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
.
Differential Calculus:
1. Show or prove that the degree of the HDE is 6.
2.Derive the integrable functions(variable substitution).
3.Find the general solution.
Calculus
Use linear algebra to solve the system of differential equations with inital values x(0) = -2, x(0) = 3.
Differential Equations
Determine in each exercise whether the function is homogenous. If it is homogenous, state the degree of the function.
a. x ln x - x ln y
b. tan x
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