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Unformatted text preview: units of measure. If we know the present altitude we
time
can tell how far the plane will glide before it touches the surface.
Here the speed functions u’(t) and v’(t) are given. How will we find the glide ratio
speed
glide
given only the horizontal position function u(t) and vertical position function v(t) ?
position
position
82 The glide ratio need not be a constant. Using our example of flight path of
glide
the ball one could ask: what is the INSTANTANEOUS RATE OF CHANGE of
RANGE with respect to HEIGHT, both functions of the same parameter t.
HEIGHT = y(t) = u.sin θ . t  1 g t 2
.
2
RANGE = x(t) = u.cos θ . t
.
,
du(x)
u (x)
=,
We have Le HOSPITAL’S RULE :
dv(x)
v (x)
Exerceise 1: Differentiate term by term the following:
Exer
erceise 1
x3 x5 x7
a) sin (x) = x   +    + . . .
3! 5! 7!
x2 x4 x6
b) cos (x) = 1   +    + . . .
2! 4! 6!
x
c) e x = 1 +  +
1! x2
x3
 +  + . . .
2!
3! Exercise 2: Differentiate term by term sin (ax), cos (ax), and eax .
Exer
ercise 2
Exer
ercise 3:
rule
Exercise 3 Differentiate sin (ax), cos (ax), and eax using the chain r ule.
It is fundamental for the student to understand how to find the INSTANTANEOUS RATE
OF CHANGE. He should feel at ease finding the INSTANTANEOUS RATE OF CHANGE of
elementary functions where the w ellbehav ed proper ties SINGLE VALUED,
ellbehav
CONTINUOUS and DIFFERENTIABLE are easily satisfied.
A function f(x) is an expression of CHANGE.
Given f(x) the expression of CHANGE,
AT
w e c a n D I F F E R E N T I AT E f ( x ) a n d f i n d f ’( x ) t h e
TA TA
AT
e x p r e s s i o n o f I N S TA N TA N E O U S R AT E O F C H A N G E .
83 17. Units of Measure A function may be dependent on more than one variable. For example f(x, y, z) may
be a function of three independent variables x, y and z along the three orthogonal
xaxis, yaxis and zaxis respectively. The function f(x, y, z) may describe the position
of an object in 3dimensional space.
With any calculation there are two par ts: the oper ation p ar t and the units
opera
oper
u nits
measure
differentia
of measu...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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