Calculus II Midterm II – Solutions
1. Find
, the nth term of the sequence, starting with
2. For the sequence in problem 1, find
3. If
, and
:
.
, then
=>
.
The quadratic forumula =>
4. Suppose
. What's this series' interval of convergence?
This is the geometric series
for
, with
, hence
for
=>
=>
The interval of convergence is thus
.
=>
5. For the function in problem 4,
=>
6. What improper integral could be used to determine the convergence or divergence of the infinite
series
?
7. Evaluate your integral from problem 6. What value do you get, and what does that say about the
convergence of the series in #6?
Let
, so
, and the integral becomes:
, but this clearly diverges.
Thus the corresponding infinite sum must also diverge.
8. Use the ratio test to determine the interval of convergence of the series
This expression → 0, as
→
.
, for all values of , so the interval of convergence is
9. Consider the series
.
Let
.
.
Using the (built-in) error bound for the alternating series,
what's the smallest that assures that
?
Extra credit:
?
This is an alternating series, and so the error from approximating the infinite sum with terms is
bounded by the absolute value of the next term, the
term. To solve this, simply observe the
values of the terms.
,
,
The third term is much smaller than
the series. Thus
.
, and so will the error be, if we only use the first two terms of
The first two terms of the series add up to:
10. Does the series converge or diverge?
(a)
The terms don't shrink to
(c)
(b)
as
→
, so both series must diverge.
(d)
These grow like the harmonic series. So (c) diverges, while (d) converges, since
it's terms shrink to 0, which implies convergence for an alternating series.
.
(e)
(f)
Both converge; just compare with the -series with
11. Find the best cubic approximation to
.
at
.
So the best-fit cubic is
12. Find the Taylor polynomial approximation (centered at x=0) of degree 10 for the function
.
We know that
, so we can substitute
in place of
to get:
=>
13. Find the Maclaurin series expansion of the function
.
.
Since
, for
, we get:
14. Find the Maclaurin series expansion of the function
We know that
, so we can substitute
.
in place of
to get:
so
.
Changing variables, let
. Then
.
NOTE: For problems 15 – 17, let
,
parametrically define a famous conic section.
15. Find any Cartesian equation for this parametric curve:
Then
and
,
. Since
, we have:
→
16. What's the name of this conic section ?
17. Find
at
Answer: This curve is called an ellipse.
for this parametric curve.
,
=>
,
Thus,
. When
Hence,
so
.
, then
.
and also
.
18. This one is intuitively obvious!
;-)
19. Find a Cartesian equation for the polar curve
Multiply both sides by to get
.
. Then
.
20. For the polar equation in problem 19, find
at the point
.
Since we have the Cartesian equation, we might as well solve for the Cartesian coordinates of
the point with polar coordinates
. We have
and
We seek
at the point
. Implicitly differentiating
.
Now just substitute
, and solve for
==>
==>
:
, we get: