Unit 1: Limits and Continuity

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ENDURING UNDERSTANDING

LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

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 Topic Name & Learning Objective: Essential Knowledge 1.1 Introducing Calculus: Can change occur in an instant? LEARNING OBJECTIVE CHA-1.A Interpret the rate of change at an instant in terms of average rates of change over intervals containing that instant. CHA-1.A.1 Calculus uses limits to understand and model dynamic change. CHA-1.A.2 Because an average rate of change divides the change in one variable by the change in another, the average rate of change is undefined at a point where the change in the independent variable would be zero. CHA-1.A.3 The limit concept allows us to define instantaneous rate of change in terms of average rates of change.

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 1.2 Defining Limits Using Limit Notation LEARNING OBJECTIVES LIM-1.A Represent limits analytically using correct notation. IM-1.B Interpret limits expressed in analytic notation. LIM-1.A.1 Given a function f, the limit of f (x) as x approaches c is a real number R if f (x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c).  If the limit exists and is a real number, then LIM-1.B.1 A limit can be expressed in multiple ways, including graphically, numerically, and analytically.

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Why Limits?

Deltas and Epsilons  Why this topic is not tested on the AP Calculus Exams.

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 1.3 Estimating Limit Values from Graphs LEARNING OBJECTIVE LIM-1.C Estimate limits of functions. LIM-1.C.1 The concept of a limit includes one sided limits. LIM-1.C.2 Graphical information about a function can be used to estimate limits. LIM-1.C.3 Because of issues of scale, graphical representations of functions may miss important function behavior. LIM-1.C.4 A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.

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Finding Limits How to…

Dominance

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 1.4 Estimating Limit Values from Tables LEARNING OBJECTIVE LIM-1.C Estimate limits of functions. LIM-1.C.5 Numerical information can be used to estimate limits.

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Finding Limits How to…

Dominance

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 1.5 Determining Limits Using Algebraic Properties of Limits LEARNING OBJECTIVE LIM-1.D Determine the limits of functions using limit theorems. LIM-1.D.1 One-sided limits can be determined analytically or graphically. LIM-1.D.2 Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.

Finding Limits  How to…

Dominance

Limit of Composite Functions

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 1.6 Determining Limits Using Algebraic Manipulation LEARNING OBJECTIVE LIM-1.E Determine the limits of functions using equivalent expressions for the function or the squeeze theorem LIM-1.E.1 It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.

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Finding Limits  How to…

Dominance

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 1.7 Selecting Procedures for Determining Limits None

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Finding Limits How to…

Dominance

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 1.8 Determining Limits Using the Squeeze Theorem LEARNING OBJECTIVE LIM-1.E Determine the limits of functions using equivalent expressions for the function or the squeeze theorem. ESSENTIAL KNOWLEDGE LIM-1.E.2 The limit of a function may be found by using the squeeze theorem.

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none

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 1.9 Connecting Multiple Representations of Limits None

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Finding Limits How to…

Dominance

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ENDURING UNDERSTANDING

LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

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 1.10 Exploring Types of Discontinuities LEARNING OBJECTIVE LIM-2.A Justify conclusions about continuity at a point using the definition. LIM-2.A.1 Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

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Continuity Should continuity come before limits?

Continuity

Continuity The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

Fun with Continuity Defined everywhere and continuous nowhere. Continuous only at a single point.

Right Answer – Wrong Question Continuity or continuity “on its domain”?

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 1.11 Defining Continuity at a Point LEARNING OBJECTIVE LIM-2.A Justify conclusions about continuity at a point using the definition. LIM-2.A.2 A function f is continuous at x = c provided that f(c) exists,  exists, and

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Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

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 1.12 Confirming Continuity on an Interval LEARNING OBJECTIVE LIM-2.B Determine intervals over which a function is continuous. LIM-2.B.1 A function is continuous on an interval if the function is continuous at each point in the interval. LIM-2.B.2 Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.

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Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

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 1.13 Removing Discontinuities LEARNING OBJECTIVE LIM-2.C Determine values of x or solve for parameters that make discontinuous functions continuous, if possible. LIM-2.C.1 If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as x approaches that point. LIM-2.C.2 In order for a piecewise-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary.

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Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

Continuity

Continuity  The definition of continuity.

Continuous Fun A fuller discussion of continuity and its definition

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 1.14 Connecting Infinite Limits and Vertical Asymptotes LEARNING OBJECTIVE LIM-2.D Interpret the behavior of functions using limits involving infinity. LIM-2.D.1 The concept of a limit can be extended to include infinite limits. LIM-2.D.2 Asymptotic and unbounded behavior of functions can be described and explained using limits.

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Right Answer – Wrong Question Is a function continuous even if it has a vertical asymptote?

Asymptotes  The graphical manifestation of certain limit

How to Tell Your Asymptote from a Hole in the Graph  From the technology series. Showing holes and asymptotes on a graphing calculator.

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 1.15 Connecting Limits at Infinity and Horizontal Asymptotes LEARNING OBJECTIVE LIM-2.D Interpret the behavior of functions using limits involving infinity. LIM-2.D.3 The concept of a limit can be extended to include limits at infinity. LIM-2.D.4 Limits at infinity describe end behavior. LIM-2.D.5 Relative magnitudes of functions and their rates of change can be compared using limits.

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Asymptotes The graphical manifestation of certain limits

Far Out  A fun limit, but best saved until after students can find maximum values and points of inflection.

Dominance

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ENDURING UNDERSTANDING

FUN-1 Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.

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 1.16 Working with the Intermediate Value Theorem LEARNING OBJECTIVE FUN-1.A Explain the behavior of a function on an interval using the Intermediate Value Theorem. FUN-1.A .1 If f is a continuous function on the closed interval [a, b] and d is a number between f (a) and f (b), then the Intermediate Value Theorem guarantees that there is at least one number c between a and b, such that f(c) = d.

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Then there is this – Existence Theorems

Theorems The Intermediate Value Theorem (IVT) and suggestions on teaching theorems

Intermediate Weather  Using the I

Then There Is This on Existence Theorems

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 Exclusion Statement EXCLUSION STATEMENT The epsilon-delta definition of a limit is not assessed on the AP Calculus AB or BC Exam. However, teachers may include this topic in the course if time permits.

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Deltas and Epsilons  Why this topic is not tested on the AP Calculus Exams.

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