Write the composite function in the form f ( g ( x ) ) . [Identify the inner function u = g ( x ) and the outer function y = f ( u ) . ] Then find the derivative d y / d x . 1. y = 5 − x 4 3
Write the composite function in the form f ( g ( x ) ) . [Identify the inner function u = g ( x ) and the outer function y = f ( u ) . ] Then find the derivative d y / d x . 1. y = 5 − x 4 3
Solution Summary: The author calculates the composite function for y=(5-x4)3, wherein the inner function and the outer function are located.
Write the composite function in the form
f
(
g
(
x
)
)
. [Identify the inner function
u
=
g
(
x
)
and the outer function
y
=
f
(
u
)
.
]
Then find the derivative
d
y
/
d
x
.
18. Using the method of variation of parameter, a particular solution to y′′ + 16y = 4 sec(4t) isyp(t) = u1(t) cos(4t) + u2(t) sin(4t). Then u2(t) is equal toA. 1 B. t C. ln | sin 4t| D. ln | cos 4t| E. sec(4t)
Question 4. Suppose you need to know an equation of the tangent plane to a
surface S at the point P(2, 1, 3). You don't have an equation for S but you know
that the curves
r1(t) = (2 + 3t, 1 — t², 3 − 4t + t²)
r2(u) = (1 + u², 2u³ − 1, 2u + 1)
both lie on S.
(a) Check that both r₁ and r2 pass through the point P.
1
(b) Give the expression of the 074 in two ways
Ət
⚫ in terms of 32 and 33 using the chain rule
მყ
⚫ in terms of t using the expression of z(t) in the curve r1
(c) Similarly, give the expression of the 22 in two ways
Əz
ди
⚫ in terms of oz and oz using the chain rule
Əz
მყ
•
in terms of u using the expression of z(u) in the curve r2
(d) Deduce the partial derivative 32 and 33 at the point P and the equation of
მე
მყ
the tangent plane
Coast Guard Patrol Search Mission The pilot of a Coast Guard patrol aircraft on a search mission had just spotted a disabled fishing trawler and
decided to go in for a closer look. Flying in a straight line at a constant altitude of 1000 ft and at a steady speed of 256 ft/s, the aircraft passed directly over
the trawler. How fast (in ft/s) was the aircraft receding from the trawler when it was 1400 ft from the trawler? (Round your answer to one decimal places.)
1000 ft
180
× ft/s
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