|Re: 2009 UH calc test: Some "old-school" answers|
Message #10 Posted by Karl Schneider on 7 Nov 2009, 2:31 p.m.,
in response to message #4 by Crawl
Thanks for your answers. Certainly, a modern calc with built-in graphing, fast processing, and mathematical functions such as prime number and maybe LCM, LCD, etc. would helpful for this exercise.
Still, even a venerable HP-34C, HP-11C, or HP-15C is useful for some of the problems requiring more than straightforward transcedental functions:
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 Euler-Mascheroni 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).
Directly-calculated solution -- the sum is not too difficult to automate using ISG:
LBL A 1.99901
RCL I STO I
1/x GSB A
x < 0? RCL I
ISG (or ISG I) (752.)
Give the sum of the x coordinates for the points of intersection of the graphs of
f(x) = sin(x) + 2x
g(x) = 5 - x2
SOLVE in the HP-34C and HP-15C 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:
LBL B 0
+ SOLVE B
5 STO 0
+ SOLVE B
Approximate the largest value of f(x) = -x8 + 787x4 + 673x3 + 521x2 + 840x + 12.
-x8 is a symmetrical term dominant for larger-magnitude values of x, reducing the sum of the remaining terms (except at x = 0). Since all the coefficients for the lower-order terms are positive, it's reasonably clear that, when x > 0, the lower-order terms all contribute positively in a monotonically-increasing 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:
LBL 0 0
* SOLVE 0
(coefficients merged for clarity)
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 HP-15C determinant and matrix solutions.
Edited: 9 Nov 2009, 1:17 a.m.