Calculus (Differentiation)
Description
Published by Marcel Pelletier, June 2023, in the MoHPC - General Software Library
Introduction
This document shows how to use a numerical approximation to the derivative of a function to calculate the arc length of a curve, the area of a surface of revolution and the volume of a solid of revolution.
Usage of the program and examples
1. Numerical Derivative (Label D)
Program D uses the registers .0 and .1. The user-defined function can be added at the end of the program memory at label A.
The SOLVE and ∫yx functions can be used with the program. The symmetric difference quotient is used to calculate the slope of the secant line with:
f(x+h) - f(x-h)
f(x) = ———————————————
2h
Example 1:
The curve y=ex•sin(x)
has a minimum in the interval [4, 6]. What is the value of the minimum?
In PRGM mode:
Delete any existing program at label A.
f LBL A
ex
g LSTx
SIN
×
g RTN
In RUN mode:
g RAD
4 ENTER → 4.0000
6 f SOLVE D → 5.4978 (Value x where f(x)=0)
f A → -172.6409 (Minimum)
2. Arc Length (Label 1)
This program uses the Arc Length Formula and the label D to calculate the length L.
a
L = ∫ √(1+(f'(x))²) dx
b
The user-defined function is located in label A.
Example 2:
Find the arc length along the curve y=x²-¹/₈ln(x)
from point A (1, 1) to point B (3, f(3)).
In PRGM mode:
Delete any existing program at label A.
f LBL A
g x²
g LSTx
g LN
8
÷
−
g RTN
In RUN mode:
1 ENTER → 1.000
3 f ∫yx 1 → 8.1373 (Arc length along the curve from 1 to 3)
3. Area of a Surface of Revolution (Label 2)
This program uses the function ∫yx to calculate the area S of the surface resulting from the rotation of the curve y=f(x),a≤x≤b
, around the x-axis as
a
S = ∫ 2π•f(x)•√(1+(f'(x))²) dx
b
The user-defined function is expected at label A.
Example 3:
The curve y=√(4−x²),-1≤x≤1
, is an arc of the circle x²+y²=4
. Find the area of the surface obtained by rotating this arc around the x-axis. The surface is a portion of a sphere of radius 2.
In PRGM mode:
Delete any existing program at label A.
f LBL A
g x²
4
x↔y
−
√x
g RTN
In RUN mode:
1 CHS ENTER → 1.0000
1 f ∫yx 2 → 25.1327 (Area of the surface.)
Example 4:
The arc of the parabola y=x²
from (1, 1) to (2, 4) is rotated around the y-axis. Find the area of the resulting surface. Since the axis of rotation is the y-axis, we use f(y)=√y
from 1 to 4.
In PRGM mode:
Delete any existing program at label A.
f LBL A
√x
g RTN
In RUN mode:
1 ENTER → 1.0000
4 f ∫yx 2 → 30.8465 (Area of the surface!)
4. Volume of a Solid of Revolution (Label 3)
This program use with ∫yx function, calculate the volume V of the solid obtained by rotating the curve y=f(x), a≤x≤b
, around the x-axis as
a
V = ∫ π(f(x))² dx
b
The user-defined function is at label A.
If the region is enclosed by the curve y₁
and y₂
where y₁>y₂, a≤x≤b
then the user function is f(x)=y₁
labeled A and g(x)=y₂
is labeled B. In this case, the volume V of the region is
a
V = ∫ π[(f(x))²−(g(x))²] dx
b
Example 5:
Find the volume of the solid obtained by rotating the region bounded by y=x³, y=8
, and x=0
around the y-axis. Since the axis of rotation is y-axis, we use f(y)=³√x
from 0 to 8.
In PRGM mode:
Delete any existing program at label A.
f LBL A
3
¹/x
yx
g RTN
In RUN mode:
g CF 1 (This is the first case!)
0 ENTER
f ∫yx 3 → 60.3185 (Volume of the solid)
Example 6:
The region enclosed by curves y=x
and y=x²
is rotated around the x-axis. Find the volume of the resulting solid.
The curves intersect at the points (0, 0) and (1, 1). The region between them is the solid of rotation. A cross-section in the plane has the shape of a washer.
We have Label A: f(x)=x
and Label B: g(x)=x²
.
In PRGM mode:
Delete any existing program at label A.
f LBL A
g RTN
f LBL B
g x²
g RTN
In RUN mode:
g SF 1 (This is the second case!)
0 ENTER
f ∫yx 3 → 0.4189 (Volume of the solid)
Program Resources
Labels
Name |
Description |
|
A |
User-defined function A |
|
B |
User-defined function B |
|
D |
Numerical Derivative |
|
1 |
Arc Length |
|
2 |
Area of a Surface of Revolution |
|
3 |
Volume of a Solid of Revolution |
|
4 |
|
|
Storage Registers
Name |
Description |
|
.0 |
Save x value for later use in subroutine D |
|
.1 |
0.0001 |
|
.2 |
Save x value for later use in subroutine 3 |
|
Flags
Program
Line |
Display |
Key Sequence |
|
Line |
Display |
Key Sequence |
|
Line |
Display |
Key Sequence |
|
000 |
|
|
|
021 |
40 |
+ |
|
042 |
43, 6, 1 |
g F? 1 |
|
001 |
42,21,14 |
f LBL D |
|
022 |
11 |
√x̅ |
|
043 |
32 4 |
GSB 4 |
|
002 |
44 .0 |
STO . 0 |
|
023 |
43 32 |
g RTN |
|
044 |
43 26 |
g π |
|
003 |
26 |
EEX |
|
024 |
42,21, 2 |
f LBL 2 |
|
045 |
20 |
× |
|
004 |
16 |
CHS |
|
025 |
32 14 |
GSB D |
|
046 |
43 32 |
g RTN |
|
005 |
4 |
4 |
|
026 |
43 11 |
g x² |
|
047 |
42,21, 4 |
f LBL 4 |
|
006 |
44 .1 |
STO . 1 |
|
027 |
1 |
1 |
|
048 |
45 .2 |
RCL . 2 |
|
007 |
40 |
+ |
|
028 |
40 |
+ |
|
049 |
32 12 |
GSB B |
|
008 |
32 11 |
GSB A |
|
029 |
11 |
√x̅ |
|
050 |
43 11 |
g x² |
|
009 |
45 .0 |
RCL . 0 |
|
030 |
45 .0 |
RCL . 0 |
|
051 |
30 |
− |
|
010 |
45,30, .1 |
RCL − . 1 |
|
031 |
32 11 |
GSB A |
|
052 |
43 32 |
g RTN |
|
011 |
32 11 |
GSB A |
|
032 |
20 |
× |
|
053 |
42,21,11 |
f LBL A |
|
012 |
30 |
− |
|
033 |
2 |
2 |
|
054 |
43 11 |
g x² |
|
013 |
2 |
2 |
|
034 |
20 |
× |
|
055 |
43 36 |
g LSTΧ |
|
014 |
45,20, .1 |
RCL × . 1 |
|
035 |
43 26 |
g π |
|
056 |
43 12 |
g LN |
|
015 |
10 |
÷ |
|
036 |
20 |
× |
|
057 |
8 |
8 |
|
016 |
43 32 |
g RTN |
|
037 |
43 32 |
g RTN |
|
058 |
10 |
÷ |
|
017 |
42,21, 1 |
f LBL 1 |
|
038 |
42,21, 3 |
f LBL 3 |
|
059 |
30 |
− |
|
018 |
32 14 |
GSB D |
|
039 |
44 .2 |
STO . 2 |
|
060 |
43 32 |
g RTN |
|
019 |
43 11 |
g x² |
|
040 |
32 11 |
GSB A |
|
|
|
|
|
020 |
1 |
1 |
|
041 |
43 11 |
g x² |
|
|
|
|
|