Name:
1
2
3
4
5
6 (Bonus):
Exam 1 : September 17, 2013
Problem 1. Let g(x) =
2
p
2
x10 + 4e3x . Then g (6) (0), the sixth derivative of g(x) at x = 0, is
(a) −729
(b) 0
(c)
8505
8
(d) 6480
(e) 272160
Problem 2. Consider the region trapped between the two curves y = 100(x − 1)2 ex−3 − 5
and y = −(x + 4)(x − 2) as pictured below.
30
20
10
-5
-4
-3
The area of this region is approximately
(a) 127.555
(b) 39.0688
(c) 35.3667
(d) 32.586
(e) 27.952
-2
-1
1
2
Exam 1 : September 17, 2013
3
Approximating π
While one can use the power series for arctan(x) and the fact that π4 = arctan(1) to approximate π, the power series for arctan(x) at x = 1 converges slowly. A better way is to use the
fact that
π
1
1
+ arctan
= arctan
4
2
3
to express π as a power series that converges quickly.
Problem 3. Using a ninth degree
polynomial
approximation for arctan(x) and the fact that
1
1
π = 4 arctan 2 + arctan 3 yields which approximation of π?
(a)
498668825
158723712
(b) 3.14085
(c)
1538665
489888
(d) 3.141561588
(e) 3.33968
Problem 4. If we use the fact that π = 4 arctan 21 + arctan 13 and a polynomial approximation for arctan(x) to approximate π to 100 decimal places, how many
terms of arctan(x) should we use?
(a) 8
(b) 19
(c) 39
(d) 101
(e) 329
Exam 1 : September 17, 2013
4
Problem 5. If we use the fact that
Z
π = 4 arctan(1) = 4
0
1
dt
1 + t2
and approximate this integral using a Riemann sum with 100 rectangles with heights calculated
via the midpoints, then we get π ≈
(a) 3.14152
(b) 3.15157
(c) 3.15166
(d) 3.16150
(e) 3.16381
Problem 6. (Bonus) When asked to produce a program to approximate the π by approximating the area of a circle, two students responded as follows:
Response 1
In[28]:=
In[29]:=
Out[29]=
rightsum!f_, "x_, a_, b_#, n_$ :!
Module!"deltax#,
deltax ! %b " a& ' n;
Sum!deltax f '. x "# a $ deltax % %i&, "i, 1, n#$
$
N!2 rightsum!Sqrt!1 " x ^ 2$, "x, " 1, 1#, 20$, 10$
3.104518326
Response 2
In[22]:=
In[32]:=
Out[32]=
trapezoid!f_, "x_, a_, b_#, n_$ :!
Module!"deltax#,
deltax ! %b " a& ' n;
deltax % %f '. x & a $ f '. x & b& ' 2 $ Sum! f '. %x "# a $ deltax % %i&&, "i, 1, n " 1#$&
$
N!4 trapezoid!Sqrt!1 " x ^ 2$, "x, 0, 1#, 10$, 10$
3.104518326
For a bonus point, explain why the two different responses produced identical results.
EXAM
Exam 1
Math 624
September 17, 2013
• You may use Mathematica (including the Mathematica
Documentation) but you must start with an empty notebook.
• Absolutely no internet use is permitted during the exam.
• Each problem is worth 1 point for a total of 5 points, with a
possible bonus point.
• Please mark your answers clearly on the answer key.
Success!