Calculus

Mrs. Dougherty’s Class

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3 Big Calculus Topics

Limits Derivatives Integrals

Chapter 2

2.1 Limits and continuity

Limits can be found

Graphically

Limits can be found

Graphically Numerically

Limits can be found

Graphically Numerically By direct substitution

Limits can be found

Graphically Numerically By direct substitution By the informal definition

Limits can be found

Graphically Numerically By direct substitution By the informal definition By the formal definition

Limits

Informal Def.

Limits

Informal Def.

Given real numbers c and L, if the values f(x) of a function approach or equal L

Limits

Informal Def.

Given real numbers c and L, if the values f(x) of a function approach or equal L as the values of x approach ( but do not equal c),

Limits

Informal Def.

Given real numbers c and L, if the values

f(x) of a function approach or equal L as the values of x approach ( but do not equal c), then f has a limit L as x approaches c.

Limits

notation

LIFE IS GOOD

Theorem 1

Constant Function

f(x)=k

Identity Function

f(x)=x

Theorem 2

Limits of polynomial functions can be found by direct substitution.

Properties of Limits

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Sum Rule: lim [f(x) + g(x)]= lim f(x) +lim g(x)=L1 + L2

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Difference Rule: lim [f(x) - g(x)]= L1 - L2

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Product Rule: lim [f(x) * g(x)]= L1 * L2

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Constant multiple Rule: lim c f(x) = c L1

Properties of Limits

If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

Quotient Rule: lim [f(x) / g(x)]= L1 / L2 , L1=0 NOT

Theorem 3

Many ( not all ) limits of rational functions can be found by direct substitution.

Right-hand and Left-hand Limits

Theorem 4

A function, f(x),

has a limit as x approaches c

Theorem 4

A function, f(x),

has a limit as x approaches c

if and only if

the right-hand and left-hand limits at c exist

Theorem 4

A function, f(x),

has a limit as x approaches c

if and only if

the right-hand and left-hand limits at c exist

and are equal.

Calculus 2.2

Continuity

Definition

f(x) is continuous at an interior point of the domain if

Definition

f(x) is continuous at an interior point of the domain if lim f(x) = f(c )

x->c

Definition

f(x) is continuous at an endpoint of the domain if

A “continuous” function is continuous at each point of its domain.

Definition

Discontinuity

If a function is not continuous at a point c, then c is called a point of discontinuity.

Types of Discontinuities

Removable

Types of Discontinuities

Removable Non-removable A) jump

Types of Discontinuities

Removable Non-removable A) jump B) oscillating

Types of Discontinuities

Removable Non-removable A) jump B) oscillating C) infinite

Test for Continuity

Test for Continuity

y=f(x) is continuous at x=c iff

1.

Test for Continuity

y=f(x) is continuous at x=c iff

1. f(c) exists

Test for Continuity

y=f(x) is continuous at x=c iff

1. f(c) exists

2. lim f(x) exists

x-> c

Test for Continuity

y=f(x) is continuous at x=c iff

1. f(c) exists

2. lim f(x) exists

x -> c

3. f(c ) = lim f(x)

x -> c

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)

2. f(x) – g(x)

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)

2. f(x) – g(x)

3. f (x) g(x)

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)

2. f(x) – g(x)

3. f (x) g(x)

4. k g(x)

Theorem 5

Properties of Continuous Functions

If f(x) and g(x) are continuous at c, then

1. f(x)+g(x)

2. f(x) – g(x)

3. f (x) g(x)

4. k g(x)

5. f(x)/g(x), g(x)/=0

are continuous

Theorem 6

If f and g are continuous at c,

Then g f and f g are continuous at c

Theorem 7If f(x) is continuous on [a ,b],then f(x) has an absolute maximum,M, and an absolute minimum,m, on [a ,b].

Intermediate Value Theorem for continuous functions

A function that is continuous on [a,b] takes on every value

between f(a) and f(b).

Calculus 2.3

The Sandwich Theorem

If g(x) < f(x) < h(x) for all x /=c

and lim g(x) = lim h(x) = L, then

lim f(x) = L.

Use sandwich theorem to findlim sin xx->0 x

Sandwich theorem examples

So you can see the light.

Calculus 2.4

Limits Involving Infinity

Limits at + infinity

are also called “end behavior” models for the function.

Definition

y=b is a horizontal asymptote of f(x) if

Horizontal Tangents

Case 1 degree of numerator < degree of denominator

Case 2 degree of numerator = degree of denominator

Case 3 degree of numerator > degree of denominator

Theorem

Polynomial End Behavior Model

Calculus 2.6

The Formal Definition of a Limit

Now this is mathematics!!!