View
0
Download
0
Category
Preview:
Citation preview
Young Won Lim6/23/14
Derivatives (1A)
Young Won Lim6/23/14
Copyright (c) 2011 - 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to youngwlim@hotmail.com.This document was produced by using OpenOffice and Octave.
mailto:youngwlim@hotmail.com
Derivatives (1A) 3 Young Won Lim6/23/14
A triangle and its slope
y = f x
f x1 h − f x1h
x1, f x1
x1 h , f x1 h
x1, f x1
http://en.wikipedia.org/wiki/Derivative
x1 x1+h
f (x1+h)
f (x1)
Derivatives (1A) 4 Young Won Lim6/23/14
Many smaller triangles and their slopes
f ( x1)
(x1, f (x1))
f ( x1+h2)
f ( x1+h1)
f ( x1+h)
f (x1 + h) − f ( x1)h
f (x1 + h1) − f (x1)h1
f (x1 + h2) − f (x1)h2
limh→0
f (x1 + h) − f (x1)h
x1 x2+h2 x2+h1 x2+h
(x1+h , f (x1+h))
h2h1
h
(x1, f (x1))
(x1+h , f (x1+h))
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 5 Young Won Lim6/23/14
The limit of triangles and their slopes
y = f x
f ' (x1) = limh→0
f (x1 + h) − f (x1)h
x1
f
The derivative of the function f at x1
function f
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 6 Young Won Lim6/23/14
The derivative as a function
y = f x x1x2x3
f x1
f x2
f x3
f x
x1x2x3
f ' x 1
f ' x 2
f ' x 3
f ' x
= limh→0
f (x + h) − f (x)h
y ' = f ' (x)
Derivative Function
Derivatives (1A) 7 Young Won Lim6/23/14
The notations of derivative functions
x1x2x3
f ' x 1
f ' x 2
f ' x 3
f ' (x)
limh→0
f (x + h) − f (x )h
y ' = f ' x Largrange's Notation
dydx
=ddx
f x
Leibniz's Notation
ẏ = ḟ x Newton's Notation
D x y = D x f x Euler's Notation
not a ratio.
x is an independent variable→ Derivative with respect to x
Derivatives (1A) 8 Young Won Lim6/23/14
Another kind of triangles and their slopes
y = f x
f
y ' = f ' x given x1
the slope of a tangent line
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 9 Young Won Lim6/23/14
Differential in calculus
x1 x1 dx
f x1
f x1 dx
dx
dy
Differential: dx, dy, … ● infinitesimals● a change in the linearization of a function● of, or relating to differentiation
the linearization of a function
function
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 10 Young Won Lim6/23/14
Differential as a function
x1
f x1
Line equation in the new coordinate.
dx
dydy = f ' x1 dx
slope = f ' x1
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 11 Young Won Lim6/23/14
Differentiation & Integration of sinusoidal functions
f x = sin x dd x
f x = cos x leads
f x = cos xdd x
f x = −sin x leads
f x = sin x ∫ f x dx = −cos x C lags
f x = cos x∫ f x dx = sin x C lags
Derivatives (1A) 12 Young Won Lim6/23/14
Derivative of sin(x)
f (x) = sin(x)
+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope
dd x
f (x ) = cos(x )
leads
Derivatives (1A) 13 Young Won Lim6/23/14
Plot of F(x,y)=cos(x)
+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope
F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)
slope=+1
F (x , y)
+1
0
-1
0
+1
0
-1
0
+1
0
-1
0
f (x ) = sin(x)
f ' (x ) = cos(x)
Derivatives (1A) 14 Young Won Lim6/23/14
Derivative of cos(x)
f x = cos x
0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope
dd x
f x = −sin x
leads
Derivatives (1A) 15 Young Won Lim6/23/14
Plot of F(x,y)=-sin(x)
F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)
slope=+1
F (x , y)
+1
-1
+1
-1
+1
-1
0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope
0 0 0 0 0 0
f (x ) = cos(x)
f ' (x ) = −sin (x)
Derivatives (1A) 16 Young Won Lim6/23/14
Integral of sin(x)
f x = sin x
0 1 2 1 0 1 2 1 0 1 2 1 area + 0
∫ f x dx = −cos x C
-1 0 +1 0 -1 0 +1 0 -1 0 +1 0 area - 1
∫0/2
sin t d t = 1
C = 0
C = 1 ∫0xsin (t) d t + 0
∫0xsin (t) d t − 1
= − cos x lags
Derivatives (1A) 17 Young Won Lim6/23/14
Integral of cos(x)
f x = cos x
0 1 0 -1 0 1 0 -1 0 1 0 -1 area
∫ f x dx = sin x C
∫0/2
cos x d x = 1
lags
∫0xcost d t
= + sin(x)
Derivatives (1A) 18 Young Won Lim6/23/14
The Euler constant e
dd x
ax = limh0
a xh − ax
h= a x lim
h0
ah − 1h
limh→0
ah − 1h
= 1 f ' 0 = 1
dd x
ex = e x
f x = e x f ' x = ex f ' ' x = ex
limh→0
ah − a0
h − 0= 1
e = 2.71828⋯
⇒ a x such a, we call e
Derivatives (1A) 19 Young Won Lim6/23/14
The Euler constant e
f ' 0 = 1
dd x
ex = e x
f x = e x
f ' x = ex
f ' ' x = ex
limh→0
ah − a0
h − 0= 1
e = 2.71828⋯
limh→0
ah − 1h
= 1 iif a = e
Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red).
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 20 Young Won Lim6/23/14
Derivative Product and Quotient Rule
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x
ddx
( f g) = d fdxg + f d g
dx
f (x )g( x)
dd x
ddx ( fg ) = ( d fdx g − f d gdx ) / g2
f ' (x )g(x ) − f (x)g ' (x )g2(x )
Derivatives (1A) 21 Young Won Lim6/23/14
Integration by parts (1)
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x
ddx
( f g) = d fdxg + f d g
dx
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)
f g = ∫ f ' g dx + ∫ f g ' dx
∫⋅d x
∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=
Derivatives (1A) 22 Young Won Lim6/23/14
Derivative of Inverse Functions
y = f (x )
a
b (a, b) = (a , f (a))
b
a(b, a) = (b , f−1(b))
y = f−1(x )= g(x )
g' (b) = 1f ' (a)
m2 = g' (b)m1 = f ' (a)
= 1f ' (g(b))
= (b ,g(b))
m1m2 = f ' (a)g ' (b)= 1
Derivatives (1A) 23 Young Won Lim6/23/14
Derivative of ln(x) (1)
y = f (x )
a
b (a, b) = (a , f (a))
b
a(b, a) = (b , f−1(b))
y = f−1(x )= g(x )
g' (b) = 1f ' (a) m = g' (b)m = f ' (a )
= 1f ' (g(b))
= (b ,g(b))
f (x )= ex g(x ) = ln x = f −1(x)
g' (x )= 1f ' (g(x))
= 1eg(x )
= 1e ln x
= 1x
Derivatives (1A) 24 Young Won Lim6/23/14
Derivative of ln |x|
y = ln x
e y = x
ddx
e y = ddx
x
e y dydx
= 1
dydx
= 1e y
= 1x
ddx
ln x = 1x
x>0
ddx
ln(−x )= 1(−x )
⋅(−1) = 1x
x
Derivatives (1A) 25 Young Won Lim6/23/14
Differential Equation
f (x ) = e3 x
f ' (x ) = 3e3 x
f ' (x ) − 3 f (x) = 3 e3x−3 e3 x = 0 f ' (x ) − 3 f (x) = 0
f (x ) ?
Young Won Lim6/23/14
References
[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”
Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26
Recommended