14.3

3.
fy y/(x+y)=x/(y+x)^2

5.
fz=xy
fz(2,3,1)=6

17.
fx=1/y
fy=-x/y^2

19.
fx=-x/(sqrt(-x^2-y^2+9))
fy=-y/(sqrt(-y^2-x^2+9))

21.
fx=cos(x)*sin(y)
fy=cos(y)*sin(x)

27.
fr=e^(r+s)
fs=e^(r+s)

31.
fx=-2*x*e^(-x^2-y^2)
fy=-2*y*e^(-y^2-x^2)

39.*
fL=(1/M-(Lt)/M^2)*e^((-Lt)/M)
fM=(t*e^((-Lt)/M)*L^2)/M^3-(e^((-Lt)/M)*L)/M^2

47.
(a)
I(T,H)=45.33+0.6845*95+5.758*50-0.00365*95^2-0,1565*50*95+0.001*95^2*50=73.1913
(b)
The partial derivative with respect to T
fT=0.6845-0.0073T-0.1565H+0.002HT=1.666

14.4

3.
f(2,1)=6
fx=2xy+y^3
fy=x^2+3xy^2
fx(2,1)=5
fy(2,1)=10
z=5(x-2)+10(y-1)+6

5.
f(4,1)=17
fx=2x
fy=-2y^-3
fx(4,1)=8
fy(4,1)=-2
z=8(x-4)-2(y-1)+17

7.
f(2,1)=5
fr=(2*r)/sqrt(s)
fs=(-r^2)/2s^(3/2)-3/s^4
fr(2,1)=4
fs(2,1)=-5
z=4(r-2)-5(s-1)+5

13.
f(2,1)=4
fx=2xy^3
fy=x^2*3y^2
fx(2,1)=4
fy(2,1)=12
L(x,y)=4+4(x-2)+12(y-1)
L(2.01,1.02)=4.28
L(1.97,1.01)=4
f(2.01,1.02)=4.287
f(1.97,1.01)=3.999

15.
f(2,1)=8
fx=3x^2*y^-4
fy=-4x^3y^-5
fx(2,1)=12
fy(2,1)=32
L(x,y)=8+12(x-2)+32(y-1)
L(2.03,0.9)=5.16
del f=2.84

17.
f(0,0)=1
fx=2xe^(x^2+y)
fy=e^(x^2+y)
fx(0,0)=0
fy(0,0)=1
L(x,y)=1+y
L(0.01,-0.02)=0.98
f(0.01,-0.02)=0.980

23.
f(x,y)=x^3y^2
f(2,1)=8
fx=3x^2y^2
fy=x^3*2y
fx(2,1)=12
fy(2,1)=16
L(x,y)=8+12(x-2)+16(y-1)
L(2.01,1.02)=8.44
f(2.01,1.02)=8.449

25.
f(x,y)=sqrt(x^2+y^2)
f(3,4)=5
fx=x/(sqrt(x^2+y^2))
fy=y/(sqrt(x^2+y^2))
fx(3,4)=3/5
fy(3,4)=4/5
L(x,y)=5+3/5(x-3)+4/5(y-4)
L(3.01,3.99)=4.998
f(3.01,3.99)=4.998

27.
f(x,y,z)=sqrt(xyz)
f(2,2,4)=4
fx=sqrt(yz)/2sqrt(x)
fy=sqrt(xz)/2sqrt(y)
fz=sqrt(xy)/2sqrt(z)
fx(2,2,4)=sqrt(8)/2sqrt(2)
fy(2,2,4)=sqrt(8)/2sqrt(2)
fz(2,2,4)=1/2
L(x,y,z)=4+sqrt(8)/2sqrt(2)(x-2)+sqrt(xz)/2sqrt(y)(y-2)+1/2*z
L(1.9,2.02,4.05)=3.945
f(1.9,2.02,4.05)=3.943

14.5

7.
fx=yz^-3
fy=xz^-3
fz=-3z^-4*xy
del f=<yz^-3,xz^-3,-3z^-4*xy>

11.
r'(t)=<-sin t, cos t>
r'(0)=<0,1>
f'(x)=2x-3y
f'(y)=-3x
t=0, x=1, y=0
f'(x=1)=2
f'(y=0)=-3
f'(r'(t))=-3

13.
r'(t)=<2e^(2t), 3e^(3t))>
r'(0)=<2,3>
f'(x)=y*cos(xy)
f'(y)=x*cos(xy)
t=0, x=1, y=1
f'(x=1)=cos(1)
f'(y=0)=cos(1)
f'(r'(t))=5cos(1)

19.
r'(t)=<e^t,1,2t>
r'(1)=<e,1,2>
f'(x)=yz^-1
f'(y)=xz^-1
f'(z)=-xyz^-2
t=1, x=e, y=1, z=1
f'(x=e)=1
f'(y=1)=e
f'(z=2)=-e
f'(r'(t))=0

27.
fx=(2x)/(x^2+y^2)
fy=(2y)/(x^2+y^2)
fx(P)=2
fy(P)=0
|v|=sqrt13
u=<3/sqrt13, -2/sqrt13>
directional derivative=6/sqrt13

31.
fx=2x
fy=8y
fx(P)=6
fy(P)=16
v=<-3,-2>
|v|=sqrt13
u=<-3/sqrt13, -2/sqrt13>
directional derivative=-50/sqrt13

33.*
fx=e^(y-z)
fx(P)=e^4
fy=xe^(y-z)
fy(P)=5*e^4
fz=-xe^(y-z)
fz(P)=-5*e^4
v=<3,9,4>
|v|=sqrt106
u=<3/sqrt106,9/sqrt106,4/sqrt106>
directional derivative=148.485

37.
|v|=sqrt14
u=<2/sqrt14,1/sqrt14,3/sqrt14>
directional derivative=sqrt14
f is increasing at P in the direction v.

39.
fx=cos(xy+z)*y
fx(P)=1
fy=cos(xy+z)*x
fy(P)=0
fz=cos(xy+z)
fz(P)=-1
|fx,fy,fz|=sqrt2
directional derivative=sqrt2*(sqrt3)/2=(sqrt6)/2

41.
fx=2x
fy=2y
fz=2z
fx(P)=6
fy(P)=2
fz(P)=4
The vector can be <6,2,4>

43.
fx=x/2
fy=(2x)/9
fz=2z
<x/2,(2x)/9,2z>=k<1,1,-2>
x/2=k,(2x)/9=k,2z=-2k
((2k)^2)/4+(((9k)/2)^2)/9+(-k)^2=1
k =  (2*sqrt(17))/17, -(2*sqrt(17))/17
P1:(4/sqrt(17), 9/sqrt(17), -2/sqrt(17))
P2:(-4/sqrt(17), -9/sqrt(17), 2/sqrt(17))