03-2 calculus powerpoint
DESCRIPTION
heavy learning calculusTRANSCRIPT
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1.2:Rates of Change & LimitsLearning Goals:2009 Mark Pickering Calculate average & instantaneous speedDefine, calculate & apply properties of limitsUse Sandwich Theorem
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Important IdeasLimits are what make calculus different from algebra and trigonometryLimits are fundamental to the study of calculusLimits are related to rate of changeRate of change is important in engineering & technology
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Theorem 1Limits have the following properties:if&then:1.
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Theorem 1Limits have the following properties:if&then:2.
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Theorem 1Limits have the following properties:if&then:3.
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Theorem 1Limits have the following properties:ifthen:4.& k a constant
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Theorem 1Limits have the following properties:if&then:5.
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Theorem 1Limits have the following properties:if&6.r & s areintegers, then:
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Theorem 1Limits have the following properties:ifwhere k is a7.constant, then:(not in your text as Th. 1)
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Theorem 2For polynomial and rational functions:a.b.Limits may be found by substitution
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ExampleSolve using limit properties and substitution:
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Try ThisSolve using limit properties and substitution:6
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ExampleSometimes limits do not exist. Consider:If substitution gives a constant divided by 0, the limit does not exist (DNE)
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ExampleTrig functions may have limits.
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Try This
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ExampleFind the limit if it exists:Try substitution
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ExampleFind the limit if it exists:Substitution doesnt workdoes this mean the limit doesnt exist?
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Important Ideaandare the same except at x=-1
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Important IdeaThe functions have the same limit as x-1
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ProcedureTry substitution Factor and cancel if substitution doesnt workTry substitution againThe factor & cancellation technique
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Try ThisFind the limit if it exists:5Isnt that easy?Did you think calculus was going to be difficult?
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Try ThisFind the limit if it exists:
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Try ThisFind the limit if it exists:The limit doesnt existConfirm by graphing
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DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.
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Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.
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Try ThisFind the limit if it exists:-5
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Try ThisGraph and on the same axes. What is the difference between these graphs?
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Why is there a hole in the graph at x=1?Analysis
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ExampleConsiderforandfor x=1
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Try ThisFind: if
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Important IdeaThe existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.
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Important IdeaWhat matters iswhat value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.
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Try ThisFind:f(0)is undefined; 2 is the limit2
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Try ThisFind:
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Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:
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Try ThisGraph and find the limit (if it exists):DNE
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Theorem 3: One-sided & Two Sided limitsif(limit from right)and(limit from left)then (overall limit)
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Theorem 3: One-sided & Two Sided limits (Converse)if(limit from right)and(limit from left)then (DNE)
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ExampleConsider What happens at x=1?Let x get close to 1 from the left:
x.75.9.99.999 f(x)
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Try ThisConsider Let x get close to 1 from the right:
x1.251.11.011.001 f(x)
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Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?
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Try ThisFind the limit if it exists:DNE
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ExampleFind the limit if it exists:
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Example1.Graph using a friendly window:2. Zoom at x=03. Wassup at x=0?
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Important IdeaIf f(x) bounces from one value to another (oscillates) as x approaches c, the limit of f(x) does not exist at c:
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Theorem 4: Sandwich (Squeeze) TheoremLet f(x) be between g(x) & h(x) in an interval containing c. Ifthen:f(x) is squeezed to L
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ExampleFind the limit if it exists:Where is in radians and in the interval
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ExampleFind the limit if it exists:Substitution gives the indeterminate form
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ExampleFind the limit if it exists:Factor and cancel doesnt work
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ExampleFind the limit if it exists:Maybethe squeeze theorem
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Exampleg()=1h()=cos
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Example&therefore
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Two Special Trig LimitsMemorize
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ExampleFind the limit if it exists:
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ExampleFind the limit if it exists:
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Try ThisFind the limit if it exists:0
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Lesson CloseName 3 ways a limit may fail to exist.
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Practice1. Sec 1.2 #1, 3, 8, 9-18, 28-38E (just find limit L), 39-42gc (graphing calculator), 43-45