AP AB Calculus
Chapter 2A
Mrs. Boddy
AP Calculus AB
Assignments – Chapter 1
Date Day Section
9/16
1
2.1
Learning Target



Find the derivative by definition
Find the derivative by alternate form
Find the equation of a tangent line at a specific point
Find where a function is differentiable
p. 104

See the relationship between functions and their
derivatives
Worksheet 2.1


PLAN/ACT PRACTICE
PARENT’S NIGHT


Differentiate using the power rule and shortcuts
Find the derivative of sine and cosine

M
9/17
2
Assignment
7, 17, 21, 23, 25ab, 29ab,
53,
83-88, 101-104
2.1
T
9/18
3
W
9/19
4
2.2
3-33 odd, 43-49 odd, 53,
59-65 odd
R

9/20
F
5
2.2
p. 115
Use derivatives to find rates of change
p. 117
87, 88, 93-99 and
AP Graded WS
9/23
6
Review
7
2.1-2.2 QUIZ
2.1 -2.2 Worksheet
M
9/24
2.1-2.2 QUIZ
T


9/25
8
2.3

Differentiate using product rule
Differentiate using quotient rule
Find higher order derivatives

Analyze graphs using product and quotient rule
W
9/26
9
2.3
p. 126
1-29 eoo, 32, 33,
39-51 eoo, 97-103 o, 117
2.3 Worksheet
R

9/27
10
Differentiate using chain rule
2.4
7-27 eoo, 45-65 eoo, 67,
69, 77a, 93, 95
F

9/30
M
11
p. 137
Differentiate using chain rule
2.4
2.3 Quiz
2.4 Worksheet
10/1
T
12
Review
Review
13
Review
Review
14
CHAPTER 2 TEST – Written
15
CHAPTER 2 TEST - MC
10/2
W
10/3
R
10/4
F
AB Calculus
Section 2.1 – Day 1
The Derivative and the Tangent Line Problem
Learning Targets: Students will be able to find a derivative by its definition and find the slope of the tangent line at a
given point. Students will be able to find a derivative by using the alternate form of derivative. Students will be able to
recognize where functions are differentiable.
The Tangent Line Problem
Given f(x) and a point P on f(x),
write the equation of the line tangent to f(x) through point P
f(x)
Q
Remember, to write the equation of a line,
you need a point (already have it) and the slope.
P
So, we must find the slope to write the equation.
Pick another point Q on f(x) such that P has coordinates ( x, f(x) )
set Q at an x-distance of h from P
then Q will have the coordinates
(
,
)
and the slope of the secant line PQ equals …
Now, move Q closer and closer to P, so the secant line becomes closer and closer to being the tangent line. The slope of
the tangent line equals the limit as h approaches 0 of the slope of the secant line:
The derivative is the general equation for the slope of the tangent line of a function.
NOTATIONS FOR DERIVATIVE:
lim
h 0
f ( x  h)  f ( x )
f ( x  x)  f ( x)
dy
 lim
= f '( x)  y ' 
x 0
h
x
dx
(AP version)
(Book version)
Also called the difference quotient.
Ex 1:
Find f '  x  for the function f ( x)  x 2  1
(f prime) (y prime)

d
d
 y 
 f ( x) 
dx
dx
d
(treat
as an operation on the fcn)
dx
Ex 2:
Find the slope of the line tangent to the function f ( x) 
1
at the point
x2
 1
 2,  by hand and then verify by
 4
calculator.
On calculator:
Graph function.
Menu, analyze graph (6),
dy
(5), type in x-value
dx
Alternate form of Derivative at a specific point (c)
lim
x c
Ex 3:
f ( x )  f (c )
xc
Write the equation of the tangent line using the alternate form of derivative of f ( x)  x2  2 x  1 at
 3, 4 .
For a function to be differentiable at c (means that it can have a derivative):
1.
Is this
differentiable
at c? Why or
why not?
It must be continuous at c
c
2.
Its one-sided limits of the derivate must be equal
(no sharp turns)
c
3.
It cannot have a vertical tangent line at c
(limit is DNE or undefined)
c
Example:
1. Where is the graph differentiable?
1
2
Is this
differentiable
at c? Why or
why not?
AB Calculus
Section 2.1 – Day 2
The Derivative and the Tangent Line Problem and Graphical Representation
Learning Targets: Students will be able to see the relationship between functions and their derivatives.
Let’s talk about lines:
What does a line with a positive slope look like?
What does a line with a negative slope look like?
What does a line with a zero slope look like?
Remember: The derivative is the slope of the tangent line! Therefore:
If the derivative is positive at a value, the function is ____________
If the derivative is negative at a value, the function is ____________
If the derivative is zero, the tangent line at that value of the function is ____________
f ( x)
f '( x)
If the derivative is zero or undefined at a value, this value is called a ________________________ of the function.
If the derivative is undefined, there may be a ____________________________ to the graph of the function at that value.
Critical points:



Ex 1:
Critical values
Points on the graph where the derivative is zero or undefined
Hint: when starting from f ( x ) , use the turning points or undefined values of the graph.
Graph the derivative of the function and fill out the sign chart.
f ( x)
f '( x)
Ex 2:
Graph the derivative of the function and fill out the sign chart.
f ( x)
f '( x)
Ex 3:
f ( x)  x 2  2 x
a. Sketch the graph
b. Identify the x-value of the turning point of the graph
c. Complete the sign chart
d. Sketch the graph of the derivative
f ( x)
f '( x)
Ex 4:
f ( x)  sin x,
a.
b.
0, 2 
Sketch the graph
At what value(s) of x is the slope of the line
tangent to the graph positive? Negative? Zero?
c. Complete the sign chart
d. Sketch the graph of the derivative.
f ( x)
f '( x)
AB Calculus
Section 2.2 – Day 1
Basic Differentiation Rules and Rates of Change
Learning Targets: Students will be able to find derivatives using derivative shortcuts and find the derivatives
of sine and cosine.
f ( x  h)  f ( x )
Definition of Derivative: lim
This formula works, but there are shortcuts that we can take so we do not
h 0
h
have to apply the definition every time we want to find a derivative.
SHORTCUTS:
1.
If f ( x )  c then f '( x)  0
Examples:
2.
f ( x)  5
f ( x)  3
f '( x ) 
f '( x) 
If f ( x)  x n then f '( x)  nx n1
No variables in the denominator!
Examples:
3.
f ( x)  x 4
f ( x)  x
f '( x) 
f '( x) 
f ( x)  4 x
f ( x) 
f '( x ) 
f '( x) 
1
x3
If c f ( x) , then c f '( x) - (c is a constant that is factored out of the function)
Examples:
f ( x)  
f '( x) 
5x
4
f ( x) 
f '( x ) 
3
x
f ( x)  6 x
f '( x )
f ( x) 
2x2
3
f ( x) 
f '( x) 
4.
1
5 4 x3
f '( x) 
If f ( x)  g ( x) , then f '( x)  g '( x)
Examples:
5.
f ( x)  x 4  3x 3  5
2 x2  3x  1
f ( x) 
x
f '( x) 
f '( x ) 
Trig Rules for Derivatives:
If
f ( x)  sin x , then f '( x)  cos x
f ( x)  cos x , then f '( x)   sin x
Examples:
f ( x)   cos x
f '( x) 
2
f ( x)  3cos x  sin x
3
f '( x) 
6. Find all x-values/points where the function has a horizontal tangent.
f ( x)  x 3  2 cos x
7. Find the equation of the tangent line to f ( x)  5 x 2  3x  2 at the point  2, 28 .
MEMORIZE!!
AB Calculus
Section 2.2 – Day 2
Basic Differentiation Rules and Rates of Change
Learning Targets: Students will be able to use derivatives to find rates of change.
So far, the derivative has been used to find the slope of the tangent line. It is also used to find the rate of change of one
variable with respect to another variable.
Average Rate of Change 
f  x1   f  x2 
y
f ( x )  f (c )


 slope of secant line!
x
x1  x2
xc
Instantaneous Rate of Change 
Ex 1:
dy
at x  slope of tangent line!
dx
Find the average rate of change over the interval and then find the instantaneous rate of change at the endpoints of
the interval.
f (t )  t 3  3
2, 2.5
An application of the derivative:
s(t )  position function
s '(t )  v(t )  velocity function (vector-valued functions: sign  direction)
Remember: Position Function Under Gravity
Ex 2:
s (t )  16t 2  vot  so
At t  0 , a diver jumps from a board 32’ above water. His initial velocity is 16’/sec. The position of the diver is
s (t )  16t 2  vot  so .
a)
When will the diver hit the water?
b)
What is his velocity immediately prior to impact?
Ex 3:
A ball is thrown straight down from the top of a 220 foot building with an initial velocity of -22 ft/second.
a)
What is the velocity after 3 seconds?
b)
What is the velocity after falling 108 feet?
Ex 4: A ball is blasted vertically upward with a velocity of 128 ft/sec from the ground.
a)
How high does the ball go?
b)
How fast is the ball traveling when it is 60 feet above the ground?
AB Calculus
Section 2.3 – Day 1
Product and Quotient Rules and Higher-Order Derivatives
Learning Targets: Students will be able to solve problems using product and quotient rule. Students will be
able to solve trigonometric derivatives. Students will be able to solve different applications using higher
ordered derivatives.
What happens when you want to take a derivative of two or more terms being multiplied together?
Product Rule:
Ex 1:
dy
 f ( x) g ( x)  f '( x) g ( x)  f ( x) g '( x)
dx
h( x)  ( x 2  1)( x3  3)
Product Rule ---------------------- vs. ---------------------------- FOIL
Ex 2:
f ( x)  2 x cos x  x sin x
What happens when you want to take a derivative of two or more terms being divided?
Quotient Rule: y 
f ( x)
g ( x) f '( x)  f ( x) g '( x)
then y ' 
g ( x)
( g ( x)) 2
How do we remember this?
Find the derivatives.
Ex 3:
f ( x) 
3x  4
x2  2
1
x
f ( x) 
x4
5
Ex 4:
MEMORIZE!!
Ex 5:
f ( x) 
sin x
1  cos x
f ( x)  sin x then f '( x)  cos x
f ( x)  cos x then f '( x)   sin x
f ( x)  tan x then f '( x)  sec 2 x
f ( x)  cot x then f '( x)   csc 2 x
f ( x)  sec x then f '( x)  sec x tan x
f ( x)  csc x then f '( x)   csc x cot x
Ex 6:
f ( x)  x sec x
Sometimes we want (need) to find 2nd, 3rd, 4th, or higher derivatives. These are called higher order derivatives.
f ( x)
Notations:
Ex 7:
y
f  4  ( x)
... f  n  ( x )
f '( x)
f ''( x)
f '''( x)
y'
y ''
y '''
y  4
...
y  n
dy
dx
d2y
dx 2
d3y
dx 3
d4y
dx 4
...
dny
dx n
4
Find f    x  for f ( x)  5cos x
Ex 8: Find y '' for y  x sec x
AB Calculus
Section 2.3 – Day 2
Product and Quotient Rules and Higher-Order Derivatives Graphically
Learning Targets: Students will be able to analyze graphs using product and quotient rule.
Remember:
s (t )  position function
s '(t )  v(t )  velocity function (vector-valued functions: sign  direction)
s "(t )  v '(t )  a(t )  acceleration function
Ex 9: The Wolf pitcher threw the ball with a velocity function (in meters per second) of v(t )  50  t 2 , 0  t  5. Find
the velocity and acceleration of the ball when t = 2.
Ex 1: Let p( x)  f ( x) g ( x) and q ( x) 
a)
p '  2 
f ( x)
. Use the graphs of f and g to find the following.
g ( x)
b) q '  0 
Ex 2: Given the following graph of f, sketch f ' and f '' . Fill out the sign chart to help you.
f ( x)
f '( x)
f '( x)
f ''( x)
AB Calculus
Section 2.4
The Chain Rule
Learning Targets: Students will be able to find derivatives using the chain rule.
3x 2  x  1 )
What happens if we have a composite function? (example
The Chain Rule:
if we are given f ( x ) and g ( x ) and h( x)  f ( g ( x)) , then the derivative is
h '( x)  f '( g ( x)) g '( x)




If y  f (u) and u  g ( x), then y  f g  x  and y '  f ' g  x   g '  x 


dy
for y  x 2  1
dx

4
Ex 2: Find the derivative of f ( x)  5 x  3x 2
Ex 1:
Find
Ex 3:
Find the derivative of f ( x)  cos 3 x 2
 
Ex 4: Find f '( x) if f ( x)  x 2 9  x 2

5
Ex 5: Find
dy
of f ( x)  5cos2  x 
dx
Ex 6: Find the slope of the tangent line at (2, 2) if
y  5 3x3  4 x
Ex 7: Find the derivative of y 
x
1
2
 3x 
2
 1 

 16 
at  4,