Solutions to MI 4 Problem set 2 F11
Introduction: This problem set continues some ideas from the MI 3 problem sets (see comments on MI 3 problem set 6 from fall 10) as well as begins some new themes. At this point in the course students are well into their unit on sequences and series, so several problems extend that work using technology. At IMSA solutions to problem sets are posted so students can study them. We encourage this for problems they may miss but also for learning new techniques or gaining insight as is shown in problem #1 below.
1) Comment: This problem begins a series of problems on partial fractions. This first attempt is given with no instruction. Most students manage to solve this with a system of equations or trial an error. When the set is returned would be a good time to demonstrate the general idea of partial fractions. At IMSA we do not ‘formally’ teach partial fractions, but find this series of problems extended over several sets is enough work on the topic.
Soln:
Let then
Let then
2) Comment: This problem continues a theme begun in MI 3 of using interval graphs to solve a problem.
Soln:
and (from original domain constraints)
or
3) Comment: This is a good example of taking advantage of CAS. IMSA students have TI-89 access. Wolframalpha will also work with the factor command.
Soln:
or or
4) Comment: This problems has students think about sequences that get generated by an iterative geometric process.
Soln: a) 2
b) or
c)
or
5) Comment: This problem asked students to think about generating a formula for a sequence of partial sums. In their unit in class IMSA students will be asked to consider the .
Soln: a) 1, 2, 4, 8, 16, 32, 64
b) 1, 3, 7, 15, 31, 63, 127
c) or
6) Comment This is a review from MI 3 log properties.
Soln:
7) Comment: This problem is more practice at sequence generation with the calculator. The are not many opportunities to use the sequence mode on the calculator in the typical class. This skill on the calculator will come to good use as the theme begun in problem 16 carries through the next few problem sets.
Soln: a) 1, 0.5403, 0.8576, 0.6543
b) 0.7391
8) Comment: A unit in MI 2 used 2X2 matrices to do some basic transformations on figures in the plane. This problem extends using matrices for other transformations found in linear algebra, although the problems sets do not go any further with this topic, it does leave that possibility.
Soln: a)
b) This transformation rotates points 90° clockwise or 270° counterclockwise.
9) Comment: More interval practice, but this time considering issues arising from the composition of functions.
Soln:
10) Comment; IMSA students learned half of their trigonometry in MI 3, including graphing sinusoidal functions. This problem is review in the context of an application.
Soln:
Amp: 220 Period: 6 phase shift: 4.75 vertical shift: 220
11) Comment: Students are practicing formulas developed in their first unit of MI 4 in this problem.
Soln: a) is geometric with 11 terms
b) is an infinite geometric series with
c)
12) Comment: This problem gets students to use CAS to do basic summations. Hopefully they will think to use it in future problem sets and calculus. Students should be surprised by their exact answer to part b. Part c opens the opportunity to discus using the finance application found on TI calculators
Soln: a) or 3.196
b)
c)
13) Comment: This problem continues what was begun in part c of problem 12. Hopefully there is some motivation given by the application as to why one might study sequences and series.
Soln: a)
5000, 10410, 16264, 22597, 29450
b) $ 1,477,408
c) we have $ 1,065,092
During his 36th year of saving.
14) Comment: This problem introduces the concept of the limit in a recursive sequence. It can lead to the idea of a fixed point which is what this limit is. This opens the possibility of further study in an application of managing a forest.
Soln: a)
10000, 9900, 9808, 9723, 9645
b) On home screen type:
c)
It appears to be approaching a steady population of 8750 trees.
15) Comment: Most students have encountered the Fibonacci sequence in middle school. This problem gives students more practice at using the recursive features of the clalculator and sets them up for a theme begun in the next problem.
Soln:
16) Comment: This problem begins a series of problems that explore ratios in Fibonacci type sequences and more generally Lucus sequences. In a future problem set it will be tied to extended fractions and eventually the explicit formula for the Fibonacci sequence and other type sequences. This theme shows some very rich mathematics can be developed through student exploration using technology.
Soln: a) 1, 2, 1.5, 1.667, 1.6
b)
c) is approaching the Golden Ratio
17) Comment: This problem is a review problem from the first half of trigonometry students did in MI 3. We would expect students to take a geometrical approach to the problem. These problems will be explored further as students do their analytical study of trig later in the term.
Soln:
a)
b)