Function similar to f x = 1 2 π e − x 2 / 2 arise in a wide variety of statistical problems. (a) Use the first derivate test to show that f has a relative maximum at x = 0 , and confirm this by using a graphing utility of graph f . (b) Sketch the graph of f x = 1 2 π e − x − μ 2 / 2 where μ is a constant, and label the coordinates of the relative extrema.
Function similar to f x = 1 2 π e − x 2 / 2 arise in a wide variety of statistical problems. (a) Use the first derivate test to show that f has a relative maximum at x = 0 , and confirm this by using a graphing utility of graph f . (b) Sketch the graph of f x = 1 2 π e − x − μ 2 / 2 where μ is a constant, and label the coordinates of the relative extrema.
Function similar to
f
x
=
1
2
π
e
−
x
2
/
2
arise in a wide variety of statistical problems.
(a) Use the first derivate test to show that
f
has a relative maximum at
x
=
0
,
and confirm this by using a graphing utility of graph
f
.
(b) Sketch the graph of
f
x
=
1
2
π
e
−
x
−
μ
2
/
2
where
μ
is a constant, and label the coordinates of the relative extrema.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
3.
Consider the sequences of functions f₁: [-π, π] → R,
sin(n²x)
An(2)
n
f pointwise as
(i) Find a function ƒ : [-T,π] → R such that fn
n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?
Justify your answer.
[10 Marks]
1. (i) Give the definition of a metric on a set X.
[5 Marks]
(ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined
as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4,
d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer.
=
(iii) Consider a metric space (R, d.), where
=
[10 Marks]
0
if x = y,
d* (x, y)
5
if xy.
In the metric space (R, d*), describe:
(a) open ball B2(0) of radius 2 centred at 0;
(b) closed ball B5(0) of radius 5 centred at 0;
(c) sphere S10 (0) of radius 10 centred at 0.
[5 Marks]
[5 Marks]
[5 Marks]
(c) sphere S10 (0) of radius 10 centred at 0.
[5 Marks]
2. Let C([a, b]) be the metric space of continuous functions on the interval
[a, b] with the metric
doo (f,g)
=
max f(x)g(x)|.
xЄ[a,b]
= 1x. Find:
Let f(x) = 1 - x² and g(x):
(i) do(f, g) in C'([0, 1]);
(ii) do(f,g) in C([−1, 1]).
[20 Marks]
[20 Marks]
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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