Linear and Quadratic Approximations The linear and quadratic approximations of 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 55-58, (a) find the specified linear and quadratic approximations of f , and (b) use a graphing utility to graph f and the approximations. f ( x ) = arctan x , a = 0
Linear and Quadratic Approximations The linear and quadratic approximations of 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 55-58, (a) find the specified linear and quadratic approximations of f , and (b) use a graphing utility to graph f and the approximations. f ( x ) = arctan x , a = 0
Solution Summary: The author explains how to find the linear and quadratic approximations of f.
Linear and Quadratic Approximations The linear and quadratic approximations of 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 55-58, (a) find the specified linear and quadratic approximations of f, and (b) use a graphing utility to graph f and the approximations.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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