Calculus Lesson, Tangents to curves
Posted: Thu Jan 22, 2015 3:00 pm
this short code display the function f(x) = x^2 , and then will find the curve slopes at any given point. when the slope approaches the curve new turn it slow down (like a car on the road turns) then change its direction and speeding up the curve.
how to find the curve slope (the Tangent angle) at a specific point on the curve?. we need the derivative of the function. such as in f(x)=x^2 the derivative of x^2 = 2*x.
if we don't know we look this universal calculator machine :
http://www.wolframalpha.com/examples/Derivatives.html
just write: derivative of sin(x) and it will give you Cos(x)
write derivative of x^3 and it will give you 3*x^2
the essential code is:
the next code is the same above but for f(x) = Sin(x) . i have amplified x,y by 3
another interesting function x*Sin(x*x)+1 , from wolfram alpha its derivative is Sin(x*x)+2*x*x*Cos(x*x)
i forgot Calculus COMPLETELY and now trying to learn again the introduction, here is a list of some of the introductory books for who may find it useful
Bob Miller's High School Calc for the Clueless
Calculus For The Utterly Confused
Calculus Know-It-ALL
Everyday Calculus_ Discovering the Hidden Math All around Us
Mathematics - Teach Yourself Visually Calculus - (Dale Johnson) Wiley 2008
The Britannica Guide to Analysis and Calculus
The Everything Guide to Calculus I
The Manga Guide to Calculus : Cartoon book
An Approach to CALCULUS (online version) http://www.themathpage.com/acalc/calculus.htm
Calculus Demystified
Calculus Made Easy :SILVANUS P. THOMPSON + Martin Gardner
Calculus Essentials For Dummies
Calculus For Dummies
Calculus II For Dummies
Paradoxes and Sophisms in Calculus
The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer
The Complete Idiot's Guide to Calculus, 2nd Ed
Calculus for Computer Graphics :John Vince :Springer-Verlag London 2013
Calculus simpleSneesyBook
Workshop Calculus with Graphing Calculators
Zombies and Calculus
E-Z Calculus : Douglas Downing
Calculus the Easy Way : Douglas Downing
What Is Calculus About: w.w. Sawyer
Discovering the Art of Mathematics: Calculus (free ebook): https://www.artofmathematics.org/books/calculus
A Gentle Introduction To Learning Calculus: http://betterexplained.com/articles/a-g ... -calculus/
Calculus for the Ambitious
From calculus to chaos
Higher Maths for Beginners – Zeldovich, Yaglom , this book is from the soviet era. many books from that era available for free from http://mirtitles.org
Calculus with Applications: Margaret L. Lial
THE MATHEMATICAL MECHANIC using physical reasoning to solve problems , Mark Levi
how to find the curve slope (the Tangent angle) at a specific point on the curve?. we need the derivative of the function. such as in f(x)=x^2 the derivative of x^2 = 2*x.
if we don't know we look this universal calculator machine :
http://www.wolframalpha.com/examples/Derivatives.html
just write: derivative of sin(x) and it will give you Cos(x)
write derivative of x^3 and it will give you 3*x^2
the essential code is:
Code: Select all
x.f + stepx
y.f = x*x
z.f=0
slope.f = 2*x ; this is the derivative of f(y)=x^2
slope = Degree(ATan(slope))
MoveEntity(200,x,y, z, #PB_Absolute)
RotateEntity(200, 0,0,slope)Code: Select all
Declare DrawPoints()
#CameraSpeed = 1
;the range of the function under study is from xMin to xMax
Global xMin.f = -2*#PI
Global xMax.f = 2*#PI
Global stepx.f = (xMax - xMin) / 1000 ; the increment of x every loop
Enumeration
#mesh
#light
#camera
#plane
EndEnumeration
ExamineDesktops()
InitEngine3D(#PB_Engine3D_DebugLog)
InitSprite()
InitKeyboard()
InitMouse()
OpenWindow(0,0,0,DesktopWidth(0), DesktopHeight(0),"curve tangent, .. press space key to start/stop",#PB_Window_ScreenCentered)
OpenWindowedScreen(WindowID(0),0,0,DesktopWidth(0), DesktopHeight(0),1,0,0,#PB_Screen_WaitSynchronization)
Add3DArchive(".", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Textures", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/fonts", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Scripts", #PB_3DArchive_FileSystem)
Parse3DScripts()
KeyboardMode(#PB_Keyboard_AllowSystemKeys)
CreateMaterial(0, LoadTexture(0, "White.jpg"))
DisableMaterialLighting(0, #True)
CreateMaterial(1, LoadTexture(1, "snow_1024.jpg"))
MaterialCullingMode(1, #PB_Material_NoCulling)
CreatePlane(#plane, 200, 200, 50, 50, 10, 10)
CreateEntity(#plane,MeshID(#plane),MaterialID(1), 0,-16,0)
CreateLight(0,RGB(245,245,205),19,13,0)
;Create Camera
CreateCamera(0,0,0,100,100)
MoveCamera(0,-5,30,43,#PB_Absolute)
CameraLookAt(0, 0, 10, 0)
CameraBackColor(0, RGB(227,239,242))
CreateText3D(0, "I'm Waiting")
Text3DColor(0, RGBA(150, 0, 250, 255))
Text3DAlignment(0, #PB_Text3D_HorizontallyCentered)
ScaleText3D(0, 2,2,2,#PB_Absolute)
AmbientColor(RGB(255,255,255))
;temp=1
CreateLine3D(#PB_Any, -20, 0, 0, #Red, 20, 0, 0, RGB(255, 0, 0)) ; Red
CreateLine3D(#PB_Any, 0,-20,0, #Green, 0, 20, 0, RGB(0, 255, 0)); Green
CreateLine3D(#PB_Any, 0,0,-20, #Blue, 0, 0, 20, RGB(0, 0, 255)) ;Blue
;-Mesh
DrawPoints()
CreateTexture(2,32,32)
StartDrawing(TextureOutput(2))
Box(0,0,32,32,RGB(255,0,0)) ; red color
StopDrawing()
CreateMaterial(2,TextureID(2))
MaterialCullingMode(2, #PB_Material_NoCulling)
CreateCube(200, 4) ; make the slope , then scale it like a line
CreateEntity(200, MeshID(200), MaterialID(2) )
ScaleEntity(200, 1,0.1,0.1)
MoveEntity(200, -2*#PI, 0, 0, #PB_Absolute)
x.f = -2*#PI
start = 0
Repeat
Repeat
event = WindowEvent()
Until event = 0
If ExamineMouse()
MouseX = -MouseDeltaX() * #CameraSpeed * 0.2
MouseY = -MouseDeltaY() * #CameraSpeed * 0.2
EndIf
ShowCursor_(0)
; Use arrow keys and mouse to rotate and fly in/out
ExamineKeyboard()
If KeyboardPushed(#PB_Key_Left)
KeyX = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Right)
KeyX = #CameraSpeed
Else
KeyX = 0
EndIf
If KeyboardPushed(#PB_Key_Up)
KeyY = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Down)
KeyY = #CameraSpeed
Else
KeyY = 0
EndIf
If KeyboardReleased(#PB_Key_Space)
start ! 1
EndIf
If start
x.f + stepx
y.f = x*x
z.f=0
slope.f = 2*x ; this is the derivative of f(y)=x^2
slope = Degree(ATan(slope))
MoveEntity(200,x,y, z, #PB_Absolute)
RotateEntity(200, 0,0,slope)
If x > 2*#PI
x = -2*#PI
EndIf
MoveText3D(0, 0, -50, 0, #PB_Absolute) ;hide the text underground when slope is busy
Else
MoveText3D(0, x, y+4, 0, #PB_Absolute) ; move the text near the slope when waiting
EndIf
RotateCamera(0, MouseY, MouseX, 0, #PB_Relative)
MoveCamera (0, KeyX, 0, KeyY)
RenderWorld()
FlipBuffers()
Until KeyboardPushed(#PB_Key_Escape) Or WindowEvent() = #PB_Event_CloseWindow
;main drawing routine
Procedure DrawPoints()
Protected.f x, y
CreateMesh(0, #PB_Mesh_PointList, #PB_Mesh_Static)
x.f = xMin: z.f = 0
While x <= 2*#PI
y.f = Pow(x,2) ; the function we want to study its tangent at several points
MeshVertexPosition(x, y, z)
MeshVertexColor(RGB(0,0,255))
x.f + stepx
Wend
FinishMesh(#True)
CreateEntity(0, MeshID(0), MaterialID(0) ,0,0,0)
EndProcedure
Code: Select all
Declare DrawPoints()
#CameraSpeed = 1
;the range of the function under study is from xMin to xMax
Global xMin.f = -2*#PI
Global xMax.f = 2*#PI
Global stepx.f = (xMax - xMin) / 1000 ; the increment of x every loop
Enumeration
#mesh
#entity
#tex
#light
#camera
#plane
EndEnumeration
ExamineDesktops()
InitEngine3D(#PB_Engine3D_DebugLog)
InitSprite()
InitKeyboard()
InitMouse()
OpenWindow(0,0,0,DesktopWidth(0), DesktopHeight(0),"curve tangent, .. press space key to start/stop",#PB_Window_ScreenCentered)
OpenWindowedScreen(WindowID(0),0,0,DesktopWidth(0), DesktopHeight(0),1,0,0,#PB_Screen_WaitSynchronization)
Add3DArchive(".", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Textures", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/fonts", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Scripts", #PB_3DArchive_FileSystem)
Parse3DScripts()
KeyboardMode(#PB_Keyboard_AllowSystemKeys)
CreateMaterial(0, LoadTexture(0, "White.jpg"))
DisableMaterialLighting(0, #True)
CreateMaterial(1, LoadTexture(1, "snow_1024.jpg"))
MaterialCullingMode(1, #PB_Material_NoCulling)
CreatePlane(#plane, 200, 200, 50, 50, 10, 10)
CreateEntity(#plane,MeshID(#plane),MaterialID(1), 0,-16,0)
CreateLight(0,RGB(245,245,205),19,13,0)
;Create Camera
CreateCamera(0,0,0,100,100)
MoveCamera(0,-50,30,33,#PB_Absolute)
CameraLookAt(0, 0, 0, 0)
CameraBackColor(0, RGB(227,239,242))
AmbientColor(RGB(255,255,255))
CreateLine3D(#PB_Any, -20, 0, 0, #Red, 20, 0, 0, RGB(255, 0, 0)) ; Red
CreateLine3D(#PB_Any, 0,-20,0, #Green, 0, 20, 0, RGB(0, 255, 0)); Green
CreateLine3D(#PB_Any, 0,0,-20, #Blue, 0, 0, 20, RGB(0, 0, 255)) ;Blue
;-Mesh
DrawPoints() ; call the function drawing procedure
CreateTexture(2,32,32)
StartDrawing(TextureOutput(2))
Box(0,0,32,32,RGB(255,0,0))
StopDrawing()
CreateMaterial(2,TextureID(2))
MaterialCullingMode(2, #PB_Material_NoCulling)
CreateCube(200, 4) ; make the slope , then scale it like a line
CreateEntity(200, MeshID(200), MaterialID(2) )
ScaleEntity(200, 1,0.1,0.1)
MoveEntity(200, -2*#PI*3, Sin(-2*#PI)*3, 0, #PB_Absolute)
x.f = xMin
start = 0
Repeat
Repeat
event = WindowEvent()
Until event = 0
If ExamineMouse()
MouseX = -MouseDeltaX() * #CameraSpeed * 0.2
MouseY = -MouseDeltaY() * #CameraSpeed * 0.2
EndIf
ShowCursor_(0)
; Use arrow keys and mouse to rotate and fly in/out
ExamineKeyboard()
If KeyboardPushed(#PB_Key_Left)
KeyX = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Right)
KeyX = #CameraSpeed
Else
KeyX = 0
EndIf
If KeyboardPushed(#PB_Key_Up)
KeyY = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Down)
KeyY = #CameraSpeed
Else
KeyY = 0
EndIf
If KeyboardReleased(#PB_Key_Space)
start ! 1
EndIf
If start
x.f + stepx
y.f = Sin(x)
z.f=0
slope.f= Cos(x) ; the derivative of sin(x) is cos(x)
slope = Degree(ATan(slope))
MoveEntity(200,x*3,y*3, z, #PB_Absolute) ; we just amplify x,y by 3
RotateEntity(200, 0,0,slope)
If x > xMax
x = xMin
EndIf
EndIf
RotateCamera(0, MouseY, MouseX, 0, #PB_Relative)
MoveCamera (0, KeyX, 0, KeyY)
RenderWorld()
FlipBuffers()
Until KeyboardPushed(#PB_Key_Escape) Or WindowEvent() = #PB_Event_CloseWindow
;function drawing
Procedure DrawPoints()
Protected.f x, y
CreateMesh(0, #PB_Mesh_PointList, #PB_Mesh_Static)
x.f = xMin: z.f = 0
While x <= 2*#PI
y.f = Sin(x)
MeshVertexPosition(x*3, y*3, z)
MeshVertexColor(RGB(0,0,255))
x.f + stepx
Wend
FinishMesh(#True)
CreateEntity(#entity, MeshID(0), MaterialID(0) ,0,0,0)
EndProcedure
Code: Select all
Declare DrawPoints()
#CameraSpeed = 1
;the range of the function under study is from xMin to xMax
Global xMin.f = -2
Global xMax.f = 4
Global stepx.f = (xMax - xMin) / 1000 ; the increment of x every loop
Enumeration
#mesh
#entity
#tex
#light
#camera
#plane
EndEnumeration
ExamineDesktops()
InitEngine3D(#PB_Engine3D_DebugLog)
InitSprite()
InitKeyboard()
InitMouse()
OpenWindow(0,0,0,DesktopWidth(0), DesktopHeight(0),"curve tangent, .. press space key to start/stop",#PB_Window_ScreenCentered)
OpenWindowedScreen(WindowID(0),0,0,DesktopWidth(0), DesktopHeight(0),1,0,0,#PB_Screen_WaitSynchronization)
Add3DArchive(".", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Textures", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/fonts", #PB_3DArchive_FileSystem)
Add3DArchive(#PB_Compiler_Home + "examples/3d/Data/Scripts", #PB_3DArchive_FileSystem)
Parse3DScripts()
KeyboardMode(#PB_Keyboard_AllowSystemKeys)
CreateMaterial(0, LoadTexture(0, "White.jpg"))
DisableMaterialLighting(0, #True)
CreateMaterial(1, LoadTexture(1, "snow_1024.jpg"))
MaterialCullingMode(1, #PB_Material_NoCulling)
CreatePlane(#plane, 200, 200, 50, 50, 10, 10)
CreateEntity(#plane,MeshID(#plane),MaterialID(1), 0,-16,0)
CreateLight(0,RGB(245,245,205),19,13,0)
;Create Camera
CreateCamera(0,0,0,100,100)
MoveCamera(0,0,10,33,#PB_Absolute)
CameraLookAt(0, 0, 5, 0)
CameraBackColor(0, RGB(227,239,242))
AmbientColor(RGB(255,255,255))
CreateLine3D(#PB_Any, -20, 0, 0, #Red, 20, 0, 0, RGB(255, 0, 0)) ; Red
CreateLine3D(#PB_Any, 0,-20,0, #Green, 0, 20, 0, RGB(0, 255, 0)); Green
;CreateLine3D(#PB_Any, 0,0,-20, #Blue, 0, 0, 20, RGB(0, 0, 255)) ;Blue
;-Mesh
DrawPoints() ; call the function drawing procedure
CreateTexture(2,32,32)
StartDrawing(TextureOutput(2))
Box(0,0,32,32,RGB(255,0,0))
StopDrawing()
CreateMaterial(2,TextureID(2))
MaterialCullingMode(2, #PB_Material_NoCulling)
CreateCube(200, 4) ; make the slope , then scale it like a line
CreateEntity(200, MeshID(200), MaterialID(2) )
ScaleEntity(200, 1,0.01,0.1)
MoveEntity(200, -2*3, 0, 0, #PB_Absolute)
x.f = xMin
start = 0
Repeat
Repeat
event = WindowEvent()
Until event = 0
If ExamineMouse()
MouseX = -MouseDeltaX() * #CameraSpeed * 0.2
MouseY = -MouseDeltaY() * #CameraSpeed * 0.2
EndIf
ShowCursor_(0)
; Use arrow keys and mouse to rotate and fly in/out
ExamineKeyboard()
If KeyboardPushed(#PB_Key_Left)
KeyX = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Right)
KeyX = #CameraSpeed
Else
KeyX = 0
EndIf
If KeyboardPushed(#PB_Key_Up)
KeyY = -#CameraSpeed
ElseIf KeyboardPushed(#PB_Key_Down)
KeyY = #CameraSpeed
Else
KeyY = 0
EndIf
If KeyboardReleased(#PB_Key_Space)
start ! 1
EndIf
If start
x.f + stepx
y.f = x*Sin(x*x)+1
z.f=0
slope.f= Sin(x*x)+2*x*x*Cos(x*x) ; the derivative of sin(x) is cos(x)
slope = Degree(ATan(slope))
MoveEntity(200,x*3,y*3, z, #PB_Absolute) ; we just amplify x,y by 3
RotateEntity(200, 0,0,slope)
If x > xMax
x = xMin
EndIf
EndIf
RotateCamera(0, MouseY, MouseX, 0, #PB_Relative)
MoveCamera (0, KeyX, 0, KeyY)
RenderWorld()
FlipBuffers()
Until KeyboardPushed(#PB_Key_Escape) Or WindowEvent() = #PB_Event_CloseWindow
;function drawing
Procedure DrawPoints()
Protected.f x, y
CreateMesh(0, #PB_Mesh_LineStrip , #PB_Mesh_Static)
x.f = xMin: z.f = 0
While x <= 4
;y.f = Sin(x)
y.f = x*Sin(x*x)+1
MeshVertexPosition(x*3, y*3, z)
MeshVertexColor(RGB(0,0,255))
x.f + stepx
Wend
FinishMesh(#True)
CreateEntity(#entity, MeshID(0), MaterialID(0) ,0,0,0)
EndProcedure
i forgot Calculus COMPLETELY and now trying to learn again the introduction, here is a list of some of the introductory books for who may find it useful
Bob Miller's High School Calc for the Clueless
Calculus For The Utterly Confused
Calculus Know-It-ALL
Everyday Calculus_ Discovering the Hidden Math All around Us
Mathematics - Teach Yourself Visually Calculus - (Dale Johnson) Wiley 2008
The Britannica Guide to Analysis and Calculus
The Everything Guide to Calculus I
The Manga Guide to Calculus : Cartoon book
An Approach to CALCULUS (online version) http://www.themathpage.com/acalc/calculus.htm
Calculus Demystified
Calculus Made Easy :SILVANUS P. THOMPSON + Martin Gardner
Calculus Essentials For Dummies
Calculus For Dummies
Calculus II For Dummies
Paradoxes and Sophisms in Calculus
The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer
The Complete Idiot's Guide to Calculus, 2nd Ed
Calculus for Computer Graphics :John Vince :Springer-Verlag London 2013
Calculus simpleSneesyBook
Workshop Calculus with Graphing Calculators
Zombies and Calculus
E-Z Calculus : Douglas Downing
Calculus the Easy Way : Douglas Downing
What Is Calculus About: w.w. Sawyer
Discovering the Art of Mathematics: Calculus (free ebook): https://www.artofmathematics.org/books/calculus
A Gentle Introduction To Learning Calculus: http://betterexplained.com/articles/a-g ... -calculus/
Calculus for the Ambitious
From calculus to chaos
Higher Maths for Beginners – Zeldovich, Yaglom , this book is from the soviet era. many books from that era available for free from http://mirtitles.org
Calculus with Applications: Margaret L. Lial
THE MATHEMATICAL MECHANIC using physical reasoning to solve problems , Mark Levi