13.3

3.
r'(t)=<2, 1/t, 2t>
|r'(t)|=2t+1/t
L=1->4|r'(t)|dt=2ln2+15

9.
r'(t)=<2t, 4t, 3t^2>
|r'(t)|=sqrt(9x^2+20)|x|
s(t)=a->t|r'(u)|du=((9t^2+20)^3/2)/27-(40*sqrt5)/27

11.
v(t)=r'(t)=<2, 4, -1>
s(t)=|v(t)|=sqrt21
s(4)=sqrt21

13.
v(t)=r'(t)=<1, 1/t, (2*lnt)/t>
s(t)=|v(t)|=sqrt(4(lnt)^2+t^2+1)/|t|
s(1)=sqrt2

15.
v(t)=r'(t)=<3cos3t, -4sin4t, -5sin5t>
s(t)=|v(t)|=sqrt(25sin(5t)^2+16sin(4t)^2+9cos(3t)^2)
s(Pi/2)=5

13.4

1.
r'(t)=<8t, 9>
|r'(t)|=sqrt(64t^2+81)
T(t)=<8t, 9>/sqrt(64t^2+81)
T(1)=<8/sqrt145, 9/sqrt145>

5.
r'(t)=<-Pi(sin(Pi*t)),Pi(cos(Pi*t)),1>
|r'(t)|=sqrt(Pi^2+1)
T(t)=<-Pi(sin(Pi*t)),Pi(cos(Pi*t)),1>/sqrt(Pi^2+1)
T(1)=<0, Pi^2/(Pi^2+1), 1/(sqrt(Pi^2+1))>

7.
r'(t)=<0,e^t,1>
|r'(t)|=sqrt(e^2t+1)
r"(t)=<0, e^t, 0>
r'(t)×r"(t)=<-e^t, 0, 0>
|r'(t)×r"(t)|=e^x
k(t)=e^x/sqrt(e^2t+1)

11.
r'(t)=<-1/t^2, -2/t^3, 2t>
|r'(t)|=sqrt(4t^8+t^2+4)/|t^3|
r"(t)=<2/x^3, 6/x^4,2>
r'(t)×r"(t)=<-16/t^3, 6/t^2, -2/t^6>
|r'(t)×r"(t)|=(2*sqrt(9t^8+64t^6+1))/t^6
k(t)=(2*sqrt(9t^8+64t^6+1)*|x^3|)/(t^6*sqrt(4x^8+x^2+4))
k(-1)=-(2*sqrt74)/3

17.
y'(t)=4t^3
|y'(t)|=4|t|^3
T(t)=sign(t)
|T(t)|=t
k(t)=4*t*|t|
k(2)=16

21.
tanh t = (exp(t)-exp(-t))/(exp(t)+exp(t)
sech t= 2/(exp(t)+exp(-t))
r'(t)=<e^(-2t)(e^(2t)-1), (-2e^t)(e^(2t)-1)/(e^(2t)+1)^2>
r"(t)=<2e^(-2x),(2e^x(e^(4x)-6e(2x)+1))/(e^(2x)+1)^3>
k(t)=

13.5

3.
v(t)=r'(t)=<3t^2,-1,8t>
a(t)=r"(t)=<6t,0,8>
s(1)=|v(1)|=sqrt74

5.
v(t)=r'(t)=<cos(theta), -sin(theta), -3sin(3theta)>
a(t)=r"(t)=<-sin(theta), -cos(theta), -9cos(3theta)>
s(Pi/3)=|v(Pi/3)|=1

15.
v(t)=v(0)+a(t)dt=<3,-2>+<t^2/2,4t>=<t^2/2+3, 4t-2>
r(t)=r(0)+r(t)dt=<0,0>+<t^3/6+3x, 2t^2-2t>=<t^3/6+3x, 2t^2-2t>

17.
v(t)=i+t^2/2k=<i, 0, t^2/2k>
r(t)=<ti, j, t^3/6k>

31.
v(0)=v-a=<10, 19, 23>
|v(0)|=3*sqrt(110)
|v|=4*sqrt(34)
|v(0)|>|v|
The particle is slowing down.

33.
r'(t)=<1, -sint, cost>
r"(t)=<0, -cost, -sint>
|r'(t)|=sqrt2
r'(t)×r"(t)=<sin^2t-cos^2t, sint, -cost>
|r'(t)×r"(t)|=sqrt((sin^2t-cos^2t)^2)+1
aN = |r′×r′′| /|r′|=sqrt(1-2*(sin(t))^2*(cos(t))^2)
aT =r′·r′′/ |r′|=<0, sin(t)*cos(t)/sqrt2, -sin(t)*cos(t)/sqrt2>

14.1

1.
f(2,2)=18
f(-1,4)=-5

3.
h(3,8,2)=6
h(3,-2,-6)=-1/6

7.
#ON THE HANDWRITTEN PDF VERSION

21.
#ON THE HANDWRITTEN PDF VERSION

23.
#ON THE HANDWRITTEN PDF VERSION

33.
#ON THE HANDWRITTEN PDF VERSION

35.
#ON THE HANDWRITTEN PDF VERSION

14.2

9.
lim(x,y)->(2,5)(g(x,y)-2f(x,y))=1

11.
lim(x,y)->(2,5)e^((f(x,y))^2-g(x,y))=e^2

15.
y=my,
(x^3+m^3x^3)/xm^2x^2=1+m3
Therefore, the resulting limit depends on m and therefore the limit does not exist.

21.
y=mx,
(mx^2)/((3+2m)x^2)=m/(3+2m)
Therefore, the resulting limit depends on m and therefore the limit does not exist.

23.
when on the x axis
y=z=0
lim(x,y,z)->(0,0,0)(x+y+z)/(x^2+y^2+z^2)=lim(x,y,z)->(0,0,0)1/x=infinite

27.
lim(z,w)->(-2,1)(z^4cos(Pi*w)/(e^(z+w)))=-16e

31.
lim(x,y)->(3,4)1/sqrt(x^2+y^2)=1/5

35.
lim(x,y)->(-3,-2)(x^2y^3+4xy)=-48