Transcript 2.2 Basic Differentiation Rules & Rates of Change

2.2 Basic Differentiation Rules
& Rates of Change
The Constant Rule
The derivative of a constant function is 0.
That is, if c is a real number, then
d
c  0
dx
Quiz
Quiz2
2.2 Basic Differentiation Rules
& Rates of Change
Function
y7
f  x  0
s t   3
y  k , k is constant
2
Derivative
dy
0
dx
f '  x  0
s ' t   0
y'  0
2.2 Basic Differentiation Rules
& Rates of Change
The Power Rule
If n is a rational number, then the function
f  x   x is differentiable and
n
d n
n 1
 x   nx .
dx
Function
y  7x
yx
3
s t   t
0
Derivative
dy
0
dx
y' 1
2
s '  t   3t
g  x  3 x
1 2 / 3
1
g' x
 2/3
3
3x
1
y 2
x
dy d 2
2
3
 x   2 x   3

dx dx
x
2.2 Basic Differentiation Rules
& Rates of Change
Find the slope of f  x   x when
4
 a  x  1  b  x  0  c  x  1
f ' x   4x
3
f '  1  4
f '  0  0
f ' 1  4
p. 104
Find an equation of the tangent line to the graph
of f  x   x when x  2.
2
AP Exam
Multiple Choice
f '  x   2 x  f '  2   4  mtangent  4
f  2   4  the tangent goes through  2, 4  .
y  y1  m  x  x1 
y  4  4  x  2 
p. 104
2.2 Basic Differentiation Rules
& Rates of Change
The Constant Multiple Rule
If f is a differentiable function and
c is a real number, then cf is also
differentiable and
d
cf  x    cf '  x  .
dx
Function
y  2/ x 2
4t
f t  
5
y2 x
y
1
3
2 x
2
3x
y
2
Derivative
dy
2
 2 / x
dx
dy

dx
8
f ' t   t
5
1
x
dy
1
  5/3
dx
3x
3
y'  
2
2.2 Basic Differentiation Rules
& Rates of Change
Function
Rewrite
5
y 3
2x
5
y
3
 2x 
5 3
y x
2
5 3
y x
8
15 4
y' 
x
2
15 4
y' 
x
8
7
y  2
3x
7
y
2
 3x 
7 2
y x
3
14
y'  x
3
y  63x
y '  126 x
2
Differentiate
Simplify
15
y'   4
2x
15
y'   4
8x
2.2 Basic Differentiation Rules
& Rates of Change
The Sum and Difference Rules
The derivative of the sum  or difference  of two
differentiable functions is differentiable and is the
sum  or difference  of their derivatives.
d
 f  x   g  x    f '  x   g '  x  Sum Rule
dx
d
 f  x   g  x    f '  x   g '  x  Difference Rule
dx
2.2 Basic Differentiation Rules
& Rates of Change
Function
f  x   x  4x  5
3
4
Derivative
f '  x   3x  4
2
x
3
2
3
g  x     3x  2 x g '  x   2x  9x  2
2
2.2 Basic Differentiation Rules
& Rates of Change
Derivatives of Sine and Cosine Functions
d
sin x  cos x
dx
d
cos x    sin x
dx
2.2 Basic Differentiation Rules
& Rates of Change
Function
f  x   2sin x
sin x
y
2
y  x  cos x
Derivative
f '  x   2cos x
1
y '  cos x
2
y '  1  sin x
Change in Distance s

Change in Time
t
Average Velocity
If a billiard ball is dropped from a height of 100 feet, its height
s at time t is given by
s  16t 2  100
Position Function
where s is measured in feet and t is measured in seconds. Find
the average velocity over each of the following time intervals.
(a) [1,2] (b) [1,1.5] (c) [1,1.1]  Instantaneous Velocity
s s  2   s 1 36  84


 48 feet per second
t
2 1
1
s s 1.5  s 1 20


 40 feet per second
t
1.5  1
0.5
s s 1.1  s 1 80.64  84


 33.6 feet per second
t
1.1  1
0.1
p. 109
The average velocity between t1 and t2 is the slope
of the secant line, and the instantaneous velocity at
Avg. Velocity
t1 is the slope of the tangent line.
At time t  0, a diver jumps from a diving board that is 32 feet
above the water. The position of the diver is given by
s  t   16t 2  16t  32
where s is measured in feet and t is measured in seconds.
(a) When does the diver hit the water?
(b) What is the diver's velocity at impact?
s  t   16t 2  16t  32  0
16  t 2  t  2   0
16  t  2  t  1  0
t 2
s '  t   32t  16
s '  2   48 feet per second
p. 109
HW
2.2/1,3,11,15,17,19,21,24,26,28,
29,33,35,37,41,45,46,49
Director: Tim Sarkela
Producer: Joey Kurz
Screen Writer: Tim Sarkela
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Tom Cruise
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Carl Marx
Old Major
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Raphael, Michelangelo, and Leonardo
Ken Poyner
Darlene Sarkela
Patty Granado
The three little pigs
Dracula
Elvis
Scorpion
Sub-zero
Cinematographer: Marin Palmer
Composer: Josiah Stocker
Editor: Emily Sherry
Executive Producer: Donkey Kong
Key grip: Raul “ricky” Marwah
Voice: Rush Limbah
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Characters:
Mr. Anderson: Tim Sarkela
Mr. Smith: Joey Kurz
Gizmo: Joey Kurz
Mr. Hoshagowa: Tim Sarkela
3rd lady: Joey Kurz
Pregnant man: Tim Sarkela
Little boy: Joey Kurz
Bruce the fun lovin moose: Tim Sarkela
Sarah: Joey Kurz
Jimmy: Tim Sarkela
Lassie: Joey Kurz
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Special thanks to:
Bob’s fish parlor
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Nintendo of America
No thanks to:
Fred’s fish parlor
The Nazi party
The Bush administration
Nintendo of Japan
And a very special thanks to Mr. Gibbons for teaching
us these crazy derivatives
Sketch the graph of a function whose
derivative is always negative.
Lesson
Find the derivative of s t   t  2t  4.
3
s ' t   3t  2
2
Lesson