Derivatives and integrals of vector functions

Last time : domain of vector functions

limit of vector functions

continuity of vector functions

Moral : work with the component functions

Let Ect) = sect) , get) , htt))

then d = Fct) = Iim Ect tht -Fct)h-70 h

.*f÷¥n""

If Fct) =L Lct) , gct) , htt ) ) , where

f. g , h are differentiable,then

F''

Ct) =L f'Ct) , g'CA) , h'Ctl)

PI E'(f) = IimEct -ist -Ect)

ss -70

= lim ( letts) - Lct)

,

gcttsl - get),

hcttsl - htt) )s -70 S s s

=L limo l't's's- Ut) , limo setts's- Sct) , limo

htt's's- htt) )s s s

Given F' (t),

we can try to compute

the derivative i''

CA) ( the tangent vector)

the unit tangent vector F'Ct)

IF '

Ct) I

the tangent line to Ect) at a

point p

Example : find the derivative of

Fct) - tt¥ , coset')

,te

-t )

find the unit tangent vector as well as an

equation for the tangent line at time t -- O

Ans :

Example : find the derivative of

Ect) - C , coset')

,

test )""

find the unit tangent vector as well as an

equation for the tangent line at time t -- O

ttt -Ztcttl)Ans : F' 'Ct) = ( (£+1,2 ,

- since'

) Cst) , e-t-te)

F''

(o)F' 'co) =L I,0,

17, if = "f¥

x- IttTco) -- LI,1.07 so tangent line is y= ,

z -- t

Interpretation :

Fct) position

F'Ct) velocity

F''

(t) acceleration

more on this later

Rules for differentiation

1.

d- Evict ) t Ict)) = Tilt) + I'

Ct)at

2. at tcuCtl) = cut'

(t)

3 . It Efctluct )) = L'Ctluct ) + fctlu'

Ct)

4. It EECt ) . Ict )]= i'

'

Ctl Ect) + Ect)Ect)

5. adz Eiict) x Ect)] = Tilt) x Ect)

tuft)xE'Ct)

6. ItEiicltl)) = ett) I'

'

Cet))

Example : show that if lrct)l=c for all t,

then F' 'Ct) is orthogonal to FCA) for all t

Ans :

Example : show that if lrct)l=c for all t,

then F' 'Ct) is orthogonal to Ect) for all t

Ans : little c ⇒ lect)T=c'

( Ect) II Ect ) - Fct)

# Erect) (t)) = # Ed ] so17

F'Ct) . Fct) + Ect) - F'(t) = 0 A

2E 'Ctl .Ect) -- OF'Lt) -Ect) - O

Integrals of vector functions

if Ect)= ( Lct) , get ) , htt))

then [ Ect) dt-ajfctldt.gg#Idt.&hCt)dt)by the fundamental theorem of calculus

,

if pitt ) = E' (t) ,

then [ Ect) at = pics) - Fica)

Example : Fct) = ( e-t

,

Fe,

cos tht))

compute of'

Ect) at

Ans :

frit) at -

- C - e-t.Et"

, sineath) + E

I Ect) at -- c - e-ti.

. It" i.

.sinI

'

! >=L - e-

'

t I, 3 , O)