Linear and Quadratic Approximations The linear and quadratic approximations or a function f at x = a are P 1 ( x ) = f ' ( a ) ( x − a ) + f ( a ) and P 2 ( x ) = 1 2 f ″ ( a ) ( x − a ) 2 + f ' ( a ) ( x − a ) + f ( a ) In Exercises 177-180, (a) find the specified linear and quadratic approximations of f , (b) use a graphing utility to graph f and the approximations, (c) determine whether P 1 or P 2 is the better approximation, and (d) state how the accuracy changes as you move farther from x = a . f ( x ) = ln x ; a = 1
Linear and Quadratic Approximations The linear and quadratic approximations or a function f at x = a are P 1 ( x ) = f ' ( a ) ( x − a ) + f ( a ) and P 2 ( x ) = 1 2 f ″ ( a ) ( x − a ) 2 + f ' ( a ) ( x − a ) + f ( a ) In Exercises 177-180, (a) find the specified linear and quadratic approximations of f , (b) use a graphing utility to graph f and the approximations, (c) determine whether P 1 or P 2 is the better approximation, and (d) state how the accuracy changes as you move farther from x = a . f ( x ) = ln x ; a = 1
Solution Summary: The author explains that the slope of the function f(x)=mathrmsinax at origin is a.
Linear and Quadratic Approximations The linear and quadratic approximations or a function f at
x
=
a
are
P
1
(
x
)
=
f
'
(
a
)
(
x
−
a
)
+
f
(
a
)
and
P
2
(
x
)
=
1
2
f
″
(
a
)
(
x
−
a
)
2
+
f
'
(
a
)
(
x
−
a
)
+
f
(
a
)
In Exercises 177-180, (a) find the specified linear and quadratic approximations of f, (b) use a graphing utility to graph f and the approximations, (c) determine whether
P
1
or
P
2
is the better approximation, and (d) state how the accuracy changes as you move farther from
x
=
a
.
Find a plane containing the point (3, -3, 1) and the line of intersection of the planes 2x + 3y - 3z = 14
and -3x - y + z = −21.
The equation of the plane is:
Determine whether the lines
L₁ : F(t) = (−2, 3, −1)t + (0,2,-3) and
L2 : ƒ(s) = (2, −3, 1)s + (−10, 17, -8)
intersect. If they do, find the point of intersection.
● They intersect at the point
They are skew lines
They are parallel or equal
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