Unformatted text preview: nimum
φ f , g = minim um ( θ f − θ g, 180 o −  θ f − θ g).
m inim And, if f(x) and g(x) intersect at right angles at a, then : m f . mg = − 1
This is the Analytical geometr y point of view. From the Analysis point of view, we
Analytical geometr
eometry
Analysis
Anal
Anal
note that at the point of intersection a :
tan
f ’(a) = m f = tan θ f
g ’(a) = mg = tan θ g
tan
So, to know if the two curves f(x) and g(x) intersect at right angles at a , all we have
to do is verify if :
f ’(a) . g’(a) = − 1 107 21.
21 . Increasing
easing,
Decreasing
Incr easing , MAXIMUM, Decr easing VA
M OT I VAT I O N
You are standing on the ground with a height measuring instrument. Your
friend is projected into the air in a module equipped with only a ver tical
speedometer. Using your height measuring instrument you can measure
the height of the module at any instant: so many meters above the ground.
Your friend, by looking at the ver tical speedometer, can know his ver tical
speed at any instant: climbing at so many meters per second or descending
at so many meters per second.
Without communicating with your friend and using only the information you
have  the height at any instant, how can you know what your friend knows the ver tical speed of the module at any instant ? Can you tell when the
ver tical speed is zero ?
Vice versa, how can your friend, with only the information he has  his ver tical
speed at any instant, find out what you know  his height at any instant ? Can
your friend tell when he has reached a maximum height and exactly what is
the maximum height ?
1 g t2 (0, 0) 2Dimension
y(t) 1Dimension u Height Height y(t) y y(t) = u . s i n θ . t  2 θ Range x(t) (0, 0) x y(t1)
h1
t1 T
/2 T t VERTICAL POSITION POSITION 108 When an object is projected into the air with a given initial velocity u and
angle of projection θ , there are two functions that describe its position.
A VERTICAL function y(t) which describes its HEIGHT at any INSTANT and a
HORIZONTAL function x(t) which describes its RANGE at any INSTANT.
We know from Dynamics that :
Range function x(t) = u. cos θ . t = u. sin θ . t   g t 2
 1
2
Geometrically
Geometricall...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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