James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

208 Chapter 3 Differentiation Rules
APPLIED Project
where should a pilot start descent?
y
y=P(x)
h
An approach path for an aircraft landing is shown in the figure and satisfies the following
conditions:
(i) The cruising altitude is h when descent starts at a horizontal distance , from touchdown
at the origin.
(ii) The pilot must maintain a constant horizontal speed v throughout descent.
(iii) The absolute value of the vertical acceleration should not exceed a constant k (which is
much less than the acceleration due to gravity).
0
x
1. Find a cubic polynomial Psxd − ax 3 1 bx 2 1 cx 1 d that satisfies condition (i) by
imposing suitable conditions on Psxd and P9sxd at the start of descent and at touchdown.
2. Use conditions (ii) and (iii) to show that
6hv 2
, 2 < k
3. Suppose that an airline decides not to allow vertical acceleration of a plane to exceed
k − 860 miyh 2 . If the cruising altitude of a plane is 35,000 ft and the speed is 300 miyh,
how far away from the airport should the pilot start descent?
;
4. Graph the approach path if the conditions stated in Problem 3 are satisfied.
The functions that we have met so far can be described by expressing one variable explicitly
in terms of another variable—for example,
y − sx 3 1 1 or y − x sin x
or, in general, y − f sxd. Some functions, however, are defined implicitly by a relation
between x and y such as
1 x 2 1 y 2 − 25
or
2 x 3 1 y 3 − 6xy
In some cases it is possible to solve such an equation for y as an explicit function (or
several functions) of x. For instance, if we solve Equation 1 for y, we get y − 6s25 2 x 2 ,
so two of the functions determined by the implicit Equation l are f sxd − s25 2 x 2 and
tsxd − 2s25 2 x 2 . The graphs of f and t are the upper and lower semicircles of the
cir cle x 2 1 y 2 − 25. (See Figure 1.)
y
y
y
0 x
0 x
0 x
FIGURE 1
(a) ≈+¥=25
(b) ƒ=œ„„„„„„ 25-≈
(c) ©=_œ„„„„„„ 25-≈
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