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
note that at the point of intersection a :
f ’(a) = m f = tan θ f
g ’(a) = mg = tan θ g
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
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) 2-Dimension
y(t) 1-Dimension u Height Height y(t) y y(t) = u . s i n θ . t -- -2 θ Range x(t) (0, 0) x y(t1)
/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
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- Fall '09
- Limit, Δx