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Calculus
Mrs. Dougherty’s Class
drivers
Start your engines
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!!!
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