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.
3. (i) Consider a mapping F: RN
Rm. Explain in your own words
the relationship between the existence of all partial derivatives of F and dif-
ferentiability of F at a point x = RN.
(ii)
[3 Marks]
Calculate the gradient of the following function f: R2 → R,
f(x) = ||x||3,
Total marks 10
where ||x|| = √√√x² + x/2.
[7 Marks]
1.
(i)
(ii)
which are not.
What does it mean to say that a set ECR2 is closed?
[1 Mark]
Identify which of the following subsets of R2 are closed and
(a)
A = [-1, 1] × (1, 3)
(b)
B = [-1, 1] x {1,3}
(c)
C = {(1/n², 1/n2) ER2 | n EN}
Provide a sketch and a brief explanation to each of your answers.
[6 Marks]
(iii) Give an example of a closed set which does not have interior
points.
[3 Marks]
A company specializing in lubrication products for vintage motors produce two
blended oils, Smaza and Nefkov. They make a profit of K5,000.00 per litre of
Smaza and K4,000.00 per litre of Nefkov. A litre of Smaza requires 0.4 litres of
heavy oil and 0.6 litres of light oil. A litre of Nefkov requires 0.8 litres of heavy oil
and 0.2 litres of light oil. The company has 100 litres of heavy oil and 80 litres of
light oil. How many litres of each product should they make to maximize profits
and what level of profit will they obtain? Show all your workings.
Chapter 3 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
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