Re: 2009 UH calc test: Some "oldschool" answers Message #10 Posted by Karl Schneider on 7 Nov 2009, 2:31 p.m., in response to message #4 by Crawl
"Crawl" 
Thanks for your answers. Certainly, a modern calc with builtin graphing, fast processing, and mathematical functions such as prime number and maybe LCM, LCD, etc. would helpful for this exercise.
Still, even a venerable HP34C, HP11C, or HP15C is useful for some of the problems requiring more than straightforward transcedental functions:
Question 3
Find the smallest natural number k so that 1 + 1/2 + 1/3 + 1/4 + ... + 1/k > 7.2
I'd forgotten all about the EulerMascheroni Constant, which had been discussed in the Forum, and was surely was intended to be utilized for analyzing this harmonic series. As "Crawl" pointed out, the "rate of divergence" equation will give a very accurate estimate of k. However, ignoring the error 'epsilon' in the equation will lead the user to assume that the correct answer is 753, not 752 (for which 'epsilon' is about 0.00072418).
Directlycalculated solution  the sum is not too difficult to automate using ISG:
Program: Execution:
LBL A 1.99901
RCL I STO I
INT 7.2
1/x GSB A
 ...
x < 0? RCL I
RTN INT
ISG (or ISG I) (752.)
GTO A
RTN
Question 5
Give the sum of the x coordinates for the points of intersection of the graphs of
f(x) = sin(x) + 2x
g(x) = 5  x^{2}
SOLVE in the HP34C and HP15C will tackle this, but they will find only one root at a time, so the guesses that define initial search ranges must be selected intelligently. Don't forget RADian mode:
Program: Execution:
LBL B 0
RAD ENTER
2 2
+ SOLVE B
* (1.247614363)
5 STO 0
 3
x<>y ENTER
SIN 4
+ SOLVE B
RTN (3.397303973)
RCL+ 0
(2.149689610)
Question 6:
Approximate the largest value of f(x) = x^{8} + 787x^{4} + 673x^{3} + 521x^{2} + 840x + 12.
x^{8} is a symmetrical term dominant for largermagnitude values of x, reducing the sum of the remaining terms (except at x = 0). Since all the coefficients for the lowerorder terms are positive, it's reasonably clear that, when x > 0, the lowerorder terms all contribute positively in a monotonicallyincreasing sum, providing the maximum function value. Therefore, there is indeed only one root for the derivative of the complete function for x > 0, indicating a local maximum; the largest value is the function evaluated at that point.
Using SOLVE between 0 and 10 to find the root of the derivative yields 4.619292584, consistent with what "Crawl" found. Programming the derivative using Horner's Method will speed things up:
Program: Execution:
LBL 0 0
* ENTER
* 10
* SOLVE 0
8 ...
* (4.619292584)
3148 ENTER
+ ENTER
* ENTER
2019 *
+ *
* *
1042 CHS
+ 787
* +
840 *
+ 673
RTN +
*
521
+
*
840
+
*
12
+
(232,366.5024)
(coefficients merged for clarity)
Question 10
Give the y coordinate of the solution to the system
31x  29y = 43
51x + 19y = 16
in reduced fraction form.
The approach, of course, is to utilize
[a b] 1 [ d b]
inv   =  *  
[c d] ad  bc [c a]
and multiply to solve for x and y.
Answers can be checked with the HP15C determinant and matrix solutions.
 KS
Edited: 9 Nov 2009, 1:17 a.m.
