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Contents Lesson 1 .............................................................................................................................. 2 Lesson#2 ............................................................................................................................. 4 Lesson 3 .............................................................................................................................. 7
OPTIONAL- where it fits in ............................................................................................... 7 Lesson #4 ............................................................................................................................ 9 Lesson #5 .......................................................................................................................... 10 Lesson #6 .......................................................................................................................... 12 Lesson 7 ............................................................................................................................ 13
Lesson #8 .......................................................................................................................... 14 Lesson #9- ......................................................................................................................... 15
Lesson #10 ........................................................................................................................ 18 Lessons 11-13 ................................................................................................................... 19
Lesson #13 ........................................................................................................................ 22
Lesson 1 u-sub from old book WS p 297 # 7-15 odd, 31, 33,44-46,
Flipped classroom
http://www.chaoticgolf.com/vodcasts/calc/lesson6_2_part1/lesson6_2_part1.html
U -substitution Use attached examples from the Greg Kelly power point- examples 1-6
show what can’t be done
When u-substitution does not work
Ex 1 dxxx 35 2
Ex 2 dxxx 22sin5
Ex 3 dxx 32
Ex 4 dxxx 322
Ex 5 dxxx 32 sin
Ex 6 22sin x
Lesson#2 and ln and e to the x-use smart notebook slide 12
cw Worksheet- hw p. 342 #25-43 odd
There's a big calculus party, and all the functions are invited. ln(x) is talking
to some trig functions, when he sees his friend ex sulking in a corner.
ln(x): "What's wrong ex?"
ex: "I'm so lonely!"
ln(x): "Well, you should go integrate yourself into the crowd!"
ex looks up and cries, "It won't make a difference!"
F(x) = eu F’(x) = u’eu chain rule
Review with examples
f(x)=e2x-1 f’(x)=2e2x-1
f(x)=2
5
xe
Examples
dxex x32
dxe x 24
1
0
xe
Integration natural log- examples in smart notebook
xx
dx
d 1ln
'
1ln u
uu
dx
d
Examples-
1.
∫𝟐𝒙
𝒙𝟐+𝟐
2. ∫𝒙𝟐
𝟑𝒙𝟑+𝟕
Don’t get tricked into thinking every integral with division is an ln u
Clarify-From page #320 of book
Because the natural log is undefined for negative number, you will often encounter expressions of the form
ln│u│. The following theorem states that you can differentiate function of the form y=ln│u│ as if the absolute
value sign were not present
dx
x
x
)47tan(
)47(sec2
dx
x
xx
3
53 23
Lesson 3 changing the limits on integration with u-sub- SEGUE FROM YESTERDAY PROBLEMS
IN cw P. 343 # 53-59,63 , 71,73,74,75, (see example #9 on page 341
Do a long division problem-notebook page 17
When u sub does not work
dxe x3
can’t do it
dxe xcos can’t do but can do dxxe xcossin
Integrate
1 3 4 3(2 5)x x dx
4 dx
x
x 3)(ln
2 dxx5
5
5
dx
x
xx 23 34
3
dxx
x29
6. 5x dx
OPTIONAL- where it fits in
Inverse functions 5-3
Reflective property of inverse functions. The graph of f contains the point (a,b) if and only if
the graph of f-1 contains the point (b,a)
Inverse functions undo each other- interchange the x and y and solve for y
F(x) = 2x3-1
G(x)= 3
2
1x
Verify that f(x) and g(x) are inverse functions
F(g(x))=g(f(x))
Listen to it-say it aloud and you can hear it
inverse functions have reciprocal slopes f(x) =2x+3 what is the inverse?
f-1(x) = 1/2x-3/2
what is the slope of f(x) - what is the slope of f-1(x)?
2
F(x) = ¼ x3+x-1
What is f-1(x) when x=3
Chart
X f(x) f’(x)
0 2 1
1 3 2
2 5 3
3 10 4
G(x) = f-1(x)
What is g’(3)
F(x) = 2x2 -3x h(x)=f-1(x) what is h’(-1)_
F’(x) = 6x2-3
What f(-1,1)
h(1,-1) plug in1 into f’(x)
F’(1)=3
So h’(-1)=1/3
Alternate lesson #1-(2012)
Return test, test corrections-
HW- watch video- slope fields and differential equations
http://www.chaoticgolf.com/vodcasts/calc/lesson6_1/lesson6_1.html
Lesson #4 CW- multiple choice- review of FRQ
HW- reverse classroom
Lesson #5
differential equations- HW FRQ 2010 p. 361 #1-5, 7,9,
(Both very good- chaotic golf- does slope fields and differential equations)
http://www.khanacademy.org/video/simple-differential-equations?topic=calculus (15 min)
http://www.chaoticgolf.com/vodcasts/calc/lesson6_1/lesson6_1.html
(20 min) watch 5-11 and 17-22 minute marker
Ex 1-solve the differential 𝑑𝑦
𝑑𝑥 = x y2 use y(1) =2 to solve for C
2) 𝑑𝑦
𝑑𝑥= 4 − 𝑦
3) sinx 𝑑𝑦
𝑑𝑥
= cos x
4) Find the particular solution y=f(x) with initial condition f(0)=-1
𝑦′ =5𝑥
𝑦
Lesson #6 Differential equations-packet- p.4 #5,11 p. 5 #1,2 and FRQ 2000,2003
HW- mr leckie- Differential equations; Growth and Decay
http://www.chaoticgolf.com/vodcasts/calc/lesson6_4/lesson6_4.html
(10 minutes)
Lesson 7 Cw/hw- from packet- Free response 1992, 1989, p4 #6,12 p. 6 #1993
∫𝑒6𝑥+1
𝑒𝑥 𝑑𝑥
2. The rate of change of y is proportional to y. When t=0 y=2 and
when t=2 y=4. What is the value of y when t=3
2 Water flows continuously from a large tank at a rate proportional to the amount of water
remaining in the tank,
There was initially 10,000 cubic feet of water in the tank and at time t=4 hour, 8000 cubic feet
of water remained, what is the value of k in the equation
To the nearest cubic foot, how much water remained in the tank at time t=8 hour
Lesson #8 Intro to slope fields-
http://www.chaoticgolf.com/vodcasts/calc/lesson6_1/lesson6_1.html
pg,2,3
Solution 2005 AP #6
Lesson #10 CW/HW - p. 377 #1-6, 11-14, 37, 39-42
1. Find the general solution to the differential equation: y = y
x
cos
sin.
(Express answer in form y = f(x).)
____________________________________________________________
Lessons 11 Go over HW- ,p. 377 #25-28,32,49
1.
Lesson 12 Reviewp. P. 380 #67-69
9. Write the equation of the curve that passes through the point (1,3) and has a slope of y/x2 at
each point (x,y)
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