Consider the function f: RR defined by 1 (a) (b) By using definition, explain whether f is differentiable on R. Deduce the continuity of f on R. Let f' be the first derivative of f computed in (a). Is is is Riemann integrable on [-1, 1]? Explain.
Consider the function f: RR defined by 1 (a) (b) By using definition, explain whether f is differentiable on R. Deduce the continuity of f on R. Let f' be the first derivative of f computed in (a). Is is is Riemann integrable on [-1, 1]? Explain.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
For a,b and c
![Consider the function f: RR defined by
f(x) = 12x³.
By using definition, explain whether f is differentiable on R. Deduce
the continuity of f on R.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
1
f'
Let f' be the first derivative of f computed in (a). Is is Riemann
integrable on [-1, 1]? Explain.
Evaluate
with the partition
S
1
f'(x)
Pn = {1, fill in, the blank
de by using the definition of Riemann integral
Hint: (1) Each (sub)interval is given by
(2) Use the formulae
i=1
[fill in, the blank], 1≤i≤n.
1
(n+i-1)2
,2}, ne N.
1
2n
3
7
+
8n² 48n³
+0
n5
1
1
3
7
=
(n+ 5)² - 2 + 80² + 45m² +0 (-1).
2n
8n2 48n³
i=1
Determine the range f(A) where A = (a, 0] for some fixed a < 0,a #
Determine the supremum sup(f(A)\{0}), if exists.
Is f(A) open in R? Explain.
Is f(A) connected? Explain.
Is f(A) compact? Explain.
x
n
For each n E N, consider n =
Define the sequence {9n} by 9n (x):= f(xn),
of {9} if exists.
with x R+ = {x ER | x>0}.
ER+. Find the limit function](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60abd4de-ba2d-4a53-b8fc-e1d2e0799cec%2F500fc7a6-74a1-4e53-b887-f6fbe16d41b5%2Fm9r3qy8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function f: RR defined by
f(x) = 12x³.
By using definition, explain whether f is differentiable on R. Deduce
the continuity of f on R.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
1
f'
Let f' be the first derivative of f computed in (a). Is is Riemann
integrable on [-1, 1]? Explain.
Evaluate
with the partition
S
1
f'(x)
Pn = {1, fill in, the blank
de by using the definition of Riemann integral
Hint: (1) Each (sub)interval is given by
(2) Use the formulae
i=1
[fill in, the blank], 1≤i≤n.
1
(n+i-1)2
,2}, ne N.
1
2n
3
7
+
8n² 48n³
+0
n5
1
1
3
7
=
(n+ 5)² - 2 + 80² + 45m² +0 (-1).
2n
8n2 48n³
i=1
Determine the range f(A) where A = (a, 0] for some fixed a < 0,a #
Determine the supremum sup(f(A)\{0}), if exists.
Is f(A) open in R? Explain.
Is f(A) connected? Explain.
Is f(A) compact? Explain.
x
n
For each n E N, consider n =
Define the sequence {9n} by 9n (x):= f(xn),
of {9} if exists.
with x R+ = {x ER | x>0}.
ER+. Find the limit function
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