Warm Up
Pick Up One
Test 3A
The first half of chapter 3
Topics
Approximations
Average rate of change
Instantaneous rate of change
Graphically, numerically, table, equation
Definition of Derivative
Generate a formula?
Generate a value?
Topics
Differentiabi
HW:
Handout AND
Website
Warm Up
Given f ( x ) = sin x find
Put sin-1x
into Y1
df 1
dx
at
(
3 2 , 3
)
2
f ( x ) = x2 x 0
df
= 2x
dx
y = x2
At x = 2:
( 2, 4 )
f ( 2 ) = 22 = 4
df
( 2) = 2 2 = 4
dx
m=4
y= x
We can find the inverse
function as follows:
y = x
Warm Up
HW p. 105 #1 23 odd, #27
What notation 2is used to
Given f ( x ) = x + 1
d Sketchyourgraph
escribe the answer to
the slope of the curve at
x 3?
=
Draw the tangent to the
Useurve derivative to
c the at x = 3
write the equation of the
tangent line a
AP#1
AP#1
NoCalculator
NoTextbooks
NoPartners
YesNotes
YesWarmupBooks
15minutes!
IVTWarmUpProblems
IVTWarmUpProblems
1)
If f(x ) is c o ntinuo us
If
is
w ith f(3)=3, f(5)=-1 and
(3)=3, (5)=-1
f(6)=0, whic h value o f f
(6)=0,
m us t b e ac hie ve d
twic e
Warm Up
1. Sketch a graph of y = ex
Draw the tangent line at x = 0
Estimate the slope of y = ex at x = 0
2. Find the derivative of
y = cos
( x)
1 3
3.9 Homework (part 1)
p. 178 #3, 6 ,9, 11 13 (all)
29, 33, 35, 52, 62
Look at the graph of
The slope at x
HW: page 84
3-17 (odds)
33, 41,43,45, 51.
Warm Up
Quiz Review
Limit Review 2
Name each TYPE of discontinuity
1
Use
lim
x
2
3
4
your calculator to evaluate:
3+ x
5 + 6x2
1
lim sin
x 0
x
Where is this fxn NOT continuous?
f(x)
Prove f(x) is not
continuo
WarmUp:Evaluate
t2
lim
t 3 t + 2
lim sin
3 2
lim e
x
HWp.76
#3and#5
#1323odd
#27,37,53,55
5x + 8x
lim 4
x 0 3 x 16 x 2
3
Iknowwhythelimit
ofaconstant
functionisthat
constant.doyou?
2
( x + 3)
lim
x 0
2
x
x 0
x2 +1
lim
x 0 1 sin x
3 sin 5 x
lim
x 0
4x
9
Warm Up
Up
ick
P
AP #2
Thursday
3.2 HW
p. 114: 3, 7, 9, 11-14 (all),
21, 25, 35, 39
ne
O
Quiz Friday
Graphically, when do you EXPECT the slopes from
the left slopes from the right?
Afunctionis
DIFFERENTIABLEifa
derivativeexistsat
everypointinthe
domain
Wh
Unit
3B
Warm Up
In the x-y plane, the line x + y = k where k is
a constant, is tangent to the graph
y = x + 3x + 1
2
a) 3
. What is k?
b) 1
c) 5
d) 6
HW:
p. 153
#1 #23 (odd)
Omit #3, 11, 15
#27, 29, 31, 45, 47,
56 (a, c, e, g)
We now have a pretty good li
Warm Up
1)
HW:
Handout
a.
1
x 1
Findtheslopeofthecurveatx=2.
b.
Writetheequationofthetangenttothecurveatx=2.
c.
Writetheequationofthenormaltothecurveatx=2.
f
Given( x ) =
There are some
that call me
Difference
Quotient
EXACT value of the slope of the tan
Unit2Limits
Areyoua
Function?
ShortUnit
NounitTESTjust
culminatingQUIZ
HWp.66
#918multiplesof3
#1929Odd
37,43,45,49,51
WarmUp
Simplify
f ( x + h) f ( x)
h
If
f ( x) = x2 4
Takeit.totheLimit
lim f ( x ) = L
x c
Asxgetsreally
really,really
closetoc
Mygraph
Warm Up
dy
d2y
Find
and
dx
dx 2
HW: pg 124 #13, 14,
1523 odd
#27, 31,42,46, 48
3 25
335
2
y = x 4 x + x + 14 6
If
7
x
Higher Order Derivatives:
dy
y =
dx
is the first derivative of y with respect to x.
dy d dy d 2 y
y =
=
=2
dx dx dx dx
dy
y =
dx
y
( 4)
Warm Up
Given
HW:
P. 135
#9, 10, 12, 19, 24, 31, 37
y = u v
u ( 4 ) = 3
u ( 4 ) = 1
5
Find y ( 4 )
u
y=
v
v( 4 ) = 4
3
v( 4 ) = 1
3.4 Velocity, Speed, and Rates of Change
Consider a graph of displacement (distance traveled) vs. time.
Average velocity ca
Unit 3A
Derivatives Part 1
OPTIONAL Ch 2 Review:
Pg. 95
#15 23 odd
#27, 29, 41, 54
Warm Up
HW: (due Wed)
p.92
#1 11 odd
#15, 17, 21,29,31
2
Given f ( x ) = x 9 and given that f ( 2 ) = 5, f ( 4) = 7
prove that somewhere on [-2, 4] f ( x ) = 0 .
I.V.T
x 2
Homework:Handout
Warm Up
Pick up one
Quiz Friday!
Section 2.4 3.2
AP #2 Thursday!
Given the graph of f, graph the
derivative fxn
u
y=x2+3
Given a piece-wise fxn, graph the
piece-wise derivative fxn
2 x, x 4
x , - 4 < x 0
f ( x ) = 5, 0 < x 4
1 x, x > 4
Warm Up
HW: p. 124
#19 odd
#25, 29, 33, 35, 39, 53, 54
a difference quotient to find f ( x ) if
f ( x) = 6x2 x
Use
Write
the equation of the tangent and the
normal lines at x = 1
You Iwill beneed that me
dont tested by
Definition AP on the
and by of Der
HW:
p. 146
Unit 3A Test
#1 riday!
F 11 (odd)
21 Topics through #25
31 (odd) omit
#37, 39, 43
todays lesson
Warm Up
Find
2x + 5
yif y =
3x 2
Find the slope of the curve at x = 2.
Write the equation of the tangent line at x = 2
Use nderiv to evaluate
y(
WarmUp
HW:
Handout
Dudethe graphs of
these equations must
be, like, CRAZY , man!
x 2 xy + y 2 = 48 has
Find the points where
horizontal and vertical tangents.
HW: p. 162
1 13 odd
17, 23, 29, 43, 45, 47
Warm Up
Find the equation of the line
x = 2 cos t
tangent to the curve at a point y = 2 sin t
at the given value of t.
t =
1.
Find (x, y) on the curve
2.
dx
dy
Find dt and dt
3.
Use the definition of the slope of
HW:
p.170 3,6,7,8,11, 15 21 odd
25,26
Warm Up
Find the derivative of y =
3
( sin x ) 2 5 x
d2y
2
3
Find dx 2 for 3 y x = 9
We can use implicit
differentiation to find:
y = sin 1 x
d
sin 1 x
dx
y = sin 1 x
y = sin x
sin y = x
d
d
sin y =
x
dx
dx
dy
cos y