AP Calculus
2.1: Rates of Change and Limits
A moving body’s average speed during an interval of time is found by
dividing the distance covered by the elapsed time.
Finding an Average Speed
Example 1. An unwanted calculus book is dropped from the top of the Comcast
Center. It falls 16t 2 feet in the first t seconds. What is its average speed during its
first 3 seconds of fall?

Finding an Instantaneous Speed
Example 2. Find the speed of the book in Example 1 at t  3.

1
LIMITS
If a function f has limit L as x approaches c, then we write
lim f x   L
x c
In other words, as x gets infinitely close to c, f(x) gets infinitely close to
L.
Assume the following:

lim x  c
x c
lim k  k (k is a constant.)

x c


Theorem 1. Properties of Limits
gx   M , then
If L, M, c, and k are real numbers, lim f x   L and lim
x c
x c
1. Sum Rule

2. Difference Rule
3. Product Rule
4. Constant Multiple Rule
5. Quotient Rule
2
6. Power Rule
Example 3. Use properties of limits to find the following limits.
2x 3  5x 2  4
(a). lim

x c

4x2  5
(b). lim
x c x 3  4

Theorem 2.
(a). If f x  is a polynomial function and c is a real number, then
lim f x   f c 


x c
(b). If f x  and gx  are polynomial functions and c is a real number,
then

lim
x c

f x  f c 

gx  gc 
3

x 
sin  .
Example 4. Use Theorem 2 to evaluate lim

x 3
2 

sin x
. What does the function
x
appear to do as x approaches 0? Is it defined at x  0?
On your graphing calculator, graph the function y 
In time we will prove this, but for now we can conjecture that


sin x
lim

x 0 x
Example 5. Use the above conjecture to evaluate the following:
(a). lim
x 0
sin 2x
x


(b). lim
x 0
tan x
x

4
x 2 1
Example 6. Evaluate lim
x 1 x  1

Example 7. Non-existent Limits
1
x 0 x
(a). lim

(b). lim
x 0
x
x

One-sided and Two-sided Limits
Right-hand: The limit of f as x approaches c from the right.
5
Left-hand: The limit of f as x approaches c from the left.
Theorem : A function f x  has a limit as x approaches c if and only if the righthand and left-hand limits at c exist and are equal. In symbols:

Example 4. Evaluate the right-hand, left-hand, and two-sided limit of the greatest
integer function (the greatest integer less than or equal to x) y  int x at x  2 , if they
exist.


(a). lim int x
x 2

(b). lim int x
x 2

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(c). lim int x
x 2
2.1: Rates of Change and Limits
2.2: Limits Involving Infinity

Objectives: To be able to evaluate more difficult limits using what we’ve learned in
2.1, and to use end-behavior to evaluate limits as x .
Evaluate the following limits.
x2  9
x 2 x 2  2x  3
1. lim



h 1
2. lim
3
h 0
1
h

sin x
x 0 2x 2  x
3. lim

x  sin x
x 0
x
4. lim

7
5. lim
x 0
3sin 4 x
sin 3x

6. lim int x
(the greatest integer function)
x  2


Finite Limits as
Graph y 
x 
1
. What appears to happen as x gets very large?
x


Now in limit notation:
The line y  0 is a (a). ________________________________ ________________________________ of
the graph of f.

8
The line y  b is a (a). ________________________________ ________________________________of the
graph of a function y  f x  if either:

Example 1: Evaluate the following:

(a). lim
x 
1
x 1

(b). lim
x 
6x  5
2x  4

The Sandwich Theorem
If gx   f x   hx  for all x  c in some interval about c, and:


Then:
9
Example 2. Use the Sandwich Theorem to evaluate lim
x 
sin x
.
x

2x  sin x
.
x 
x
Example 3. Use your finding in Example 2 to evaluate lim

10
2.2: Limits Involving Infinity, continued.
Example 1. Find the vertical asymptote(s) of the function, then express it in limit
notation from both sides.
(a). f x  
1
x 1
2

(b). f x   sec x over 0,  


End-Behavior Models
The function g is
(a). a right end behavior model for f iff:
(b). a left end behavior model for f iff:
11
If f x  
px 
, where p and q are both polynomial functions, the following is true.
qx 
The end behavior model can be found by simplifying: __________________________________

If deg p  degq , lim f x  
x 


If deg p  degq , lim f x  
x 

 degq , lim f x 
If deg p

x 

 2. Evaluate lim f x for the following:
Example

x 
(a). f x  
x
x  x  2

(b). f x  
2x  3
3x  100

(c). f x  
x2  5
x  10

3
Using Substitution
12
1
Example 3. Evaluate lim cos .
x 
x
2.3: Continuity

A continuous function is one whose outputs vary continuously with the
inputs and do not jump from one value to another without taking on the
values in between.
1  x , 0  x  1

, 1 x  2
 1
Example 1. Let f x    2 , x  2 . Find the points at which f is
x 1 , 2  x  3

5  x , 3  x  4
continuous, and the points and which f is discontinuous.

Continuity at a Point
Interior Point:
A function y  f x  is continuous at an interior point c of its domain iff :

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Endpoint:
A function y  f x  is continuous at a left endpoint a or is continuous at a right
end point b of its domain if

If f is not continuous at a point c, then f is discontinuous at c and c is a point of
discontinuity.
Types of Discontinuity
(a).
(b).
(c).
Removing a Discontinuity
Let f (x) 
x 2  3x 10
.
x2  4
(a). Where are the discontinuities (where is f undefined?)

14
(b). Let’s focus on x = 2. How should f be defined at that x-value to remove the
discontinuity?
(c). Write the extended function:
This is a continuous extension of f.
A function is continuous on an interval iff it is continuous at every point of the
interval. A continuous function is one that is continuous at every point of its
domain.
Is f (x) 
1
continuous over 1,1?
x2


Is it a continuous function?
Properties of Continuous Functions
If the functions f and g are continuous at x  c , then the following combinations are
continuous at x  c :
1. f  g
2. f g


3. f  g


15
4. k  f where k is a constant
5.


f
if gc   0
g
6. If f is continuous at c and g is continuous at f c  then g o f is continuous at c.

Example 2. State whether the function is continuous at the given value of x. Justify
your answer.


x 2 1 if x  0
(a) f x   
,x=0
x  2 if x  0

5x  6 if
(b) f x   
 x  2 if
x 1
,x=1
x 1

(c) f x  
x2  9
,x=3
x3

16
x2  9
(d) f x  
, x  3
x3


2.4: Rates of Change and Tangent Lines
The average rate of change of a quantity over a period of time is the
amount of change divided by the amount of time it takes.
Example 1. Find the average rate of change of f (x)  x 3  1 over the interval 1,4 .


We can think of average rate of change as the slope of a (a).____________________________.
Finding a Tangent to a Curve at Point P
1. Start with the slope of a (a). __________________________ drawn through P and a
point Q on the curve.
2. Find the limit of the secant slope as Q  P .
3. This number is the slope of the curve at P. The tangent to the curve at P is the
line through P with this slope.

Example 2. Find the slope of the parabola y  x 2 at x  a.


17
What is the equation of the tangent line to y  x 2 when x  3 ?


Slope of a Curve (a.ka. slope of a tangent line) at a Point
The slope of the curve y  f x  at the point P a, f a, is the number:


provided the limit exists.
Example 3. Let f x  
1
.
x
(a). Find the slope of the curve at x  a.


18
(b). Determine for what value(s) of a will the slope equal 
of the tangent line through a, f a.
1
and write an equation
2


Normal to a Curve
The normal line to a curve at a point is the line perpendicular to the tangent at that
point.
Example 4. Write an equation for the normal line to the curve f x  3 x 2 at
x  2 .


19