Name _
AP Calculus AB
Chapter 2 Day 1
2.1 Limits
dfCalculus: (almost) everything that we do will be looked at 4 ways
Rule of Four:
Analytic - This is the "x's and y's" equation manipulation that most students think
of when they think "math."
Numeric - You
Name _
AP Calculus AB
Chapter 2 Day 2
2.2 Limits Involving Infinity
1
x
Given: f ( x) =
Observe the behavior of
1
=
x x
1
as x
x
1
=
x x
lim
and lim
We can say that the line y=_ is a _ of the graph of f.
Horizontal Asymptote
The line y = b is a horizonta
AP CALCULUS
SUMMER PACKET ANSWERS
1.
2.
3.
4.
a)
a)
a)
a)
5. a)
b)
b)
d)
c)
b)
b)
c)
c)
b)
d)
e)
f)
g)
h)
d)
c) 25
6. a)
7. a)
c)
b)
b)
c)
c)
e)
d)
f)
8. a)
b)
9. a)
b)
c)
c)
d)
10. a)
b)
11. a)
b)
12. a)
c)
d)
c)
13. a)
e)
b)
d)
e)
14. a)
b)
c)
b)
d)
c)
Name _
AP Calculus AB
Chapter 2 Day 3
2.3 Continuity
Continuity
A continuous function is one in which the outputs vary continuously with the
inputs and do not jump from one value to another without taking the values in
between.
Example 1
Example 2
The gra
Name _
AP Calculus AB
Chapter 2 Day 4
2.3 Continuity
Continuous Extension Function of f(x) : A function identical to f except that it is continuous at
one or more points where f is not.
Example 1:
Given:
f ( x) =
x3 7x 6
x 3
Graph the function. What do yo
Name _
AP Calculus AB
Chapter 2 Day 5
2.4 Rates of Change and Tangent lines
2.4 Rates of Change and Tangent Lines
Average rate of change - speed (mph), growth rates (% per year) monthly rainfall
(inches per month)
The average rate of change of a function
Name _
AP Calculus AB
Chapter 3 Day 1
3.1 Derivative of a Function
The study of rates of change of functions is called differential calculus . Finding the
derivative was a 17th century breakthrough and well later talk about the applications but
for now we
Name _
AP Calculus AB
Chapter 3 Day 3
One Sided Derivatives
A function y = f(x) is differentiable on a closed interval [a,b] if it has a derivative at
every point of the interval and if the limits,
[the right hand derivative at a]
[the left hand derivativ
Name _
AP Calculus AB
3.2 Differentiability
Chapter 3 day 5
How f(a) Might Fail to Exist
A function will not have a derivative at a point P( a, f ( a ) ) where the slopes of the secant
f ( x) f ( a)
lines,
, fail to approach a limit as x approaches a.
xa
Name _
AP Calculus AB
3.3 Rules for Differentiation
Chapter 3 day 7
5) The Product Rule: The product of two differentiable functions u and v is also
differentiable and
d
dv
du
(uv ) = uv '+vu ' = u
+v
dx
dx
dx
(
fg ) =
In words:
Example: Find f ' ( x) if
Name _
3.2 and 3.3 worksheet
Chapter 3 day 8
AP Calculus AB
Chapter 3 day 6
AB Calculus 3.2 and 3.3 review
(Review your homework 3.2 3.3)
Find the derivative of each function.
1.
2.
3.
4.
Name _
3.2 and 3.3 worksheet
Chapter 3 day 8
AP Calculus AB
Chapter
Name _
AB Calculus
3.3 Practice
Find the derivative of each function.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Given find:
19.
20.
21.
22.
23. Find the equation of the tangent line to at .
24. Find the equation of the tangent line to
Name _
AP Calculus AB
3.4 Velocity and Rates of Change
Chapter 3 day 10
Derivatives represent instantaneous rates of change of a function with respect to a certain
variable.
Example 1:
a) Find the rate of change of the area of a circle with respect to the
Name _
3.4 Rectilinear Motion
Notes about motion along a straight line:
Given a function, s(t) that is moving along a straight line
Average velocity over the interval [a,b]=
Instantaneous velocity at time t=
Instantaneous speed at time t =
Instantaneous a
1. The height of an object t seconds after it is launched vertically into the air is given by the
position function
a)
b)
c)
d)
e)
Sketch the graph if the function.
Find an equation for its velocity, .
For which times t is the object rising?
For which tim
Name _
AP Calculus AB
Unit 3: Chapter 3 Part II
Day 1
3.5 Derivatives of Trigonometric Functions
Trigonometric functions are important because so many of the phenomena we want
information about are periodic (heart rhythms, earthquakes, tides, weather).
Sk
Name
_
AB Calculus
Trig/Chain Rule Practice
1.
2.
3.
4.
5.
is
a) 1
b) 0
Given , then
a)
b)
d)
e)
If , then
a) 1
d)
e) 0
Find the derivative:
a)
b)
d)
e)
Find the derivative of
a)
b)
d)
e)
6. Find when given and .
a)
b) 2
Find the derivatives of the follow
Name _
Chapter 3 Day 4
AP Calculus AB
Implicit Differentiation (3.7)
We often have equations of graphs that dont come in the convenient explicit form
y = f ( x ) . For example: x 3 4 y 3 + 6 y = 8 . In such a case we must use what is called
implicit diffe
Name _
Chapter 3 Day 5
More! Implicit Differentiation (3.7)
1. If , then =
A.
B.
D.
E.
2. for the equation
A.
B.
D.
AP Calculus AB
E.
3.
A.
B.
D.
C. 3
C.
C.
E.
4. If , then the value of at x=2 is
A. 1
D.
B. 2
E.
5. Given ,
A) show that the derivative is .
Name _
AP Calculus AB
Unit 3 Day 7 Inverse Functions
Derivatives of Inverse Functions ( 3.8)
If (a,b) is a point on f, then (b, a) is a point on g.
The graph of is obtained by reflecting the graph of f about the line y = x.
In order for a function to have
Name _
Rules Quiz: Differentiation
AP Calculus AB
Rules for Differentiation: For all problems, a is a constant. u and v are a differentiable functions of x
1.
4. =
a=
0
2. =
5. =
3. =
6. =
7. =
8. =
10. =
11. = 12.
13. =
14. =
16. =
17. =
9. =
15. =
18. =