Calculus

Section 1.1 A Preview of Calculus

What is Calculus?

Calculus is the mathematics of change

Two classic types of problems:The Tangent Line ProblemArea Under A Curve Problem

The Tangent Line Problem

Secant line - a line that intersects the graph at two points (easy to find the slope given two points)

Tangent line - a line that intersects the graph at one point

Using the understanding of limits, one can limit the distance between two points to achieve a tangent line, thus finding the slope of a curve

These are Derivatives See fig. 1.1 & fig. 1.2 on page 45 of textbook

The Area Problem

With the understanding of limits, one can limit the width of rectangles to increase the number of rectangles to more accurately find the area between a curve and the x-axis

These are Integrals See fig. 1.3 & fig. 1.4 on page 46 of textbook

Calculus

Section 1.2 Finding Limits Graphically and Numerically

An Introduction to Limits

Notation:

Read as “the limit as x approaches c of f(x) is L” If f(x) becomes arbitrarily close to a single

number, L, as x approaches c from either side, then the limit is L.

limx c

f (x) L

Evaluating Limits Numerically

Example:

What is the domain of the function? Domain: (-∞, 2) (2, ∞) So, what happens at 2? Set up a table

f (x) x 2 4

x 2

x 1.5 1.9 1.999 2 2.001 2.01 2.5

f(x)

Graphing Calculator: Table

Type function into y= Go to 2nd TBLSET, Ind - Ask Go to TABLE Type in values for the independent variable

Evaluating Limits Numerically

x 1.5 1.9 1.999 2 2.001 2.01 2.5

f(x) 3.5 3.9 3.999 ? 4.001 4.01 4.5

So, as x approaches 2 from either side, the function value approaches 4 (CONVERGE)

We say “the limit as x approaches 2 is 4”

limx 2

x 2 4

x 24

Limits That Fail to Exist

Common types of behavior associated with nonexistence of a limit f(x) approaches a different value from the left and from

the right (they do not CONVERGE ) f(x) increases or decreases without bound as x

approaches c (asymptotes) f(x) oscillates between two fixed values as x

approaches c

Limits and Their Properties

Section 1.3 Evaluating Limits Analytically

Properties of Limits

Theorem 1.1 Some Basic Limits

limx c

b b

limx c

x c

limx c

x n c n

Properties of Limits

Theorem 1.2 Properties of Limits

limx c

bf (x) bL

limx c

f (x) g(x) L K

limx c

f (x)g(x) LK

limx c

f (x)

g(x)

L

K provided K 0

limx c

f (x) n Ln

Given :

limx c

f (x) L

limx c

g(x) K

Properties of Limits

Theorem 1.3 If p(x) is a polynomial function and c is a real number,

then

In r(x) is a rational function, given by r(x)=p(x)/q(x) and c is a real number such that q(c)≠0, then

limx c

p(x) p(c)

limx c

r(x) r(c)

In Other Words…

DIRECT SUBSTITUTE if the function is algebraic (polynomial, rational, radical, etc.) and the function is continuous at c (meaning the function is defined to the left AND to the right of the value c)

Cannot DIRECT SUBSTITUTE when…

the function is not defined at c, or c is not part of the domain.

Therefore, ask: “Is c part of my domain?” AND “Is the function defined to the left AND

to the right of c?”

Strategies for Finding Limits

If f(x) is not defined at c, then find a new function, g(x), that agrees with the original function for all values except at c

Two techniques:Dividing Out TechniqueRationalization Technique

Dividing Out Technique

Factor both numerator and denominatorCancel like factorsAsk, “Is c now part of my domain?”This is your new function, g(x)DIRECT SUBSTITUTE to find the limit

Example 1

Find:

limx 2

x 2 x 2

x 2

(x 2)(x 1)

(x 2)• Factor:

•Cancel:

(x 2)(x 1)

(x 2)(x 1)

Example 1 cont.

So, the “new” function is

g(x) (x 1)

• Now, take the limit of the “new” function

limx 2

(x 1)

limx 2

(x 1) 2 13

Example 1 cont.

Therefore, we conclude that:

limx 2

x 2 x 2

x 23

Rationalization Technique

For functions with radicalsMultiply both numerator and

denominator by the conjugateCancel like factorsKeep denominator factored

(DO NOT MULTIPLY OUT)

Example 2

Find:

limx 9

3 x

9 x

• Rationalize:

limx 9

3 x

9 x

(3 x )

(3 x )

limx 9

(9 x)

(9 x)(3 x )

limx 9

(9 x)

(9 x)(3 x )

1

3 x

Example 2 cont.

Now, take the limit of

g(x) 1

3 x

limx 9

1

3 x

1

3 9

1

6

Example 2 cont.

Therefore, we conclude that:

limx 9

3 x

9 x

1

6