HP 15c Manual

 Section 13: Finding the Roots of an Equation 191 
 
By  making  the  height  1.5  decimeters,  a 
5.0×1.0×1.5-decimeter box is specified. 
If  you  ignore  the  upper  limit  on  the 
height  and  use  initial  estimates  of  3  and 
4  decimeters  (still  less  than  the  width), 
you  will  obtain  a  height  of  4.2026 
decimeters – a  root  that  is  physically 
meaningless.  If  you  use  small  initial 
estimates  such  as  0  and  1 decimeter, 
you  will  obtain  a  height  of 0.2974 
decimeter –...

192 Section 13: Finding the Roots of an Equation 
 
 
 Many  functions  exhibit  special  behavior  when  their  arguments 
approach zero. You can check your function to determine values of x 
for which any argument within your function becomes zero, and then 
specify estimates at or near those values. 
Although  two  different  initial  estimates  are  usually  supplied  when  using 
_, you can also use _ with the same estimate in both the X- and 
Y-registers. If the two estimates are identical, a...

 Section 13: Finding the Roots of an Equation 193 
 
Restriction on the Use of _ 
The  one  restriction  regarding  the  use  of _ is  that _ cannot  be 
used  recursively.  That  is,  you  cannot  use _ in  a  subroutine  that  is 
called  during  the  execution  of _.  If  this  situation  occurs,  execution 
stops  and Error 7 is  displayed.  It is possible,  however,  to  use _ with 
f thereby using the advanced capabilities of both of these keys. 
Memory Requirements 
_ requires  five  registers  to...

 
194 
Section 14 
Numerical Integration 
Many  problems  in  mathematics,  science,  and 
engineering  require  calculating  the  definite 
integral  of  a  function.  If  the  function  is 
denoted  by f(x) and  the  interval  of  integration 
is a to b, the  integral  can  be  expressed 
mathematically as 
 
The  quantity I can  be  interpreted 
geometrically  as  the  area  of  a  region  bounded  by  the  graph  of f(x), the 
x-axis, and the limits x = a and x = b.* 
When  an  integral  is...

 Section 14: Numerical Integration 195 
 
 
 In Run mode: 
2. Key  the  lower  limit  of  integration  (a)  into  the  X-register,  then  press 
v to lift it into the Y-register. 
3. Key the upper limit of integration (b) in to the X-register. 
4. Press ´ f followed by the label of your subroutine. 
Example: Certain  problems  in  physics  and  engineering  require  calculating 
Bessel  functions. The  Bessel  function  of  the  first  kind  of  order  0  can  be 
expressed as 
. 
Find 
. 
In Program...

196 Section 14: Numerical Integration 
 
Keystrokes Display  
| ¥  Run mode. 
0 v 0.0000 Key lower limit, 0, into Y-
register. 
| $ 3.1416 Key upper limit, π, into X-
register. 
|R 3.1416 Specify Radians mode for 
trigonometric functions. 
Now  you  are  ready  to  press ´f 0 to  calculate  the  integral.  When  you 
do so, youll find that – just as with _ – the calculator will not display 
the  result  right  away,  as  it  does  with  other  operations.  The  HP-15C 
calculates  integrals  using  a...

 Section 14: Numerical Integration 197 
 
Before  calling  the  subroutine  you  provide  to  evaluate f(x), the f 
algorithm – just like the _ algorithm – places the value of x in the X-, 
Y-,  Z-,  and  T-registers.  Because  every  stack  register  contains  the x-value, 
your  subroutine  can  calculate  with  this  number  without  having  to  recall  it 
from  a  storage  register.  The  subroutines  in  the  next  two  examples  take 
advantage  of this  feature. (A  polynomial evaluation...

198 Section 14: Numerical Integration 
 
 
Keystrokes Display  
[ 002– 23 Calculate sin θ. 
- 003– 30 Since a value of θ will be 
placed into the Y-register by 
the f algorithm before it 
executes this subroutine, the 
- operation at this point 
will calculate  
(θ – sin θ). 
\  004–  24 Calculate cos (θ – sin θ). 
|n 005–  43 32  
In  Run  mode,  key  the  limits  of  integration  into  the  X- and  Y-registers.  Be 
sure  that  the  trigonometric  mode  is set  to  Radians, then press ´f 1 to...

 Section 14: Numerical Integration 199 
 
Find Si(2). 
Key in the following subroutine to evaluate the function f(x) = (sin x) / x.* 
Keystrokes Display  
|¥ 000– Program mode. 
´ b .2 001–42,21, .2 Begin subroutine with a b 
instruction. 
[ 002– 23 Calculate sin x. 
® 003– 34 Since a value of x will be 
placed in the Y-register by the 
f algorithm before it 
executes this subroutine, the 
® operation at this point 
will return x to the X-register 
and move sin x to the Y-
register. 
÷ 004– 10 Divide sin...

200 Section 14: Numerical Integration 
 
Accuracy of f 
The accuracy of the integral of any function depends on the accuracy of the 
function  itself.  Therefore,  the  accuracy  of  an  integral  calculated  using f 
is limited by the accuracy of the function calculated by your subroutine.* To 
specify  the  accuracy  of  the  function,  set  the  display  format  so  that  the 
display shows no more than the number of digits that  you consider accurate 
in  the  functions  values.†=If= you  specify...