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Unformatted text preview: units of measure. If we know the present altitude we
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
given only the horizontal position function u(t) and vertical position function v(t) ?
82 The glide ratio need not be a constant. Using our example of flight path of
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
RANGE = x(t) = u.cos θ . t
We have Le HOSPITAL’S RULE :
Exerceise 1: Differentiate term by term the following:
x3 x5 x7
a) sin (x) = x -- --- + --- -- --- + . . .
3! 5! 7!
x2 x4 x6
b) cos (x) = 1 -- --- + --- -- --- + . . .
2! 4! 6!
c) e x = 1 + --- +
--- + --- + . . .
3! Exercise 2: Differentiate term by term sin (ax), cos (ax), and eax .
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 ell-behav ed proper ties SINGLE VALUED,
CONTINUOUS and DIFFERENTIABLE are easily satisfied.
A function f(x) is an expression of CHANGE.
Given f(x) the expression of CHANGE,
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
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
x-axis, y-axis and z-axis respectively. The function f(x, y, z) may describe the position
of an object in 3-dimensional space.
With any calculation there are two par ts: the oper ation p ar t and the units
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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