256 CHAPTER 14. VECTOR VALUED FUNCTIONS OF ONE VARIABLE

Example 14.2.5 Let f (x) = c where c is a constant. Find f ′ (x).

The difference quotient,

f (x+h)−f (x)h

=c−c

h= 0

Therefore,

limh→0

f (x+h)−f (x)h

= limh→0

0= 0

Example 14.2.6 Let f (t) = (at,bt) where a,b are constants. Find f ′ (t).

From the above discussion this derivative is just the vector valued functions whosecomponents consist of the derivatives of the components of f . Thus f ′ (t) = (a,b).

14.2.1 Geometric And Physical Significance Of The DerivativeSuppose r is a vector valued function of a parameter t not necessarily time and considerthe following picture of the points traced out by r.

r(t)r(t +h)

In this picture there are unit vectors in the direction of the vector from r (t) to r (t +h).You can see that it is reasonable to suppose these unit vectors, if they converge, convergeto a unit vector T which is tangent to the curve at the point r (t). Now each of these unitvectors is of the form

r (t +h)−r (t)|r (t +h)−r (t)|

≡ T h.

Thus T h→ T, a unit tangent vector to the curve at the point r (t). Therefore,

r′ (t) ≡ limh→0

r (t +h)−r (t)h

= limh→0

|r (t +h)−r (t)|h

r (t +h)−r (t)|r (t +h)−r (t)|

= limh→0

|r (t +h)−r (t)|h

T h =∣∣r′ (t)∣∣T.

In the case that t is time, the expression |r (t +h)−r (t)| is a good approximation forthe distance traveled by the object on the time interval [t, t +h]. The real distance would bethe length of the curve joining the two points but if h is very small, this is essentially equalto |r (t +h)−r (t)| as suggested by the picture below.