HP 49G Guide For:
"Precalculus with Limits:
A Graphing Approach"
by Larson, Hostetler, Edwards (Third Edition)
Guide Authored by Micah Croff, ©2001 (v 1.0)
For Math 27, Santa Rosa Junior College
Santa Rosa, CA, USA
Table Of ContentsIntroduction Copyright Information Chapter 1 Chapter 1.1 - Functions Chapter 1.2 - Graphs of Functions Chapter 1.4 - Combinations Of Functions Chapter 2 Chapter 2.2 - Polynomial Functions of Higher Degree Chapter 2.4 - Complex Numbers Chapter 3 Chapter 3.3 - Properties of Logarithms Chapter 3.4 - Solving Exponential and Logarithmic Equations Chapter 4 Chapter 4.1 - Radian and Degree Measure Chapter 4.3 - Evaluating Trigonometric Functions Chapter 4.8 - Applications And Models Chapter 5 Chapter 5.1 - Simplifying a Trigonometric Expression Chapter 5.2 - Verifying Trigonometric Identities Chapter 5.3 - Solving Trigonometric Equations Chapter 6 Chapter 6.1 - The Law Of Sines Chapter 6.2 - The Law Of Cosines Chapter 6.3 - Vectors in the Plane Chapter 6.4 - Vectors and Dot Products Chapter 9 Chapter 9.1 - Sequences and Series Chapter 9.5 - The Binomial Theorem Chapter 10 Chapter 10.5 - Parametric Equations Chapter 12 Chapter 12.1 - Introduction To Limits Chapter 12.2 - Techniques for Evaluating Limits
This is a guide to aid students in becoming more familiar with their new HP-49G and is not designed to be a replacement for HP's manuals but rather, accompany them. Furthermore, I am assuming that you have read at least the standard manual that has come with all new HP-49Gs since its release. You should also have your ROM updated to the current version, which is 1.19-6 at this time. While this ROM is technically BETA software it has many, many bug fixes and enhancements that the "stable" version does not. I have been using my HP-49G for a year now with BETA ROMs without any errors. I would like to suggest that you also pickup the books on the HP-49G by Dr. Urroz that are available at: http://www.greatunpublished.com for about $25 a piece. A great place for calculator information is: http://www.hpcalc.org where many programs are available for download. I would like to personally recommend the programs written by Steen Schmidt without which I would be lost. They are some of the best pieces of coding (on any platform) that I have ever seen. All the current programs can be found at: http://www.hpcalc.org Thanks, Steen, for all of your help as well as your wonderful programs!
Evaluating a function:
If you have an equation such as: g(x) = -x^2 + 4x + 1 and you would like to
find values for x you can use the [SUBST] function under the ALG menu. If
you want to evaluate for g(2) you would do the following:
Example:
Enter -x^2 + 4x + 1
Enter "X=2" (Note: if you use the '' marks the calculator
won't EVAL the arguments. You would probably want to enter the
above as: 'X = 2' to save a few key strokes)
Apply [SUBST]
This yields on the stack:
"-(2)^2 + 4(2) + 1"
Apply [EVAL]
Result:
5
A Piecewise-Defined Function:
To provide domain restrictions you will need to divide the equation by them.
If you want to graph the following:
f(x) = {X^2, X < 0}
{X-1, X >= 0}
You will have to enter two separate equations. The first being:
(X^2 + 1) / (X < 0)
the second:
(X - 1) / (X >= 0)
Note: You can access the greater/less than or equal to on the calculator
but ASCII doesn't directly support those characters.
Evaluating the Difference Quotient:
You can approach this several ways. The easiest, I would say, would be to set
the equation to a variable and use the [SUBST] as directed above to substitute
for X + h. If you enter the equation on the stack and press X [STO] the X
variable will contain the equation. To remove the equation from the "X" variable
you must "PURGE" it. Enter 'X' then hit the PURGE button (F5 in RPN). You can
then enter it on the stack via the 'X' key and SUBST "X=X+H" for the first
then enter the equation again via the 'X' key and press the [-] button. You
will then need to enter an "h" which you can get by pressing the
"[ALPHA] [<-] H" then the divide. Using the [EVAL] will reduce and give the
final result.
Note: In the difference quotient several things should reduce. If you have
a very lengthy answer you probably entered something incorrectly. When graphing,
under the [2D/3D] screen you have H-Ticks and V-Ticks which are equivalent to the
TI's "scale" parameters. You can't enter "PI" so you must approximate the
values. i.e. 3.14159 To access graphing functions in RPN mode you must HOLD
the left shift [<-] when when pressing the F1-F6 graphing screens.
Relative Minimums and Maximums:
You can approximate the Relative min/max by graphing and using the [EXTR] function.
Graph the equation as usual and then move the cursor to the general area where you
want to find the Relative min/max. (If it is closer to another Relative min/max
you'll get the wrong answer!) You then access the [FCN] [EXTR] which will yield the
correct point. It will also push the result onto the stack so that you can view it
once you leave graph mode.
Graphing Step Functions:
To graph the greatest integer function you can use the FLOOR() function which is only
accessible through the [CAT] button.
Example:
f(x) = [[X]]
Graph FLOOR(X)
NOTE: Under the [2D/3D] menu you must UNCHECK the "Connect" box otherwise it will
appear as if the graph is continuous.
To find the composite of a function the [SUBST] is indispensable.
Example:
In the "f(x)" function you want to replace every "x" with the function, "g(x)"
Enter "X^2 - 9"
Enter "X"
Enter "sqrt(9-X^2)"
Enter [->] [=]
You have now told it that "X" is equal to the "g(x)" function
Apply [SUBST]
Apply [EVAL]
Result:
-X^2
To find roots of a polynomial you can set the equation equal to 0 and use
[SOLVEVX] or [SOLVE].
Example:
Find the roots of "x^3 - x^2 - 2x"
Enter "x^3 - x^2 - 2x"
Enter "0"
Apply [->] [=]
Apply [<-] [S.SLV] [SOLVEVX]
Result:
{X=0 X=2 X=-1}
Complex number can be manipulated just as real numbers can be. To switch to
complex mode press and HOLD [<-] then press and release [i].
You'll see the "R" at the top of the screen change to "C". To change back
just repeat the process.
Example:
Evaluate: (3 - i) + (2 + 3i)
Enter "3 - i"
Enter "2 + 3*i"
Apply [+]
Result:
5 + 2i
There isn't much in this chapter that can't be found in the standard User's
Manual from HP. However, you can have the calculator do some rewriting for
you with ln and log.
Example:
Rewrite ln(5x)
Enter "ln(5x)"
Apply [<-] [EXP&LN] [TEXPAND]
Result:
ln(5) + ln(x)
To solve exp and ln equation you will need to use either [SOVLE] or [SOLVEVX]. Go here to see more.
Converting Degrees to Radians and vice versa:
You can access the conversion under "[MTH] [REAL]" menu. There are "D->R"
and "R->D" functions present. Simply enter what you need to convert and
apply the function.
Example:
Convert 135° to radians:
Enter "135"
Apply the "D->R" function
Result: 2.35619449019
If you then type "XQ" it will convert to a fraction, which is probably what
you want.
Apply "XQ"
Result:
(3*pi)/4
To evaluate a trig function simply enter the value to evaluate and apply the
appropriate trig function. To apply a CSC(x) you'll need to enter the following:
Example (Evaluate for Pi/4):
Enter PI/4
Apply [SIN]
Apply [1/X]
Apply [EVAL]
Result:
sqrt(2)
Note: If you need to evaluate an angle in degree you either need to switch modes
to degrees (not recommended because you'll forget to switch back), convert from degrees
to radians (described above), or tell the calculator that the number is in degrees.
Example:
Enter "30_°" (° accessible from the [UNITS] [ANGLE] menus. "_" is [->] [-])
Apply trig function
Degrees-Minutes-Seconds:
To convert to/from you can utilize the [->HMS] and [HMS->], respectively.
Both are accessible from the [->] [TIME] [Tools...] menu.
Example:
To convert: 5° 40' 12" to degrees.
Enter "5.4012"
Apply [HMS->]
Result:
5.67 (NOTE: The calculator doesn't display the ° sign)
The HP-49G supports symbolic logic which makes things much easier when you are attempting
to solve an equation for a given variable.
Example:
If you are given "tan(A) = a/b" and you want to solve for "a".
Enter: "tan(A) = a/b"
Enter: "a" (Note: If you enter "A" instead the answer will be incorrect
You can access the lower-case letters by [ALPHA] [<-])
Apply the [SOLVE] function which is under [<-] [S.SLV]
Result:
a = b*tan(A)
NOTE: There is also the [SOLVEVX] function which eliminates the need to enter the variable
you wish to solve for and will use the default independent variable instead. Usually "X".
There are several functions under the [->] [TRIG] menu that are quite useful for manipulating
Trig equations. The most useful (in my opinion) are the [TRIGSIN], [TRIGCOS], and [TRIGTAN].
When you apply the [TRIGSIN] the calculator will attempt to put everything in terms of the sin(x)
function. The others work the same way but for the different trig functions.
Example:
If you have "sin(x)*cos(x)^2 - sin(x)" you can very easily simply this.
Enter "sin(x)*cos(x)^2 - sin(x)"
Apply [TRIGSIN]
Result:
-sin(x)X^3
This is probably the hardest part of the course (in my humble opinion) and your calculator can help somewhat but you still should be able to do this by hand. If you just rely on the calculator to expand and convert for you life will be much more difficult in Calculus. The [TRIG] function will apply the basic Pythagorean Identity of "sin(x)^2 + cos(x)^2 = 1" to the equation. The [TAN2SC] will convert all "tan(x)" functions into "sin(x)/cos(x)" which can be very helpful in seeing how something might factor or reduce out. The [TAN2SC2] will convert all "tan(x)" to "sin(2x)/cos(2x) + 1".
Probably the best way to solve these is to graph them and find the roots (X-intercepts).
You can access the [ROOT] under the [FCN] menu in the graphing window. Conversion to a
Fraction can be done by applying the "XQ()" function. You will have the decimal on the
stack and if you press [ALPHA] [ALPHA](twice locks the keyboard in Alpha mode) and type "XQ"
then press enter it will convert to a fraction.
You can also use the [SOLVE] function to isolate a variable.
Example:
Enter the equation: "sin(x) + sqrt(2) = -sin(x)"
Because you want to isolate "sin(x)" type that in on the stack.
Apply [SOLVE]
Result:
sin(x) = -(sqrt(2)/2)
You can then take the "arcsin(-(sqrt(2)/2))" which will yield: "pi/4". You'll be
responsible for determining the correct values and which quadrant they are in depending
on the interval given. For the interval [0,2pi) are: x = (5*pi)/4 and x = (7*pi)/4
To solve a triangle using "The Law Of Sines" is rather simplistic but I will include it,
none the less. You will want to use the [SOLVE] function to isolate the angle or side you want.
Example:
Enter "a/(sin(52_°) = b/(sin(88_°)"
Enter "a"
Apply [SOLVE]
Result:
a = (b*sin(52_°))/sin(88_°)
You will then be able to use the [SUBST] (as described above Chapter 1)
You can use the same method with the [SOLVE] as described above (Chapter 6.1) to generate the Alternative Form of the law of cosines to find an angle instead of the side. Just solve for "cos(A)".
You can manipulate vectors using the standard arithmetic operators but you must first
learn the correct method of telling the calculator that you are working with a vector.
Example:
Find the magnitude of the vector R-S where, R = (1,2) and S = (4,4).
Enter the top vector "[4,4]" (Note: The [] show the calculator it is a vector)
Enter the second vector "[1,2]"
Apply [-]
Apply [<-] [ABS]
Result:
sqrt(13)
You can also find the argument (angle) of the vector.
Example:
Find the angle of "u = 3i + 3j"
Enter "(3,3)"
Apply [->] [CMPLX] [ARG]
Returns: .785398163397
Apply [<-] [MTH] [REAL] [R->D] (converts measurement. See above)
Result:
45. (In degrees)
To find the dot product of two vectors simply enter the two vectors (see above)
and apply [<-] [MTH] [VECTOR...] [DOT] function.
Example:
Enter "[4,5]"
Enter "[2,3]"
Apply [<-] [MTH] [VECTOR...] [DOT]
Result:
23
The same is true for the cross product only apply the [CROSS] function.
Finding the factorial of a number is fairly easy.
Example: 5!
Enter 5
Apply [<-] [MTH] [PROBABILITY...] [!]
Result:
120
You can also display the terms of a sequence (as an array) in the
calculator. The order in which you enter the arguments on the stack are follows:
1. The equation
2. The variable in the equation (i.e. 'X')
3. Starting value
4. The ending value
5. The function step value (1 would count 1,2,etc where as 1/2 would count in halves)
Example:
Find the first five terms of: "-1^X/(2*x - 1)"
Enter "-1^X/(2*X - 1)"
Enter "X"
Enter 1
Enter 5
Enter 1
Apply [<-] [PRG] [LIST...] [PROCEDURES...] [SEQ]
Result:
{-1 1/9 -1/5 1/7 -1/9}
To find the sum of a sequence such as, "3x" when x = 1-5:
You should enter the information in the following order:
1. Variable to be used
2. Starting Value
3. Ending Value
4. The equation
Example:
Enter "X"
Enter "1"
Enter "5"
Enter "3*X"
Apply the sigma ([->] [EQW])
Result:
45
To find binomial coefficients you will need to use the [COMB] function.
This is the same as the nCr on the TI's.
Example: Find 8C2
Enter "8"
Enter "2"
Apply [<-] [MTH] [PROBABILITY...] [COMB]
Result:
28
Expanding a binomial is quite simple as well. You can use the [EXPAND] function.
Example: Find (x + 1)^3
Enter "(X + 1)^3"
Apply [->] [ALG] [EXPAND]
Result:
X^3 + 3x^2 + 3*x + 1
NOTE: If you have the "Incr Pow" box under the CAS settings in the
MODE screen your answer will be displayed backwards.
You will need to turn on complex mode (see Chap 2.4) to plot Parametric equations.
The way that the HP-49G plots a Parametric you will need to combine the two functions
into one. The way that this is accomplished is by multiplying the entire second equation
by i. You can then add the two equations together. You will also have to change the
Type under the [2D/3D] menu to Parametric.
Example: Plot the parametric equation: "x = 1 - t"
(Note: You can change the variable under [2D/3D] to something other than 'X'
such as 't' which is normally used for Parametric equations but for the purpose of this
document we will assume the the variable is 'X'.)
Enter "(1 - x) + (2x - x^2)i" in plot window.
Apply [ERASE] [DRAW]
Result:
Limits of all kinds can be found with the HP-49G. The syntax is as follows:
1. Enter the equation
2. Enter the variable and what it is approaching
3. Apply the [lim] function
Example:
Find the limit of "(x^3 - x^2 + x - 1)/x - 1" as x approaches 1
Enter "(x^3 - x^2 + x - 1)/x -1"
Enter "x=1" (Note: If you are using another variable you'll need to substitute it for 'x')
Apply [<-] [CALC] [LIMITS & SERIES...] [lim]
Result:
2
If the left hand and the right hand limits are not equal to one another the limit is said
to "not exist". This is not to say that a description of the graph might not be infinity
but if such is the case the limit technically "does not exist". You can evaluate the
left and right hand limits as by substituting the 'X=1' with 'X=1+0'
for the right side and 'X=1-0' for the left.
Example: Evaluate "|2x|/x" when 'x' approaches 0 from the left.
Enter "|2x|/x"
Enter "x=0-0"
Apply [<-] [CALC] [LIMITS & SERIES...] [lim]
Result:
-2
NOTE: If rigorous mode is UNCHECKED in the CAS MODE settings the absolute value
will NOT operate and, therefore, yield incorrect results!
Parts of this guide are copyright © 2001 Houghton Mifflin used here for academic purposes. HP and HP-49g are copyrighted © 2000 This guide may be reproduced and distributed freely but may not be modified in any way. No fee may be charged for the distribution of this guide.