calculus mrs. dougherty’s class. drivers start your engines
TRANSCRIPT
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Calculus
Mrs. Dougherty’s Class
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drivers
Start your engines
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3 Big Calculus Topics
Limits Derivatives Integrals
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Chapter 2
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2.1 Limits and continuity
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Limits can be found
Graphically
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Limits can be found
Graphically Numerically
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Limits can be found
Graphically Numerically By direct substitution
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Limits can be found
Graphically Numerically By direct substitution By the informal definition
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Limits can be found
Graphically Numerically By direct substitution By the informal definition By the formal definition
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Limits
Informal Def.
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Limits
Informal Def.
Given real numbers c and L, if the values f(x) of a function approach or equal L
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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),
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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.
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Limits
notation
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LIFE IS GOOD
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Theorem 1
Constant Function
f(x)=k
Identity Function
f(x)=x
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Theorem 2
Limits of polynomial functions can be found by direct substitution.
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Properties of Limits
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Properties of Limits
If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c
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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
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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
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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
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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
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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
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Theorem 3
Many ( not all ) limits of rational functions can be found by direct substitution.
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Right-hand and Left-hand Limits
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Theorem 4
A function, f(x),
has a limit as x approaches c
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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
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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.
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Calculus 2.2
Continuity
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Definition
f(x) is continuous at an interior point of the domain if
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Definition
f(x) is continuous at an interior point of the domain if lim f(x) = f(c )
x->c
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Definition
f(x) is continuous at an endpoint of the domain if
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A “continuous” function is continuous at each point of its domain.
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Definition
Discontinuity
If a function is not continuous at a point c, then c is called a point of discontinuity.
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Types of Discontinuities
Removable
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Types of Discontinuities
Removable Non-removable A) jump
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Types of Discontinuities
Removable Non-removable A) jump B) oscillating
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Types of Discontinuities
Removable Non-removable A) jump B) oscillating C) infinite
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Test for Continuity
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Test for Continuity
y=f(x) is continuous at x=c iff
1.
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Test for Continuity
y=f(x) is continuous at x=c iff
1. f(c) exists
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Test for Continuity
y=f(x) is continuous at x=c iff
1. f(c) exists
2. lim f(x) exists
x-> c
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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
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Theorem 5
Properties of Continuous Functions
If f(x) and g(x) are continuous at c, then
1. f(x)+g(x)
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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)
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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)
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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)
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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
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Theorem 6
If f and g are continuous at c,
Then g f and f g are continuous at c
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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].
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Intermediate Value Theorem for continuous functions
A function that is continuous on [a,b] takes on every value
between f(a) and f(b).
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Calculus 2.3
The Sandwich Theorem
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If g(x) < f(x) < h(x) for all x /=c
and lim g(x) = lim h(x) = L, then
lim f(x) = L.
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Use sandwich theorem to findlim sin xx->0 x
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Sandwich theorem examples
So you can see the light.
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Calculus 2.4
Limits Involving Infinity
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Limits at + infinity
are also called “end behavior” models for the function.
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Definition
y=b is a horizontal asymptote of f(x) if
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Horizontal Tangents
Case 1 degree of numerator < degree of denominator
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Case 2 degree of numerator = degree of denominator
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Case 3 degree of numerator > degree of denominator
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Theorem
Polynomial End Behavior Model
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Calculus 2.6
The Formal Definition of a Limit
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Now this is mathematics!!!