Unformatted text preview: efor zer neg tiv after.
ero
positi v e bef or e , z er o , ne g a ti v e after.
What can you say about f ”(a ) ?
a
Exercise
Exercis e 1: Given that you know the initial velocity u and the angle of
projection θ , how long will it take to reach a maximum height ?
Exercise 2: What is the duration of flight ?
Exercise 3: What is the range ? 112 2 2. Decr easing, MINIMUM, Incr easing
y P1 P3 P2 φ1
x1 x2 = 0 φ3
x3 x Consider the function y = x 2
dy
 = y ’(x) = 2x
dx
The tang ent to the cur ve of y = x 2 a t P 1 w ill inter sect the x axis
tangent
at obtuse angle φ 1 .
The SIGN of tan φ 1 i s < 0, negative.
The f ir st deri v a ti v e of y = x 2 e valuated at instant x 1
irst deriv tiv
y ’(x 1 ) = tan φ 1 = 2 x 1 must be negative.
negative.
decreasing
easing.
When x < 0 we note that the function y = x 2 i s decr easing. We also note
that the SIGN of the tangent or equivalently the value of the f ir st deri v a ti v e
tangent
irst deriv tiv
y ’(x) evaluated at any instant x < 0 is negative.
113 We can say: if y(x) is decr easing then y ’(x) is ne g a ti v e (<0).
decreasing
neg tiv
decreasing
easing.
Conversely : if y ’(x) is ne g a ti v e then y(x) must be decr easing
neg tiv
At P 2 , where y = x 2 i s minim um the tangent to the cur ve is PARALLEL to
minimum
um,
tangent
the xaxis. Hence φ 2 = 0 . So the f ir st deri v a ti v e of y = x 2 e valuated at
irst deriv tiv
instant x 2 is :
y ’(x 2 ) = tan φ 2 = 2 x 2 = 0
The tang ent to the cur ve of y = x 2 a t P 3 w ill inter sect the xaxis
tangent
at acute angle φ 3 .
positiv
The SIGN of tan φ 3 i s > O, positi v e .
The f ir st deri v a ti v e of y = x 2 e valuated at instant x 3
irst deriv tiv
y ’(x 3 ) = tan φ 3 = 2 x 3 must be positive
positive.
increasing
easing.
When x > 0 we note that the function y = x 2 i s incr easing We also note
irst deriv tiv
that the SIGN of the tangent or equivalently the value of the f ir st deri v a ti v e
y ’(x) evaluated at any instant x > 0 is positive
positive.
We can say : if y(x) is incr easing then y ’(x) is posi...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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