

0

r

1 .

r

h1

1

2

dS

dr = 2r +(1 r2)12 r2(1 r2)12 = 0

2r(1 r2)12 = 2r2 1 4r2(1 r2) =(2r2 1)2 8r4 8r2 +1 = 0 r2 =

8

32

16

2r(1 r2)12 = 2r2 1 r2 12

r2 =2 +

2

4 r =

2 +

2

2

S=(1+2)

r =

2 +

2

2

S=0

r =0

S=2 r =1

2 >1 1 +

2 >2

(1 +2)


29/141

ABCD

P

AB

C

Q

BC R

CD

S

R

AB

L

QR

x =C Q

x

x

L

A B

CD

S P

Q

R

P Q

P B =RP

RS

P BQ

RSP

RP = 20x

x2

(20 x

)2 = 20x

40x 400

L2 = RQ2 = RP2 + P Q2 =

400x2(40x 400) + x2 = x3(x 10) x

x

RP = 30

20x40x 400 = 30

x = 45 15

5

L

L2 =x3(x 10) 45 155 x 20

2LdLdx = 3x2(x 10) x3(x 10)2

dLdx = 0

x =15

x =15

L =15

3

x=

20

L=

202 x =45155 L =1518 65

15

3

w w 2

23

3

3w

4

w =210

=297

w =8.5

=11


30/141

a

b

K

K = a

b1

a

b(a + b

) 0 a


31/141

a =5

5 +

17,5

5

17, 5

5 +

17, 5

5

17 .

5

2

0


32/141

4

3

3

b3 =1 a3

K =

a2(1 2a3)1 a3

0 a

13

2,

0

K =a2 2a5

1 a3

0 a 13

2

dK

da =

(2a 10a4

)(1 a3

)

(a2 2a5

)(3a2

)(1 a3

)2

,

dKda =0 4a7 9a4 + 2a =0 4a7 9a4 + 2a =a(a3 2)(4a3 1)

a =0

a = 3

2

a =1 34

a =1 34

K = 1(3 32) > 0

a = 1 34 K = 0

a = 0

a =13

2

a = 1

3

4

b = 3

3

4

h

8

3

9

3

12

3

h

h 2h33

9 3

12 3

8 3

h1


33/141

=

12

(

) + 8 (

)+

9

(

)=12

2

3 h3

+

8

(1 h2)h +9

2

3 13

2

3 h3

(1 h2)h=6 h + 3h3

=6 h + 3h3

0 h 1.

d(

)dh = 1+9h2

h =1

3

h = 1

3

[0, 1

]

52

9

h = 0

=6

h = 1

=8

h = 13

h =1

r

r

h

200

2

r

200

2= =2 (r + 2r)h + 2

1

2r2

+1

2 2r rtop


34/141

h =

3100

r r 0


35/141

d2y

dx2(x,y)=(0,0)

y

x

x+y

0

et2

dt = xy

x

x+y

0

et2

dt = xyddx

d

dx

x+y

0

et2

dt =d

dx(xy)

e(x+y)2 d

dx

(x + y

)=

d

dx

(xy

)e(x+y)21 + dy

dx = y + x dy

dx ()

x =0, y = 0

dy

dx = 1 (x, y) =(0, 0)

x

e(x+y)21 + dydx =y + x dy

dxddx

d

dxe(x+y)21 + dy

dx = d

dxy + x dy

dx

e(x+y)2

(2

(x + y

))1 +

dy

dx2

+ e(x+y)2d2y

dx2 =

dy

dx+

dy

dx+ x

d2y

dx2x = 0, y = 0, dydx = 1

d2y

dx2 = 2

(x, y) =(0, 0)


36/141

f

f(x) = x x0

f(t)dt + x3

x

c

f

(c

)=1

f

(c

)

c

d

dxf(x) = x

0

f(t)dt + xddx

x

0

f(t)dt + 3x3 = x0

f(t)dt + xf(x) + 3x2

x = c

f(c) = c c0

f(t)dt + c3

f(c) = 1

f

(c

)=

c

0

f

(t

)dt + cf

(c

)+ 3c2 =

f(c) c3c

+ c + 3c2 =1

c+ c + 2c2

f(x) = 2x(ex22 1)

limx0

x0 sin(xt3)dtx5

t = x13u

dt =x13du

x

0

sin(xt3)dt = x13 x430

sin(u3)du

limx0

x0 sin(xt3)dtx5

=limx0

x13 x43

0 sin(u3)dux5

=limx0

x43

0 sin(u3)du

x163

= lim

x0

d

dx x43

0

sin(u3

)du163 x133

= lim

x0

sin(x4) 43 x13163 x133

=limx0

sin(x4) 43 x13163 x133

=1

4limx0

sin(x4)x4

=1

4 1 =

1

4


37/141

f

f(0) = 2 f(0) =11 f(0) =

8

f(2) = 5

f(2) = 3

f(2) = 7

g(x) = 1x

x

0

f(t)dt

x = 2

g

x = 2

d

dxg(x) = d

dx1

x

x

0

f(t)dt

=

1

x2

x

0

f(t)dt + 1x

d

dx

x

0

f(t)dt=

1

x2

x

0

f

(t

)dt +

1

xf

(x

)

g(2) = 0

14 20

f(t)dt + 12 f(2) =0

2

0

f(t)dt = 2f(2) = 10

d2

dx2g

(x

)=

d

dx

1

x2

x

0

f

(t

)dt +

1

xf

(x

) = 2x3

x

0

f(t)dt 1x2

d

dx

x

0

f(t)dt 1x2

f(x) + 1x

f(x)=

2

x3

x

0

f(t)dt 1x2

f(x) 1x2

f(x) + 1x

f(x)

x =2

g(2) = 14

2

0

f(t)dt 12

f(2) + 12

f(2) = 32


38/141

A(c) = c0

f(t)dt

B(c) = 5c

f(t)dt ,

c

A

(c

) = f

(c

)

B

(c

) = f

(c

)

A

(3

)= f

(3

)

B

(3

)= f

(3

)

d

dcR(c) = d

dc

A(c)B(c) = A(c)B(c) A(c)B(c)B(c)2

c =3

7 =d

dcR(c)

c=3

=A(3)B(3) A(3)B(3)

B(3)2 = f(3)B(3) + A(3)f(3)B(3)2 = f(3)2

f

(3

)=14

f

g(x) = 11

f(t) x tdt .

g(x)

f(x) 1 < x


39/141

limn

n 1(2n + 1)2 + 1(2n + 3)2 + + 1(4n 1)2

f(x) = 1x2

[2, 4]

n

2

n

xk =2 +2k

n 0 k n

ck =2 +2k 1

n

1 k n

n

k=1

f(ck)xk = nk=1

f2 + 2k 1n 2

n =

n

k=1

2n(2n + 2k 1)2

limn

n

k=1

2n

(2n + 2k 1

)2 =

4

2

dx

x2 =

1

x

4

2

= 1

4+

1

2 =

1

4 .

limn

n 1(2n + 1)2 + 1(2n + 3)2 + + 1(4n 1)2 = 18 .

p

(x

)

x

2

N


40/141

p(x) = 3x

x

10 2x

x+x x x x

x+x

4

(10 2x

)p

(x

) x

N

N n

i=1

4 (10 2xi) p(xi ) xi

0 =x0


41/141

x sin(x2) cos(x2)dx

1

0 x1 x dx

u =sin(x2)

du =2x cos(x2)dx

x sin(x2) cos(x2)dx = 12 u du = 1

2

u2

2 + C=

1

4 sin2(x2) + C .

u = 1 x

du = dx

x = 0 u =1

x =1 u = 0

1

0 x1 x dx = 0

1 (1 u)u12 (du) = 1

0 (u12 u32)du=u32

32 u525210

=2

3

2

5 =

4

15

u =cos(x2)

x sin(x

2

) cos(x2

)dx = 1

4 cos2(x

2

) + C;


42/141

sin(x2) cos(x2) = 1

2 sin(2x2)

u = sin(2x2)

x sin(x2) cos(x2)dx = 18

cos(2x2) + C .

x sin(x2) cos(x2)dx = 14 sin2(x2) + C1 x sin(x2) cos(x2)dx = 1

4 cos2(x2) + C2

x sin(x2) cos(x2)dx = 18

cos(2x2) + C3

C2 =C1 +1

4

C3 =C1 +1

8

a

0

f

(x

)f(x) + f(a x) dx =a

2 [0, a]

I=

a

0

f(x)f(x) + f(a x) dx

u =a x

du = dx

x =0 u = a

x = a u = 0

I= a

0

f(x)f(x) + f(a x) dx = 0a f(a u)f(a u) + f(u)(du) = a0 f(a x)f(x) + f(a x) dx .

2I= a

0

f(x)f(x) + f(a x) dx + a0 f(a x)f(x) + f(a x) dx

= a

0

f(x) + f(a x)f(x) + f(a x) dx

= a

0

dx =a

I=a

2


43/141

x

y

a

y = f(x)(f(x) +f(a x))

(a2, 12)

a

I

I=a

2

R

y =x x2

x

V

R

x

V

V

x

x =0

x =1

V = 1

0

R(x)2 dx = 10

(x x2)2 dx

y

V =2 dc ( )( )dy


44/141

x

c = 0 d = 14 y

x

y

x

y =xx2 x x =

1 +

1 4y

2

x =

1

1 4y

2

V =2 14

0

y1 +1 4y2

1

1 4y

2 dy .


45/141

R

y2 =x2 x4 V

R

x

W

R

y

V

W

V

W

V

R

x

x

V

x2 x4

V =2

1

0 ( )2 dx = 2

1

0 (x2 x4)

2 dx

W

R

y

y

W

y2 =x2x4 x

(x2

)2x2+y2 =0

x2 =(11 4y2

)2


46/141

x =

(1 +1 4y2)2 x =(1 1 4y2)2

W =2 1

2

0 ( )2 ( )2dy=2

12

0

1 +1 4y2

2 2

1 1 4y22 2dy

y

W

x2 x4

W =2 2 1

0

(

)(

)dx =2 2 10

x

x2 x4 dx

x

V

(1 +

1 4y2

)2

(1

1 4y2

)2

V =2 2 12

0

(

)(

)dy=2 2

12

0

y 1 +1 4y2

2

1 1 4y22 dy

V

V =2 1

0 (x2 x4)dx =2 x33 x5510 = 415


47/141

W

W =2 12

0

1 +1 4y22

1

1 4y2

2 dy

=2 12

0

1 4y2 dy

=2 2

0

cos 12

cos d

= 2

0

cos2 d

= 2

0

1 + cos2

2 d

= 2+

sin2

42

0

=2

4

y =1

2sin

dy =1

2cos d

a

0 a


48/141

V(h)

h

V(h) = h

0 g(y)2 dy

t

dV

dt =g(h)2dh

dt

1

dVdt = k1h k1 2 dhdt = k2 k2 g

(h

) = k h14

k

a = g

(0

)= 0

g

(5

)= 3

k =3

514

g

(h

)= 3h14

514

V(5) = 50

3514

y142dy = 9512

y32

32 50

=30

3


49/141

V

x4 + y4 =1

x = 52

V

V

1

0

u34(1 u)14 du

V

A = 1

0

u34(1 u)14 du

x (52) (1 x4)14 ((1 x4)14)

V =2

1

1 (

)

(

)dx = 2

1

1 x +

5

2

(1 x4

)14

(

(1 x4

)14

dx

52+(1y4)14

52(1y4)14

V = 1

1

(

)2 (

)2dy=

1

1

5

2+

(1 y4

)14

2

5

2

(1 y4

)14

2

dy

0


50/141


51/141

f(x) =(x2 3)ex

[2, 2]

f(x) = 2xex + (x2 3)ex =(x2 + 2x 3)ex

f(x) = 0

(x2 + 2x 3

)ex =0 x2 + 2x 3 =0 x = 1,3 .

3

[2, 2]

x = 1

f

x =1 2 2

f(1) = 2e

f(2) = e2

f(2) = e2

e >1

e2 >e2 > 2e

e2

2e

x1x

y =x1x ln y =ln x

x x

1

y

dy

dx =

d

dxln y =

d

dx

ln x

x =

1

x x ln x 1

x2 =

1 ln x

x2

d

dxx1x =x1x

1 ln x

x2 .

1 ln x >0

0


52/141

n

k=1

2kn =2(n+1)n 21n

21n 1 =

21n

21n 1 ,

limn 1n nk=1 2kn = limn 21nn(21n 1) = limt0+ t2t2t 1

= lim

t0+

2t + t ln 2 2t

ln 2 2t =

1ln 2

.

limx0sin x

x1x2

limx0

sin x

x =1 lim

x0

1

x2 =

1

y =sin x

x1x2

ln y =

lnsin xx

x2 x 0

0

0

limx0

ln y =limx0

ln(sin x) ln xx2

= lim

x0

cos x

sin x

1

x2x

=limx0

x cos x sin x

2x2 sin x

= lim

x0

cos x x sin x cos x

4x sin x + 2x2 cos x =lim

x0

sin x

4sin x + 2x cos x

=limx0

sin x

x

4sin x

x + 2cos x

=1

4 + 2 =

1

6

limx

0sin x

x1x2

=limx

0y =lim

x0

elny =e16


53/141

limx0

cos(2x) e2x2sin4 x

limx0

cos(2x) e2x2sin4 x

=limx0cos(2x) e2x2

x4

x4

sin4 x

=limx0

cos(2x) e2x2x4

limx0

x

sin x4

=limx0

cos(2x) e2x2x4

= lim

x0

2sin(2x) + 4xe2x24x3

= limx0

4cos

(2x

)+ 4e2x

2

16x2e2x2

12x2

= lim

x0

8sin(2x) 48xe2x2 + 64x3e2x224x

=limx0sin(2x)

3x 2e2x

2

+8

3x2e2x

2=

2

3 2 + 0 =

4

3


54/141

limx0

cos(2x) e2x2sin4 x

=limx0

1 (2x)2

2! +(2x)4

4! (2x)6

6! +

1 +

(2x2

)+(2x2)2

2! +

(2x2)33!

+

x x3

3! + 4

=limx0

4

3x4 +

56

45x6 +

x4 2

3x6 +

=limx0

4

3+

56

45x2 +

1 2

3x2 +

= 4

3

a

limx0

sin(x + ax3) xx5

a

limx0

sin(x + ax3) xx5

= lim

x0

cos

(x + ax3

)(1 + 3ax2

) 1

5x4

= lim

x0

sin(x + ax3)(1 + 3ax2)2 + cos(x + ax3)(6ax)20x3

= lim

x0

cos(x + ax3)(1 + 3ax2)3 sin(x + ax3) 3(1 + 3ax2)(6ax) + cos(x + ax3)(6a)60x2

.

1+6a

x 0

a = 16

a = 16

limx0

sin(x + x36) xx5

=limx0

cos(x + x3

6)(1 + x2

2)3 sin(x + x

3

6) 3(1 + x2

2)x + cos(x + x3

6)60x2

limx0

cos(x + x36)(1 (1 + x22)3)60x2

= limx0

cos(x + x36)(32 + 3x24 + x48)60

= 1

40

limx0

sin

(x + x3

6

) 3

(1 + x2

2

)x

60x2

= limx0

sin(x + x36)x + x36 limx0 (1 + x22)(1 + x26)20 = 120 .


55/141

limx0

sin(x + x36) xx5

= 3

40 .

sin(x + ax3) =(x + ax3) (x + ax3)33!

+(x + ax3)5

5!

=x + a 16 x3 + 1

120

1

2ax5 +

a = 16

340

b >a >0

(x b

)2 + y2 =a2 y

=2

d

cx

1 + dxdy2 dy

y

x = b +a2 y2 x = b

a2 y2

dx

dy =

y

a2 y2

1 +

dx

dy

2

=a

a2 y2

.

=2 a

a(b +a2 y2) a

a2 y2dy

+ 2 a

a(b a2 y2) a

a2 y2dy

=4ab a

a

1a2 y2

dy

=4ab arcsin

y

aa

a

=4ab (arcsin1 arcsin(1)) = 4ab 2

2 =42ab


56/141

y(1)

dy

dx =xy2

y(0) = 1

dy

dx =xy2

dy

y2 =x dx dy

y2 = x dx 1

y =

1

2x2 + C .

y(0) =1 1 = C

y =

2

2 x2

y(1) = 2

y(2)

Q

12

t = 0

625

t =3

Q

t = 15

dQdt = kQ2 k dQQ2 =

k dt

1Q = kt + C

C

t =0

1

(1

2

)= 1

Q

(0

)= C

C= 2

t =3 1(625) = 1Q(3) = k 3 2 k =1318

Q = 18(13t + 36)

Q(15) =677

f

f

(0

)= 4

f

(1

)=3

f

(0

)=5

f

(1

)= 7

f

(0

)=8

f

(1

)=11

1

0

f(x)f(x)dx 1

u =f(x) dv =f(x)dx

du = f(x)dx

v =f(x)

1

0

f(x)f(x)dx =f(x)f(x)10

1

0

(f(x))2 dx

f(1)f(1) f(0)f(0) =3 7 4 5 =1 (f(x))2 0


57/141

1

0

f(x)f(x)dx


58/141

u =sin2 x du =2 sin x cos x dx =sin 2x dx

sin3 xsin2x dx = u32 du=

2

5u52 + C

=

2

5 sin5

x + C

sin3 x =(3sin xsin3x)4

sin3 xsin2x dx = 14(3sin x sin3x)sin2x dx

=1

4(3sin x sin2x sin3x sin2x)dx

=1

8

3

(cos x cos3x

)

(cos x cos5x

)dx

= 18 (cos5x 3cos3x + 2cos x)dx

=1

40 sin5x

1

8 sin3x +

1

4 sin x + C

sin =(ei ei)(2i)

sin3 xsin2x dx = eix eix2i 3 e2ix e2ix

2i dx

=1

16(

e5ix + e5ix 3e3ix 3e3ix + 2eix + 2eix

)dx

=1

16e5ix e5ix

5i

e3ix e3ix

i + 2

eix eix

i + C

=1

40 sin5x

1

8 sin3x +

1

4 sin x + C

e

1

ln x

x

dx = e

1

ln x d

(2

x

)=2xln xe1 e1 2xdxx=2

e 2 e

1

dxx

=2

e 4 xe1

=2

e 4

e + 4

=4 2

e


59/141

e

1

ln xx

dx = e

1

1x

d(xln x x)=

xln x x

x

e

1

+1

2

e

1

xln x x

x32 dx

=1 + 12

e

1

ln xx dx xe1=2

e +

1

2

e

1

ln xx

dx

1

2

e

1

ln xx

dx = 2

e

e

1

ln x

x dx = 4 2

e .

x =eu

dx = eu du

e

1

ln xx

dx = 1

0

ueu

eu du

= 1

0

u eu2 du

= 1

0

u d

(2eu2

)=2ueu21

0 2

1

0 eu

2

du

=2e12 4eu210

=2e12 4e12 + 4=4 2

e

ex dx

1 x2 dx

dx(x2 + 1)2

x =t2

dx = 2tdt

ex dx =2 tet dt


60/141

u = t,dv =et dt du = dt, v =et

=2tet 2 et dt=2tet 2et + C

=2

xe

x 2e

x+ C

x =sin

2

2

dx = cos d

1 x2 =

1 sin2 =

cos2 = cos = cos cos 0

2

2

1 x2 dx = cos cos d= cos2 d=

1

2 (1 + cos2)d=

1

2 +

1

4 sin2 + C

=1

2 +

1

2 sin cos + C

=1

2arcsin x +

1

2x

1 x2 + C

1 x2

x

1

x =tan

2


61/141

1

x

x2 + 1

12

12

1 x

1 + xarcsin x dx

1

2

12 dxx +1 x2

0

dx

1 + ex

12

12

1 x

1 + xarcsin x dx =

12

121

1 x2

arcsin x dx + 12

12x

1 x2

arcsin x dx

u = arcsin x

dv =x

1 x2dx

du =1

1 x2

v =1 x2

12

12arcsin x

x1 x2

dx =arcsin x1 x21212

12

12dx =

2

3 1

12

12 1 x

1 + xarcsin x dx=

23 1


62/141

t = sin dt =cos d

12

12dx

x +

1 x2=

6

6cos d

sin + cos

= 6

6cos

(cos sin

)cos2 sin2

d

= 12

66

1 + cos2 sin2cos2

d

=1

2

6

6(sec2 + 1 tan2)d

=1

21

2ln tan2 + sec2 + 1

2ln sec26

6

=1

2ln(3 + 2) +

6

u =e

x + 1

du = e

x dx

dx1 + ex

= ex dx

ex + 1 = du

u = ln u + C= ln(1 + ex) + C

0

dx

1 + ex = lim

c

c

0

dx

1 + ex = lim

c[ln(1 + ex)]c0 = lim

c(ln(1 + ec) ln 2) = ln 2

u = ex

du = ex dx

dx1 + ex

= ex dx

ex + (ex)2= du

u + u2

= 1u

1

1 + udu

=ln

u

ln

1 + u

+ C

=x ln

(1 + ex

)+ C

0

dx

1 + ex = lim

c

c

0

dx

1 + ex

= limc[x ln(1 + ex)]c0

= limc(c ln(1 + ec) + ln 2)

= limc( ln(ec) ln(1 + ec) + ln 2)

= limc(

ln

(ec + 1

)+ ln 2

)=ln 2


63/141

0

dx

(ax + 1

)(x2 + 1

) a

1(ax + 1)(x2 + 1) = 1a2 + 1 a2ax + 1 ax 1x2 + 1.

dx

(ax + 1

)(x2 + 1

) =

1

a2 + 1

a

2

ax + 1dx ax 1

x2 + 1 dx

= 1a2 + 1a ln ax + 1 a2 ln(x2 + 1) + arctan(x) + C .

0

dx(ax + 1)(x2 + 1) = limc c0 dx(ax + 1)(x2 + 1)=

1

a2 + 1 limc

a ln ax + 1 a2

ln(x2 + 1) + arctan(x)c0

=1

a2 +

1

limc

a ln

(ac + 1

)

a

2

ln

(c2 + 1

)+ arctan c

a ln 1 +a

2 ln 1 arctan0

=a

a2 + 1 limc

ln ac + 1c2 + 1

+ 1a2 + 1

limc

arctan c

=1

a2 + 1a ln a +

2

limc

ac + 1

c2 + 1

=a limc

arctan c =

2

y = 1x

x 1

x

S

x

x 1 x

D

D

S

=

1

R(x)2 dx = 1

dx

x2 .


64/141

p =2 > 1

=2

1

y

1 +

(y

)2 dx =2

1

1

x

1 +

1

x4 dx .

1

x

1 +

1

x4

1

x 0 x 1

1

dx

x =

S

S

D

S

D

n

0

tnet dt = n!

n

n = 0

0

et dt =1 = 0!

n > 0

0

tn1et dt =(n 1)!

0

tnet dt = limc

c

0

tnet dt

= limc([tnet]c

0+ n

c

0

tn1et dt)= lim

c

[tnet

]c

0+ n

0

tn1et dt

=n

0

tn1et dt

=n (n 1)!=n!

limc

cnec =0

n

(x) = 0 tx1et dt


65/141

x > 0 x >0

0 < x 0

(x) =(x + 1)x

(n+1) = n! n

x! = (x + 1)

12! = (12) =

0

t12et dt =2

0

eu2

du = 2

2 =

.

n

r

n2rn(n2)! .

n =1

[r, r] n =2 r n = 3 r

n =4

0ln x

x2 + 1dx = 0

0

ln x

x2 + 1dx =

1

0

ln x

x2 + 1dx +

1

ln x

x2 + 1dx

1

ln x

x2 + 1 dx 0

ln x

x2 + 1 ln x

x2 x 1

1

ln x

x2 dx =

0

tet dt

= limc

c

0

tet dt

= limc

c

0

t d(et)= lim

c

tet

c

0+

c

0

et dt

= limccec e

t

c

0= limc

(cec ec + 1)=1

x =et

dx =et dt

limc

cec = limc

c

ec

= lim

c

1

ec =0 .

1

ln x

x2 + 1 dx


66/141

1

0

ln x

x2 + 1dx

1

0

ln x

x2 + 1dx =

1

ln(1u)(1u)2 + 1 duu2 = 1 ln u1 + u2 du

x = 1

u

dx = du

u2

0

ln xx2 + 1

dx = 1

0

ln xx2 + 1

dx +

1

ln xx2 + 1

dx

=

1

ln x

x2 + 1dx +

1

ln x

x2 + 1dx =0

1

ln x

x2 dx

0

ln x

x2

1

x32

x 1

1

dx

x32

p =3

2 >1

ln xx2

1

x32

x 1

f(x) = x ln x [1,)

f(x) =1(2x) 1x =(x 2)(2x)

x =4

f(x) 4 f(4) =4 ln 4 =2 2 l n2 =2(1 ln 2) >0

f

[1,)

0

dx

ex ex

0

dxex ex

= 1

0

1ex ex

+

1

dxex ex

1

0

dx

ex ex

ex ex (1 + x) (1 x) = 2x ,

x = 0 1

(ex ex

)

1

(2x

)

L = limx0+

1(ex ex)1x = limx0+ xex ex = limx0+ 1ex + ex = 12 .

0 < L <

1

0

dx

x

p = 1 1

1

0

dx

ex ex

0

dx

ex ex


67/141

L = limx

1(ex ex)ex

= limx

1

1 e2x =1 .

0


68/141

P(3,5, 1)

L x = 2t 1

y = t + 2

z= 2t < t <

P

L

P

L

L

v =2ij 2k

n =2i j 2k

2 (x 3) + (1) (y (5)) + (2) (z 1) = 0 2x y 2z=9

n

L

P

v

#

P Q

Q

Q(1, 2, 0)

t = 0

#

P Q = 4i + 7j k v

n = v #

P Q =

i j k

2 1 2

4 7 1

=15i + 10j + 10k ,

n =3i + 2j + 2k

3 (x 3) + 2 (y (5)) + 2 (z 1) =0 ,

3x + 2y + 2z=1 .

P 3x 4y + z=10

P(2, 3,1)

Q(1, 2, 2)

P

P

P

Q

P

n =3i 4j+k

v =3i4j+k

x = 3t + 2

y = 4t + 3

z=t 1 < t <

P n

n = 3i4j+k P P Q

n

#

P Q = i j + 3k n

n =n #

P Q = i j k

3 4 11 1 3 = 11i 10j 7k ,


69/141

n =11i + 10j + 7k

11 (x 2) + 10 (y 3) + 7 (z (1)) = 0 ,

11x + 10y + 7z=45 .

L

L1 x = 2t 1, y = t + 2, z=3t + 1

L2 x = s + 5, y =2s + 3, z= s

1 =2i j+ 3k v2 =i + 2j k L1

L2

v = i j k

2 1 31 2 1 = 5i + 5j + 5k

L

v =i j k

n = vv1 =

i j k

1 1 1

2 1 3

= 4i 5j + k

P L L1 P1(1, 2, 1) P

P

4 (x (1)) + (5) (y 2) + 1 (z 1) = 0 ,

4x + 5y z=5 P0 P L2 s

4 (s + 5) + 5 (2s + 3) (s) = 5

s = 2 L2 P0(3,1, 2)

L

x = t + 3 , y = t 1 , z= t + 2 ;

( < t <

).


70/141

c

r =ect cos t i + ect sin tj

< t <

r = ect cos t i + ect sin tj

v =dr

dt =(c ect cos t ect sin t)i + (c ect sin t + ect cos t)j .

r =((ect cos t)2 + (ect sin t)2)12 =ect ,v

=

((c ect cos t ect sin t

)2+

(c ect sin t + ect cos t

)2

)12

=

c2 + 1 ect ,

r v =ect cos t (c ect cos t ect sin t) + ect sin t (c ect sin t + ect cos t) = c e2t .

r v

cos =r vr v = c e2tect c2 + 1 ect = cc2 + 1 ,


71/141

xyz

2x + y 2z=1,

t

r =t i + t2j + t3 k

< t <

t

t

n =

2i +j 2k

v=

i + 2tj + 3t2

k

n v =0

n v =2 + 2t 6t2

3t2 t 1 =0

t =(1 13)6

n

v

12 = 2t1 = 3t2(2)

t2 =

13

x = t

y = t2

z = t3

2x + y 2z = 1 2t3 t2 2t + 1 = 0

2t3 t2 2t + 1 =(2t 1)(t 1)(t + 1) t =12

t = 1

t = 1

(x,y,z) =(12, 14, 18)(1, 1, 1)

(1, 1,1)


72/141

lim

(x,y)(0,0)xy2

x6 + y2 =0

lim(x,y)(0,0)

xy

x6 + y2

0 y2 x6 + y2

(x, y) 0 xy2

x6 + y2 =x y2

x6 + y2 x 1 =x

(x, y) =(0, 0)

lim(x,y)(0,0)

x = 0

lim(x,y)(0,0)

xy2

x6 + y2 =0.

x

lim(x,y)(0,0)

x

xy

x6 + y2 =lim

x0

x 0

x6 + 02 =lim

x00 =0,

y =x

lim(x,y)(0,0)

y =x

xy

x6 + y2 =lim

x0

x x

x6 + x2 =lim

x0

1

x4 + 1 =1.

lim

(x,y)(0,0)xy

x6 + y2

a

b

c

d

a

c+

b

d >1

lim

(x,y)(0,0)xayb

x

c +

y

d =0

ac+ b

d 1 lim(x,y)(0,0) xaybxc + yd

a

b

c

d

a

c+

b

d >1

lim

(x,y)(0,0)xayb

xc + yd =0

a

c+

b

d 1

lim

(x,y)(0,0)xayb

xc + yd


73/141

n

>0

lim(x,y)(0,0) x2

y3x3 + y

lim(x,y)(0,0)

x =y3

x2y3x3 + y =limy0 (y3)2y3y33 + y =limy0 yy limy0 11 + y9 .limy0

11 + y9 1 > 9 12 = 9 limy0 yy

x2y3x3 + y x = y3

lim(x,y)(0,0)

x2y3x3 + y 9


74/141

(x, y) =(0, 0) 1 2k > 0 x12k 0 (x, y) (0, 0)

lim(x,y)(0,0)

x(x2 + y2)k =0

k = 12

lim(x,y)(0,0)

x

x(x2 + y2)12 =limx0 x(x2 + 02)12 =limx0 1 =1 ,

lim(x,y)(0,0)

y

x(x2 + y2)12 =limy0 0(02 + y2)12 =limx0 0 =0 .

lim(x,y)(0,0)

x

(x2 + y2)12

k >12

lim(x,y)(0,0)

x

x(x2 + y2)k =limx0 x(x2 + 02)k =limx0 x12k =

lim(x,y)(0,0)

x(

x2 + y2

)k

f(x, y) =

xayb

x4 + y6

(x, y) =(0, 0)0

(x, y) =(0, 0)

a

b

a

b

f

f(x, y) (0, 0)

f(x, y)

(x, y)

(0, 0)

y =x

f(x, y)

1

(x, y)

(0, 0)

y = x

f(x, y)

(x, y)

(0, 0)

lim

(x,y)(0,0)f(x, y)

f(x, y)

(x, y)

(0, 0)

y

f

(x, y

)

(x, y

)

(0, 0

)

y

fx(0, 0) fy(0, 0) f(x, y) (0, 0)


75/141

a =4 b =1 f (0, 0)

0 f(x, y) = x4yx4 + y6

x4x4 + y6

y 1 y y

(x, y

) =0

f

(x, y

)

(0, 0

) 0 = f

(0, 0

)

a = 3 b = 1

lim(x,y)(0,0)

y=x

f(x, y) = limx0

f(x, x) = limx0

x4

x4 + x6 =lim

x0

1

1 + x2 =1

lim(x,y)(0,0)

y=x

f(x, y) = limx0

f(x,x) =limx0

x4

x4 + x6 =lim

x0

1

1 + x2 = 1 .

a =2

b = 3

lim(x,y)(0,0)

y=mx

f(x, y) = limx0

f(x,mx) = limx0

m3x5

x4 + m6x6 =lim

x0

m3x

1 + m6x2 =0

lim(x,y)(0,0)

y

f(x, y) = limy0

f(0, y) = limy0

0 = 0 .

lim(x,y)(0,0)

y=x23

f(x, y) = limx0 f(x, x23) = limx0 x4x4 + x4 =limx0 12 = 12=0

f(x, y) (0, 0)

a = 0

b = 6

lim(x,y)(0,0)

y=mx

f(x, y) = limx0

f(x,mx) = limx0

m6x6

x4 + m6x6 =lim

x0

m6x2

1 + m6x2 =0

lim(x,y)(0,0)

y

f(x, y) = limy0

f(0, y) =limy0

y6

y6 =lim

y01 =1 .

a = 1

b = 1

f(x, y)

(0, 0)

y = x

fx(0, 0) fy(0, 0) f

(a, b)


76/141

(a, b)

3a + 2b > 12

(3, 1)

(1, 3)

(1, 4)

(2, 3)

(0, 6

) (0, 7) (1, 1)(1, 2) (1, 3)(1, 4) (1, 5)(2, 1) (2, 2)(2, 3) (2, 4)(3, 1)(3, 2)(4, 1)

p

t =p (1 p) + 2p

x2

p(x, t) x t

(a, b)

p(x, t) = 1(1 + eax+bt)2

pt = 2(1 + eax+bt)3 eax+bt bpx = 2(1 + eax+bt)3 eax+bt a

pxx =6(1 + eax+bt)4 (eax+bt a)2 2(1 + eax+bt)3 eax+bt a2p(1 p) =(1 + eax+bt)4 (2eax+bt + (eax+bt)2)

2b(1 + eax+bt) =2 + eax+bt + 6a2eax+bt 2a2(1 + eax+bt)

2a2 2b 2 =(4a2 + 2b + 1)eax+bt

eax+bt

4a2 +2b+ 1 =0

eax+bt

a = 0

b =0 2 =1

2a22b2 = 0

4a2+2b+1 = 0

2a2 2b 2 =0

6a2 =1

b = 56

(a, b) =(16,56)

(16,56)


77/141

z

w

x

y

xw3 +

yz2 + z3 = 1

zw3 xz3 + y2w =1

z

x

(x,y,z,w) =(1,1,1, 1)

x

z

w

x

y

w3 + x 3w2wx + y 2zzx + 3z2zx =0 ,

zxw3+ z 3w2wx z

3 x 3z2zx + y

2wx =0 .

x =1 y = 1 z= 1 w =1 5zx + 3wx = 1 2zx + 2wx =1

zx

z

x =

5

4

(x,y,z,w) =(1,1,1, 1) .

xy

xy

xy

P0

v1 =2i+j 1

v2 = i 5j

1

v3 =i +j 2


78/141

z=f(x, y)

f(3, 3) =1

fx(3, 3) = 2 fy(3, 3) = 11 f

(2, 5

)=1

fx

(2, 5

)= 7

fy

(2, 5

)= 3.

w

u

v

f(w, w) = f(uv,u2 + v2)

(u, v)

w

u

(u,v,w) =(1, 2, 3)

f(w, w) = f(uv,u2 + v2)

u

fx(w, w)wu + fy(w, w)wu =fx(uv,u2 + v2)(uv)u + fy(uv,u2 + v2)(u2 + v2)u


79/141

(fx(w, w) + fy(w, w))wu

=fx(uv,u2 + v2) v + fy(uv,u2 + v2) 2u

(fx

(3, 3

)+ fy

(3, 3

))w

u =2fx

(2, 5

)+ 2fy

(2, 5

) (u,v,w) =(1, 2, 3) fx(3, 3) = 2 fy(3, 3) = 11fx(2, 5) = 7 fy(2, 5) = 3 w

u =

8

9 (u,v,w) =(1, 2, 3) .

u =x + y + z v =xy + yz+ zx w =xyz f

(u,v,w

)

f

(u,v,w

)= x4 + y4 + z4

(x,y,z

)

fu

(2,1,2

)

(x,y,z) =(1,1, 2) (u,v,w) =(2,1,2)

f(u,v,w) = x4+y4+z4 x y z fu ux + fv vx + fw wx =4x

3

fu uy + fv vy + fw wy =4y3

fu uz + fv vz + fw wz =4z3

u =x + y + z v =xy + yz+ zx w =xyz

fu

1+

fv (y + z) + fw yz=4x

3

fu 1 + fv (x + z) + fw xz=4y3fu 1 + fv (x + y) + fw xy =4z3

(x,y,z) =(1,1, 2)

fu + fv 2fw =4

fu + 3fv + 2fw = 4

fu fw =32

2fu + 8fw = 16

6fu =240

fu(2,1,2) =40.

f

f(u,v,w) =u4 4u2 + 2v2 + 4uw

(u,v,w)

(x,y,z)

T3 uT2 + vTw =0

z=f(x, y)

x = r cos

y =r sin

2f

x2+

2f

y2 =

2z

r2+

1

r

z

r

+1

r2

2z

2

.


80/141

z=F(x, y) x y

z

r =

F

x

x

r +

F

y

y

r =Fx cos + Fy sin ,

z

=F

x x

+F

y y

=Fx (r sin ) + Fy (r cos ) .

F =f

z

r =fx cos + fy sin .

2z

r2 =

r(fx) cos +

r(fy) sin .

r (fx)

r (fy) F =fx F =fy 2zr2

=(fxx cos + fxysin ) cos + (fyx cos + fyysin ) sin =fxx cos

2 + 2fxycos sin + fyysin2

F =f

z

=fx (r sin ) + fy (r cos ) ,

2z

2 =

(fx) (r sin ) + fx

(r sin )

+

(fy) (r cos ) + fy

(r cos )

=(fxx(r sin ) + fxyr cos )(r sin ) + fx(r cos )+ (fyx(r sin ) + fyyr cos )(r cos ) + fy(r sin )

=fxxr2 sin2 2fxyr

2 cos sin + fyyr2 cos2

r

(fx cos + fy sin

)

F =fx

F =fy

1r

1r2

fxx + fyy

f(x, y)

f xx2 + y2

, y

x2 + y2 =f(x, y)

(x, y

) =

(0, 0

)

fxx

(3

10, 1

10

)

fx

(3, 1

)= 8

fy

(3, 1

)= 7

fxx

(3, 1

)=2

fxy

(3, 1

)=

5 fyy(3, 1) = 4


81/141

fx(x, y) = x

f xx2 + y2

, y

x2 + y2

=fx

x

x2 + y2,

y

x2 + y2

x

x

x2 + y2

+ fy xx2 + y2 , yx2 + y2 x yx2 + y2=fx x

x2 + y2,

y

x2 + y2 1 (x2 + y2) x 2x(x2 + y2)2

+ fy xx2 + y2

, y

x2 + y2 2xy(x2 + y2)2

fxx(x, y) = xfx xx2 + y2 , yx2 + y2 y2 x2(x2 + y2)2

+ fy xx2 + y2

, y

x2 + y2 2xy(x2 + y2)2

=fxx xx2 + y2

, y

x2 + y2 y2 x2(x2 + y2)22

+ fxy xx2 + y2

, y

x2 + y2 2xy(x2 + y2)2 y2 x2(x2 + y2)2

+ fx x

x2

+ y2

, y

x2

+ y2

x y2 x2

(x2 + y2)2+ fyx x

x2 + y2,

y

x2 + y2 y2 x2(x2 + y2)2 2xy(x2 + y2)2

+ fyy xx2 + y2

, y

x2 + y2 2xy(x2 + y2)22

+ fy xx2 + y2

, y

x2 + y2

x 2xy(x2 + y2)2

=fxx

x

x2 + y2,

y

x2 + y2

y2 x2

(x2 + y2)

2

2

+ 2fxy xx2 + y2

, y

x2 + y2 y2 x2(x2 + y2)2 2xy(x2 + y2)2

+ fyy xx2 + y2

, y

x2 + y2 2xy(x2 + y2)22

+ fx xx2 + y2

, y

x2 + y2 2x(x2 3y2)(x2 + y2)3

+ fy

x

x2 + y2,

y

x2 + y2

2y

(3x2 y2

)(x2 + y2

)3


82/141

(x, y) =(310, 110)

fxx(310, 110) =fxx(3, 10) (8)2 + 2fxy(3, 10) (8) (6) + fyy(3, 10) (6)2+ fx(3, 10) 36 + fy(3, 10) 52

=2 64 + 2 5 48 +

(4

) 36 +

(8

) 36 + 7 52

=540

f(x, y)

f(2, 1) = 8

f(x, y) = 8

xy

(2, 1

) 3x 5y =1

P z=f(x, y) (2, 1, 8)

P

P P

f(3, 2) = 11

fx(2, 1) = 1

d

dtf(t2 + 1, t3)

t=1

=6

x =4t+2 , y =2t+1 , z=t+8 , ( < t < )

P

x = t + 2 , y =2t + 1 , z=t + 8 ,

( < t <

)

P

fx(2, 1) =3c fy(2, 1) = 5c

c

c

c

fx(2, 1) = 1 c = 13 fy(2, 1) = 53

P

P 3x 5y + 3z = 25

f

(x, y

)= x + 5

3 y + 25

3


83/141

d

dtf(t2 + 1, t3) =fx(t2 + 1, t3) 2t + fy(t2 + 1, t3) 3t2

t =1

3c 2+

(5c

)3 =6

c = 2

3

fx

(2, 1

)= 2 fy

(2, 1

)=10

3

P

P 6x 10y + 3z=26

f(x, y) = 2x + 103 y + 263

P

z=3c(x2)+(5c) (y 1) + 8 x = 4t + 2 , y = 2t + 1 , z =t+8 , ( < t < )

P

1 =(3c) 4+(5c) 2

c =12

fx(2, 1) = 32 fy(2, 1) = 52

P

P 3x 5y 2z= 15

f

(x, y

)= 3

2 x 5

2 y + 15

2

P

3c (x 2)+ (5c) (y 1) (z8) = 0

x = t + 2 , y =

2t+1 , z=t+8 , (


84/141

(g)P0 z=xy (f)P0 (g)P0 =

i j k

2 4 4

2 1 1

= 8i + 10j 6k ,

P0

f(x, y)

f(0, 0)

fx fy fxx fyy fxy


85/141

f(0, 0) f

(0, 0)

f

f(0, 0) f

f =

1

0.44

f

10.44 2.27

f(0, 0)


86/141

f(0, 0) fx(0, 0) >0

fy(0, 0) < 0 x

fx(0, 0) >0

y

f

fy

(0, 0

)< 0

f

f

fxx(0, 0) < 0 fx(0, 0) >0 fyy(0, 0) > 0

fy(0, 0) < 0


87/141

fx

y

fxy(0, 0)


88/141

a

f(x, y) = x3 3axy + y3

fx =3x2 3ay =0 fy = 3ax + 3y2 =0 a =0

3x2 = 0 x = 0

3y2 = 0 y = 0

(0, 0)

a

=0

y =x2

a

x4a3x = 0

x =0

x = a

y =x2

a

(0, 0) (a, a)

=fxx fxyfyz fyy

= 6x 3a3a 6y

(a, a) = 27a2 fxx(a, a) = 6a(a, a) a > 0

a < 0

(0, 0) = 9a2

(0, 0)

a

=0

(0, 0) a = 0 (0, 0) = 0

f(x, y) = x3 + y3 x

f(x, 0) = x3

x =0

f(x, y)

(0, 0)

(0, 0)

a =0

f(x, y) =2x3+2xy2xy2

D =

{(x, y

) x2 + y2 1

}

f

(x, y

)

D

fx = 6x2 +2y2 1 = 0 fy = 4xy 2y = 0

y =0

x =12

y = 16

(x, y) =(16, 0)

(16, 0)

D

D

x2 +y2 = 1

y

y =

1 x2

1 x 1

f

(x,

1 x2

) = x2 +x 1

1 x 1

d

dx

f

(x,

1 x2

)=2x + 1 = 0 x =

1

2

(x, y

) =

(12,32) (12,32) f D

x = 1

x = 1 (x, y) =(1, 0)

(1, 0)

f

D

(16, 0),(16, 0),(12,32),(12,32),(1, 0),(1, 0) .

f

1323, 13 23, 54, 54, 1 ,1 ,


89/141

54

f

D

x = cos t

y = sin t


90/141

f(x, y) = 2(x2 + y2 1)2 + x2 y2

D ={(x, y) x2 + y2 1}

f

D

fx =4(x2 + y2 1) 2x + 2x = 0 fy =4(x2 + y2 1) 2y 2y =0 .

x =0 4x2 + 4y2 3 =0

y =0

4x2 + 4y2 5 =0 x = 0 y =0 4x2 + 4y2 3 =0 y = 0

x =0

4x2 + 4y2 5 = 0

4x2 + 4y2 3 =0

4x2 + 4y2 5 =0

(0, 0)

(32, 0)

(0,52)

D

x2 + y2 =5

4 >1

f

D

y = 1 x2 f(x,1 x2) = 2x21

1 x 1 (ddx)f(x,1 x2) =4x

x =0

(x, y) =(0, 1) (0,1) x =1 x = 1

1 x 1

(x, y) =(1, 0)

(1, 0)

f

f(0, 0) = 2f(32, 0) =f(32) = 78f(0, 1) = f(0,1) = 1f(1, 0) = f(1, 0) = 1

f

D

2 1


91/141

x =cos t y =sin t


92/141

h(x, y) =1 x2 +x2 y2 + y

D ={(x, y) 0 y x 1}

h

D

hx = x

1 x2+

xx2 y2

=0

hy = y

x2 y2+ 1 = 0 .

y2 =x2y2 x2 =2y2 x =

2y

x >0

y >0

12y2 =2y2y2 3y2 =1 y =13 y >0

x =

23

h

(x, y) =(23, 13)

D 0 13 23 1

h

D

y = 0 0 x 1

h(x, 0) =1 x2 + x 0 x 1

d

dxh(x, 0) = x

1 x2+ 1 =0 x2 =1 x2 2x2 =1 x =

12

x >0 x =0 x =1

(x, y) =(12, 0), (0, 0), (1, 0)

x = 1

0 y 1

h

(1, y

)=

1 y2 + y

0 y 1

(x, y) =(1, 12), (1, 0), (1, 1)

y =x

0 x 1

h(x, x) = 1 x2 +x 0 x 1

(x, y) =(12, 12), (0, 0), (1, 1)


93/141

h

h(0, 0) = h(1, 0) = h(1, 1) = 1h(12, 0) =h(1, 12) =f(12, 12) = 2h

(2

3, 1

3

)=

3

3

k

k

k 1

k + 1

r

H(r) = kr + 1 r2

0 r 1

H(r)

k

r = 0

r = 1

k + 1

r =

k

(k + 1

)

n

n n 1

f(x,y,z) =x3 + yz

x2 + y2 + z2 =1

g(x,y,z) =x2 + y2 +z2 1

f = g

g =

0

fx = gxfy = gy

fz =

gzg = 0

3x2 = 2x 1

z = 2y 2

y =

2z 3

x2 + y2 + z2 = 1 4

2

3

z=42z

z=0

= 12

= 12

z =0

3

y =0

4

x = 1

(1, 0, 0)

= 12

1

3

3x2 = x

y = z

x =0 y = z = 1

2

4

x =1

3

y =z= 2

3

4 (

0,1

2,1

2

) (13,23,23)

= 12 1 3 3x2 = x y = z

(0,12,12)

(13,23,23)

(1, 0, 0), (0,12,12), (13,23,23),(0,12,12), (13,23,23)

f

1, 12,12, 1327,1327 1

1


94/141

1

0

1

y

sin(x3)dxdy

Ry2ex

2

dA

R =

{(x, y

) 0 y x

} 20

2yy2

0

xy

x2 + y2dxdy

0

0

dydx(x2 + y2)2 + 1

x

x =

y

x =1

xy

y

y =0

y =1

x R

y =x2 x = 1 x x

0 y 1

1

0

1

y

sin(x3)dxdy =R

sin(x3)dA .

y

y

y =0

y =x2


95/141

1

0

1

y

sin(x3)dxdy =R

sin(x3)dA=

1

0

x2

0

sin

(x3

)dy dx

= 1

0sin(x3)y]y=x2y=0 dx

= 1

0

sin(x3)x2 dx=

1

3 cos(x3)1

0

=2

3

y

R

y2ex2

dA =

0

x

0

y2ex2

dy dx

=1

3

0

x3ex2

dx

=1

6

0

tet dt

=1

6

0

tet dt =

0

t d(et)= lim

c[tet]c0 + c

0

et dt= lim

c

c

eclim

c[et]c0

= lim

c

1

eclim

c(ec 1)

=1

R

x

x = 0 x =

2y y2

x =

2y y2 x2 = 2y y2

x2+(y1)2 =12

x =

2y y2

x2+(y1)2 =1

y

y =0 y =2 R

x2+

(y1

)21

x2+

(y1

)2=1

r =2 sin =0 =2


96/141

2

0

2yy2

0

xy

x2 + y2dxdy =

R

xy

x2 + y2 dA

= 2

0

2sin

0

r cos r sin

r2 rdrd

= 2

0 2sin

0 sin cos r d r d

= 2

0

sin cos r22r=2 sin

r=0

drd

= 2

0

2sin3 cos d

=sin4

22

0

=1

2

R ={(x, y) x 0

y 0}

0

0

dydx(x2 + y2)2 + 1 =R 1(x2 + y2)2 + 1 dA=

2

0

0

rdrd

r4 + 1


97/141

= 2

0

1

2limc

arctan(r2)r=cr=0

d

= 2

0

4 d

=2

8

R

1(x2 + y2)2 dA R

R

y =x

x

R

1(x2 + y2)2 dA =8R 1(x2 + y2)2 dA=8

4

0

2 sec

sec

1(r2)2 r dr d=8

4

0

1

2r2

r=2 sec

r=sec

d

=3 4

0

cos2 d

=3 4

0

1 + cos2

2 d

=3

2 + sin2

24

0

=3

8 +

3

4


98/141

R1

x2y cos(y52)dA

R1

R2

x2y cos

(y5

2

)dA

R2

x

y

R1

x = 1 y = x

y =x

y

x

R2

y = 1

y = xy =x

R1 x

f(x, y) =x2y cos(y52)

x

f(x,y) = f(x, y)

R1

x2y cos(y52)dA =0

x

y

R2

x2y cos(y52)dA = 10

y

yx2y cos(y52)dxdy = 1

0

13

x3y cos(y52)x=yx=y

dy

=2

3

1

0

y4 cos

(y5

2

)dy =

2

3

2

5sin

(y5

2

)1

0

=4

15


99/141

R(1 + x y)dA

R ={(x, y) x y 23

0

x 1

0 y 1}

R(1 + x y)dA =R dA +R(x y)dA

R

y =x

f(x, y) = xy

f(y, x) = f(x, y)

R

12

(1

3

)2=8

9

R(1 + x y)dA =R dA +R(x y)dA = 89 + 0 = 89


100/141

S={(x, y) x 5

y 5}

xy

p(x, y) (x, y) 2

N


101/141

r2 =2 cos(2)

z=

2 r2

R

4R2 r2 dA = 4 4

0

2cos2

02 r2 rdrd

=4 4

0

13(2 r2)32r=2cos2

r=0

d =4

3

4

0

(232 (2 2cos2)32)d=

8

2

3

4

0

1 (2sin2 )32d = 223

32

3

4

0

sin3 d

=2

2

3

32

3

4

0

(1 cos2 ) sin d = 223

32

3

1

12(1 u2)du

=2

2

3

32

3u u3

31

12=

2

2

3

32

32

3

5

62=

223

64

9 +

4029

.


102/141

D

y +z = 1

y = x2

xy

V

D

dz dy dx

dxdydz

7/24/2019 Sample Cal

Sample Calculus problems

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