HP 35s User Manual

							 
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HP 35s  Using the formula solver – part 2 
 
hp calculators - 7 - HP 35s  Using the formula solver – part 2 - Version 1.0 
 A quick look at this expression shows that one solution is r = 0. This is not a useful solution so it would be 
helpful to provide a guess to direct the Solver away from zero. One guess is the value already in R. Type 8 
on the lower line of the display as the second guess. 
  
Go to equation mode by typing !. Find the old equation by moving up or down through the equation list 
with the 5 and 6 keys. 
  
 Begin editing the formula by pressing the left cursor key 7. The cursor appears at the end of the formula. 
  
 Figure 12 
 
 Press 8 to delete the V. Then type the formula for the volume of a sphere. 
  
 9:;$#$%&;, 
  
 Press the right-arrow < key a few times to see the last part of the changed formula. It should look as in 
Figure 13. 
  
 Figure 13 
 
 To solve the equation, press the 0 key. The Solver asks which variable to solve for: The unknown 
variable is R so press &. The Solver now asks for the value of the known variable H. 
  
 Figure 14 
  
 This value is to stay unchanged, so press 1. The Solver looks for a solution. 
  
 Figure 15 
 
Answer: If both the radius of the sphere and that of the base of the cylindrical can are 7.5 then the sphere and the 
can will have the same volume.  
 
The Solver can be used for many kinds of problems, further information is given in the HP 35s manual, and a detailed 
description of the Solver is provided in Appendix D of the manual.   
						

							 
 
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HP 35s  Solving numeric integration problems 
 
 
 
 
Numeric integration 
 
Using the integration function  
 
Practice solving numeric integration problems 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Solving numeric integration problems 
 
hp calculators - 2 - HP 35s  Solving numeric integration problems - Version 1.0 
Numeric integration 
 
Numeric integration has many uses in different areas of science. One of the more common ways to visualize integration 
is that of the area under a curve to the X-axis between two points. 
 
Using the integration function 
 
The HP 35s has a very powerful numeric integrator built into the calculator. This function is found above the ! key 
and is access by pressing !. The method used in this training aid will be to enter the function to integrate as an 
equation and then to integrate it between an upper and lower limit of integration. 
 
The general approach to integrate an equation will be: 
 
Step 1: If the equation that defines the integrands function isnt stored in the equation list, key it in and leave 
Equation mode. The equation usually contains just an expression. 
 
Step 2: Enter the limits of integration:  
 
 in RPN mode, key in the lower limit and press #, then key in the upper limit;  
 
 in algebraic mode, key in the lower limit, press $, then key in the upper limit. 
 
Step 3: Display the equation: Press ! and, if necessary, scroll through the equation list (press the % or 
& cursor keys) to display the desired equation. 
 
Step 4: Select the variable of integration: Press ! and then press the appropriate key on the HP 35s to 
indicate the proper variable. This starts the calculation. 
 
Note that using the integration function uses much more of the calculators memory than any other operation and, 
although highly unlikely, if a MEMORY FULL message is shown, refer to appendix B in the HP 35s manual for more 
information on what steps to take. 
 
You can halt a running integration calculation by pressing  or (. However, no information about the integration is 
available until the calculation finishes normally. 
 
The display format setting chosen through the !) menu affects the level of accuracy assumed for your function 
and used for the result. The integration is more precise but takes much longer in the ALL setting (!)*) and in 
the FIX (!)+), SCI (!),), and ENG (!)-) modes with more digits displayed. The 
uncertainty of the result ends up in the Y–register, pushing the limits of integration up into the T– and Z–registers. 
 
This training aid cannot begin to illustrate the wide range of applications available using the built-in numeric integration 
function, but it can illustrate some of the more common uses. For additional information, see chapters 8 and 15 of the HP 
35s User’s Guide.   
						

							 
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HP 35s  Solving numeric integration problems 
 
hp calculators - 3 - HP 35s  Solving numeric integration problems - Version 1.0 
Practice solving numeric integration problems  
Example 1: Integrate the function 1/X from 1 to 10. Use FIX 4 as the display setting.  
Solution: In either RPN or algebraic mode: !)+*!./0#1
1The display should look similar to the one shown in Figure 1. Note, if you have other equations already in 
the HP 35s calculator, the top line of the display may not indicate 3*3 lin. solve but may show another 
equation.1 
 Figure 1 
 
 To show the checksum and length of this equation, press the following in RPN or algebraic mode  
1
1In RPN or algebraic mode: !21
 
 Figure 2 
 
 If the checksum of the equation just entered does not equal B3AA, then you have not entered it correctly. 
To exit equation mode, press:   
 In RPN or algebraic mode:1!1 
 Now enter the lower and upper limits of the integration. 
 
1In RPN mode: +#+3!1
1
1In algebraic mode: +$+3!1
 
 Integrate the function using X as the variable of integration.  
 !01 
  After a few moments, the HP 35s will display the answer shown below. 
 
 Figure 3 
  
 Now view the uncertainty of the result. 
 
1In RPN mode: 1
1
1In algebraic mode: 41
   
						

							 
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HP 35s  Solving numeric integration problems 
 
hp calculators - 4 - HP 35s  Solving numeric integration problems - Version 1.0 
 Figure 4 
 
Answer: The area under the 1/X curve from 1 to 10 is approximately 2.3026. Figure 4 shows the uncertainty of the 
result assuming algebraic mode. In RPN mode, the uncertainty is shown in the second level of the stack. 
 
Example 2: Integrate the function Sin2(X) from 0 to !. Use FIX 4 as the display setting. Make sure the HP 35s is in 
radians mode.  
Solution: In either RPN or algebraic mode:  !)+*9,1
111!5/06,#1
1
1Note: It is possible to write the equation using the 7 function, but the equation displayed using the 6 
function is may be clearer to read. The display should look similar to the one shown in Figure 5. 1
 
 Figure 5 
 
 To show the checksum and length of this equation, press the following in RPN or algebraic mode. Note that 
the symbol 81means to press the right arrow cursor key.  
1
1In RPN or algebraic mode: 921
 
 Figure 6 
 
 If the checksum of the equation just entered does not equal C615, then you have not entered it correctly. 
To exit equation mode, press:  
 
 In RPN or algebraic mode:1!1
 
 Now enter the lower and upper limits of the integration. 
 
1In RPN mode: 3#9:!1
1
1In algebraic mode: 3$9:!1
 
 Integrate the function using X as the variable of integration. 
 
 901
 
  After a few moments, the HP 35s will display the answer shown below. 
   
						

							 
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HP 35s  Solving numeric integration problems 
 
hp calculators - 5 - HP 35s  Solving numeric integration problems - Version 1.0 
 Figure 7 
  
 Now view the uncertainty of the result. 
 
1In RPN mode: 1
1
1In algebraic mode: 41
 
 Figure 8 
 
Answer: The area under Sin2(X) from 0 to ! is approximately 1.5708. The uncertainty of the result is 0.0002, as 
shown in the Y level of the stack, in Figure 8 (assuming RPN mode).  
Example 3: Integrate the function shown below from 0 to 2!. Use FIX 4 as the display setting. Make sure the HP 35s 
is in radians mode. 
 0.25 COS(X)-1
1
    Figure 9 
 
Solution: In either RPN or algebraic mode:  9)+*9,1
111!+;1
1114+3?,@1
111#1
1
1The display should look similar to the one shown in Figure 10. 1
 
 Figure 10 
  
 To show the checksum and length of this equation, press the following in RPN or algebraic mode. Note that 
the symbol 81means to press the right arrow cursor key. 
1
1In RPN or algebraic mode: 921
 
 Figure 11 
 
 If the checksum of the equation just entered does not equal BB03, then you have not entered it correctly. 
To exit equation mode, press:  
 
 In RPN or algebraic mode:1!1  
						

							 
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HP 35s  Solving numeric integration problems 
 
hp calculators - 6 - HP 35s  Solving numeric integration problems - Version 1.0 
Now enter the lower and upper limits of the integration. Note that the algebraic keystrokes are to allow for 
 
In RPN mode: 3#9:,A!1
In algebraic mode: ,A9:#$3$!1
Integrate the function using X as the variable of integration. 
901
 After a few moments, the HP 35s will display the answer shown below. 
 
 
the computation of the upper limit of integration. 
1
1
1
 
 
 
 
 
 
 
 Figure 12 
 
Now view the uncertainty of the result. 
In RPN mode: 1
In algebraic mode: 41
 
 
 
1
1
1
 Figure 13 
 
nswer:A The area under the function from 0 to 2! is approximately 8.3776. The uncertainty in the result is 0.0008. 
 
Figure 13 assumes algebraic mode. In RPN mode, the uncertainty is shown in the second level of the 
stack.   
						

							 
 
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HP 35s  Solving Trigonometry Problems 
 
 
 
 
The trigonometric functions 
 
Trigonometric modes 
 
Practice working problems involving trig functions 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Solving Trigonometry Problems 
 
hp calculators - 2 - HP 35s  Solving Trigonometry Problems - Version 1.0 
The trigonometric functions 
 
The trigonometric functions, sine, cosine, tangent, and related functions, are used in geometry, surveying, and design. 
They also occur in solutions to orbital mechanics, integration, and other advanced applications. 
 
The HP 35s provides the three basic functions, and their inverse, or “arc” functions. These work in degrees, radians and 
gradians modes. In addition, ! is provided as a function on the left-shifted “cos” key, and the sign function is found in the 
INTG menu on the left-shifted “tan” key. 
 
The secant, cosecant and cotangent functions are easily calculated using the !, , and # keys respectively, 
followed by $. To help remember whether the secant function corresponds to the inverse sine or cosine, it can be 
helpful to note that the first letters of “secant” and “cosecant” are inverted in relation to those of “sine” and “cosine”, just 
as the secant and cosecant are the inverted cosine and sine functions. 
 
The display mode can be changed to show either rectangular and radial coordinates. This can therefore be useful in 
some trigonometric calculations. 
 
Trigonometric modes 
 
The HP 35s can calculate trigonometric functions in any of these three modes: Degrees, Radians or Gradians. 
 
Practice working problems involving trig functions 
 
Example 1: Select the appropriate angle mode. 
 
Solution: Press the 9 key.%
 
 Figure 1 
 
 Press &,  or ( to select DEGrees, RADians or GRADians mode, or use the arrow keys ), *, 
+ and , to select the required mode and then press -. For example, to select RAD, press .%
 
Answer: The selected trigonometric mode is displayed at the top of the screen if it is RAD or GRAD. If no angle 
mode is shown, then it is degrees. The 9 command works the same way in algebraic and in RPN 
modes. 
 
 There are 360 degrees, or 2 ! radians in a circle. Gradians mode divides each quarter of a circle into 100 
parts, in a sort of decimal system, making 400 gradians in a circle. 
 
 Note: It is very easy to forget that one angle mode is set but angles are being entered in a different mode. 
It is a good policy to make it a habit to check the angle mode before every calculation. The commands 
DEG, RAD and GRAD can be entered into programs, and it is worth using them to ensure that a program 
will work as required. 
 
Example 2: What is the sine of !/2 radians?