Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Graded Question #1: Unit B Exercise 18 Page 93
Write each of the numbers in scientific notation.
a. 3578
b. 984.35
c. 0.0058
d. 624.87
e. 0.0003005
f. 98.180004
Graded Question #2
Compare the numbers in each pair and give the factor by which the numbers differ.
Example:
is
, or 100, times as large as
.
a.
b.
c. 1 million, 1 billion
Graded Question #3: Unit C Exercise 20 Page 106
State the number of significant digits and explain your answer.
350,000 yr
Graded Question #4: Unit C Exercise 22 Page 106
State the number of significant digits and explain your answer.
0.004502 mGraded Question #5: Unit C Exercise 26 Page 106
State the number of significant digits and explain your answer.
mi
Graded Question #6: Unit C Exercise 56 Page 107
Multiply 12 cubic feet by 62.4 lb./cubic foot. Use the appropriate rounding rules to do the
calculation. Express the result with the correct number of significant digits.
Graded Question #7: Unit C Exercise 62 Page 107
Garden mulch costs $46 per cubic yard. How much does it cost to fill your trucks 1.25 cubic yard
cargo area with mulch? Use the appropriate rounding rules to do the calculation. Express the result
with the correct number of significant digits.
referring to the image;
comment on the math assignment
Requirements: Step by Step solution | .doc file
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Complete Solution
(a) Continuity at
=
0
x=0
For
0
x
=0,
?
(
)
?
=
?
2
sin
(
1
/
2
)
?
?
2
?
?f(x)?=?x
2
sin(1/x
2
)??x
2
?
Since:
lim
0
2
=
0
x0
lim
x
2
=0
By the Squeeze Theorem:
lim
0
(
)
=
0
=
(
0
)
x0
lim
f(x)=0=f(0)
f is continuous at
0
0.
(b) Differentiability at
=
0
x=0
Using the definition:
(
0
)
=
lim
0
(
)
(
0
)
=
lim
0
2
sin
(
1
/
2
)
=
lim
0
sin
(
1
/
2
)
f
(0)=
h0
lim
h
f(h)f(0)
=
h0
lim
h
h
2
sin(1/h
2
)
=
h0
lim
hsin(1/h
2
)
Since:
1
sin
(
1
/
2
)
1
?
?
sin
(
1
/
2
)
?
?
1sin(1/h
2
)1?h?hsin(1/h
2
)?h?
Thus:
lim
0
sin
(
1
/
2
)
=
0
h0
lim
hsin(1/h
2
)=0
f is differentiable at
0
0, and
(
0
)
=
0
f
(0)=0.
(c) Formula for
(
)
f
(x) for
0
x
=0
Differentiate:
(
)
=
2
sin
(
1
/
2
)
f(x)=x
2
sin(1/x
2
)
Using product and chain rules:
(
)
=
2
sin
(
1
/
2
)
+
2
cos
(
1
/
2
)
(
2
3
)
f
(x)=2xsin(1/x
2
)+x
2
cos(1/x
2
)(
x
3
2
)
Simplifying:
(
)
=
2
sin
(
1
/
2
)
2
cos
(
1
/
2
)
f
(x)=2xsin(1/x
2
)
x
2
cos(1/x
2
)
(d) Boundedness of
(
)
f
(x) near
0
0
Consider:
(
)
=
2
sin
(
1
/
2
)
2
cos
(
1
/
2
)
f
(x)=2xsin(1/x
2
)
x
2
cos(1/x
2
)
The first term
2
sin
(
1
/
2
)
0
2xsin(1/x
2
)0.
The second term:
?
2
cos
(
1
/
2
)
?
=
2
?
?
x
2
cos(1/x
2
)
=
?x?
2
which diverges to infinity as
0
x0.
(
)
f
(x) is unbounded in every neighborhood of
0
0.
(e) Riemann integrability of
(
)
f
(x) on
[
1
,
1
]
[1,1]
Although
(
)
f
(x) is unbounded near
0
0, it is discontinuous only at a single point
=
0
x=0.
A function that is:
bounded on compact intervals except possibly at finitely many points, and
has a finite number of discontinuities,
is Riemann integrable.
Alternatively, note:
1
1
(
)
=
(
1
)
(
1
)
1
1
f
(x)dx=f(1)f(1)
by the Fundamental Theorem of Calculus, since
f is differentiable everywhere and continuous on
[
1
,
1
]
[1,1].
Thus:
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].
Requirements:
Define the function
(
)
=
{
2
sin
?
(
1
2
)
,
0
,
0
,
=
0.
f(x)=
x
2
sin(
x
2
1
),
0,
x
=0,
x=0.
(a)
Prove that
(
)
f(x) is continuous at
=
0
x=0.
(b)
Determine whether
(
)
f(x) is differentiable at
=
0
x=0. If so, find
(
0
)
f
(0).
(c)
Find an explicit formula for
(
)
f
(x) for
0
x
=0.
(d)
Decide whether
(
)
f
(x) is bounded on any neighborhood of
=
0
x=0.
(e)
Determine whether
(
)
f
(x) is Riemann integrable on
[
1
,
1
]
[1,1].v
Requirements: