What is Calculus, anyway????
Calculus is …..
 the mathematics of change – velocities and accelerations.
 the mathematics of tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures,
and a variety of other concepts that have enabled scientists, engineers, and economists to
model real-life situations.
What is the difference between Pre-Calculus and Calculus?
 Pre-Calculus is more _________________________.
 Calculus is more ____________________________.
 Examples:
o An object traveling at a ______________ velocity can be analyzed with Pre-Calculus.
To analyze the velocity of an _________________ object, you need Calculus.
o The slope of a ______________ can be analyzed with Pre-Calculus. To analyze the
slope of a __________________, you need Calculus.
o A tangent line to a _______________ can be analyzed with Pre-Calculus. To analyze
a tangent line to a _____________________, you need Calculus.
o The area of a ____________________ can be analyzed with Pre-Calculus. To
analyze the area under a general ______________, you need Calculus
Calculus can also be described as….
 A limit machine that involves three stages.
o The first stage is ____________________________.
o The second stage is the ______________________________.
o The third stage is a _______________________________________________, such
as a derivative or integral.
Limits



The notion of a _________________ is fundamental to the study of Calculus.
The _________________________________________ is one of the classic problems in
Calculus that requires the use of limits.
In this problem, you are given a function f and a point P on its graph, and are asked to find an
equation of the tangent to the graph at point P, as shown in the figure below.


Except for cases involving a vertical tangent line, the problem of finding the tangent line at a
point P is equivalent to finding the __________________ of the tangent line at P.
You can approximate this slope by using a line through the point of tangency and a second
point on the curve, as shown below. Such a line is called a _________________________.

As point Q approaches point P, the slope of the secant line will approach the slope of the
tangent line.

When such a limiting position exists, the slope of the tangent line is said to be the limit of the
slope of the secant line.
Don’t worry!!! That example makes limits seem complicated, but they are actually pretty simple.
An Introduction to Limits


Suppose you are asked to sketch the graph of the function f given by ___________________.
For all values other than x  1 , you can use standard graphing techniques. However, at
x  1 , it is not clear what to expect.

To get an idea of the behavior of the graph of f near x  1 , you can use two sets of x-values –
one set that approaches 1 from the left and one set that approaches 1 from the right.
x approaches 1 from the left
x
f(x)
0.75
0.9
0.99
x approaches 1 from the right
0.999
1
?
1.001
1.01
1.1
1.25
f(x) approaches __________
f(x) approaches ________

As x moves arbitrarily close to 1, f(x) moves arbitrarily close to _______________________.

Using limit notation you can write lim f ( x)  _____________ .
x 1
This is read as “the limit of
f(x) as x approaches 1 is 3.”
(The discussion above gives an example of how you can estimate a limit numerically by constructing
a table and graphically by drawing a graph.)

Examples:
o Use a table and graph to find lim
x 2
x
f(x)
x 2  3x  2
.
x2
2
?
o Use a table and graph to find lim
x 0
x
f(x)
x
.
x 1 1
0
?
1, x  2
o Use a table and graph to find lim f ( x ) if f ( x)  
.
x2
0, x  2
x
f(x)
2
?
Does every function have a limit????
Limits that fail to exist:
 Here are three kinds of behavior that cause limits to not exist:
o Behavior that differs from the right and the left
x
 Example: lim
x 0 x
x
f(x)
0
?
Therefore, the limit does not exist.
o Unbounded behavior

Example: lim
x 0
x
f(x)
1
x2
0
?
Therefore, the limit does not exist.
o Oscillating behavior

1
Example: lim sin   (Graph the function on your calculator.)
x 0
x
Therefore, the limit does not exist.