Contents
Lesson #1 ............................................................................................................................ 2
Lesson #2 ............................................................................................................................ 5
Lesson 3 .............................................................................................................................. 7
Lesson 4 .............................................................................................................................. 8
Lesson 5 .............................................................................................................................. 9
LESSON 6 ........................................................................................................................ 12
Lesson #8 .......................................................................................................................... 13
LESSON #9 ...................................................................................................................... 14
Lesson #10 ........................................................................................................................ 15
LESSON 11 ...................................................................................................................... 16
Lesson 12 .......................................................................................................................... 17
LESSON 13 ...................................................................................................................... 19
Lesson #1
HW 2-4 Worksheet, (don’t use mr leckie, he doesn’t use u-sub)
Chain rule-function within a functionhttp://www.calculus-help.com/funstuff/phobe.html
Find the derivative
f(x)=(x2+x)2
Show students the website of worked out u-sub problems
Use the product rule to find the derivative
find f’(x)
3
F(x) =
x
2
 1
2
2x
F(x) = 3
x2  4
f ( x )  ( x 3  2 x 2  5 )( x 2  4)
Lesson #2
Chain rule- p.158 1-31 odd
1 Find an equation of the tangent line
𝑦 = 𝑥𝑐𝑜𝑠 4𝑥
𝑎𝑡 𝑥 = 2𝜋
2 G(x)= tan2x2 find G’(x)
Lesson 3
4.4 Derivatives of exponential and logarithmic functions
p. 183 1-20 , 37-40
2. Find all points on the graph F(x) = √(𝒙𝟐 − 𝟏)𝟐 for which f ’(x)=0 and
where f ’(x) does not exist
𝟑
SET A-(if it’s anything but an x use u-substitution) derivative of eu= u’eu
F(x) = ex
F’(x) = ex (ln e)
F(x) = e3x
F’(x) =3e3x(ln e)
f(x)=3x
f’(x) = 3xln 3
f(x)=34x
f’(x)= 34xln3(4)
f(x) = e√x
F(x) = (e-x+ex)3
1
SET B (if it’s anything but an x use u-substitution) derivative of ln u=u’
𝑢
f(x)= lnx
1
f’(x) =
𝑥
1.
f(x)=ln(3x+2)
3
F’(x)=
3𝑥+2
2.
f(x)=ln(x3+x)
F’(x)=
3𝑥 2 +1
𝑥 3 +𝑥
3.
f(x)=(lnx)3
F’(x)=
4
F(x) = e-x ln x
Lesson 4
Go over 100 pt test
Multiple choice packet # 1-35
Quiz on chain rule tomorrow
Lesson 5
18 pts chain rule test-
Implicit differentiation
HW 4.2 p. 167 1-12 ,p. 169 59-62
(if time do a second derivative problem and hw 27-30)
So far the functions we have been working with have been
explicitly expressed in terms of one variable.
Y=sin x f(x)=3x-5
To understand how to find dy/dx implicitly, you must realize that
the differentiation is taking place with respect to x. This means
that when you differentiate terms involving x alone, you must
differentiate as usual. However, when you differentiate terms
involving y you must apply the chain rule because you are
assuming that y is defined implicitly as a differentiable function of x
Ex 1: Ask students to find the derivative, they will solve for y and
then take itCircle- which is not a function
x  y  25 find dy/dx
2
2
what happens if we can’t solve for y like-
x 5  x 4 y 2  y 4  10 xy  11
y 2  sin( xy)  2 cos(ln x)
then we must learn another way-implicit differentiation- do the
circle by implicit
if time allows – do a second derivative
Find the second derivative
2x2+y2=16
(hand out)
Guidelines for implicit differentiation
1.
Differentiate both sides of the equation with respect to x
2.
Collect all terms involving y’ on the left side of the equation
and move all others to the right.
3.
Factor y’ out of the left side of the equation
4.
Solve for y’ by dividing both sides of the equation by the left
hand
EX 1- find dy/dx
x2  2y3  x2 y  2
EX 2- find dy/dx
2 y 3  5 xy 2  x 4  10
EX 3 - find dy/dx
2y = x2+siny
EX 4
Write the equation of the tangent line to the graph of
y 3  yx 2  x 2  3 y 2  1 at the point (1,2)
LESSON 6
4.2 Implicit differentiation p. 167-169 # 27-30, 52,63,64
Hw- AP question 2004,2005
( watch related rates with Mr Leckie
http://www.chaoticgolf.com/vodcasts/calc/lesson4_6b/lesson4_6b.html
Scoring guide
http://apcentral.collegeboard.com/apc/public/repository/_ap05_sg_calculus_
ab__46570.pdf
Determine the slope of the graph when x=1
2x 2  4 y 2  9
Find the second derivative
2x2+y2=16
Lesson #8
Related rates – CW workshseet, HW- worksheet
P. 255 #11,13,19, -chapter Test-- Tues 11/5HW -worksheet – solutions worked out on website
Differentiating with respect to time dy/dt or dx/dt
1 y=x2+3 find dy/dx
2
then find dy/dt when x=1 , given dx/dt =2
3
y=√𝑥 find dy/dt when x=4, given dx/dt =3
4
y=2(x2-3x) find dy/dt when x=3 given dx/dt=2
LESSON #9
Mr leckies- problems (seniors follow along at home)
HW-AP 2002 #5AND 2002 form B#6
Multiple choice packet # 43-51
Lesson #10
CW- 2002 related rates AP problems
HW -p. 255 # 9,11,13,17,19
Show how to get down to one variable- 2002 #5 and inverted circular cone, relationship
between height and radius
Show the difference between dy/dt and dy/dx
1
Find dy/dx 3x2+2y3=8 find dy/dt 3x2+2y3=8
2
A pebble is dropped into a calm pond, causing ripples in the form of
concentric circles. The radius r of the outer circle is increasing at a rate of
1ft per second. When the radius is 4 ft, at what rate is the total area A of the
disturbed water changing?
2. A cone-shaped icicle is dripping from the roof.
The radius of the icicle is decreasing at a rate of 0.2 cm/hour, while the
length is increasing at a rate of 0.8 cm/hour.
If the icicle is currently 4 cm in radius and 20 cm long, is the volume of the
icicle increasing or decreasing, and at what rate?
3
A 15 foot ladder is resting against the wall. The bottom is initially 10
feet away from the wall and is being pushed towards the wall at a rate of
.25ft/sec. How fast is the top of the ladder moving up the wall 12 seconds
after we start pushing?
LESSON 11
Review sheet- 2 sided
1.
2.
Find slope for the following curve at the point
(1,0)
x2-3y3=4xy
Find the
x2+y2=25
d2y
d 2x
for the curve defined by
Lesson 12
Go over all HW- cw.review sheet (other side of 2002 AP problems
HWpp. 186
1-13 odd, 35,39,47,
1. xy=10 find dy/dt when x=8 given that dx/dt =5
d2y
2. Find the dx
1-xy=x-y
3. Find dy/dx of the curve y=cos3(x2)
Test on related rates
2012- Hurricane Sandy missed a week of school- (so test was 11/13- chapter test 11/20
More review if needed
Review for chapter 4 test
1. Find the derivative
y=√𝑥 2 + 1 sin(2𝑥)
3. Find dy/dx of the curve y=cos3(x2)
5
Find the velocity of the function if it’s position is s(t) = 4x 2sin(x3-1).
6. Assume x and y are both differentiable functions of time.
If 3x2 – 4y3= -32 find dx/dt when x = 0 and dy/dt = 3.
7. Find the equation of the tangent line to the curve at the point (1,-1)
x2y+3x=y2+1
Topics for chapter 4 test
Derivatives-power, product, quotient, chain,
u-substitution -trig function, ex, lnx)
2nd derivative
Implicit differentiation
Related rates
1.
Find dy/dx for the following curve
x 2 y  y 2 x  2
3.
Find the equation of the tangent line to the curve
y=xsin4x when x=π
4.
Find the slope of the curve y2+yx+3x-6y=-3 at the
point(s) when x=1
Smart board responder questionshttp://exchange.smarttech.com/search.html?m=05&q
=calculus+derivative
LESSON 13
Test