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Unit 1: Limits and Continuity

Syllabus
2020
Section
—
Level
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Exam analysis

1.1 Introducing Calculus: Can Change Occur at an Instant?

Syllabus
2020
Topic
1.1
Level
—

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 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.
  • Enduring understanding CHA-1: Calculus allows us to generalize knowledge about motion to diverse problems involving change.

1.2 Defining Limits and Using Limit Notation

Syllabus
2020
Topic
1.2
Level
—

LIM-1.A—Represent limits analytically using correct notation

  • LIM-1.A Represent limits analytically using correct 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 the common notation is lim x→c f(x) = R.
    • 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.
  • Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

LIM-1.B—Interpret limits expressed in analytic notation

  • LIM-1.B Interpret limits expressed in analytic notation.
  • LIM-1.B.1 A limit can be expressed in multiple ways, including graphically, numerically, and analytically.
  • Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

1.3 Estimating Limit Values from Graphs

Syllabus
2020
Topic
1.3
Level
—

LIM-1.C—Estimate limits of functions

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

1.4 Estimating Limit Values from Tables

Syllabus
2020
Topic
1.4
Level
—

LIM-1.C—Estimate limits of functions—Topic 1.4

  • LIM-1.C Estimate limits of functions.
  • LIM-1.C.5 Numerical information can be used to estimate limits.
  • Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

1.5 Determining Limits Using Algebraic Properties of Limits

Syllabus
2020
Topic
1.5
Level
—

LIM-1.D—Determine the limits of functions using limit theorems

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

1.6 Determining Limits Using Algebraic Manipulation

Syllabus
2020
Topic
1.6
Level
—

LIM-1.E—Determine the limits of functions using equivalent expressions for the function or the squeeze theorem

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

1.7 Selecting Procedures for Determining Limits

Syllabus
2020
Topic
1.7
Level
—

1.7—Selecting Procedures for Determining Limits—Topic 1.7

  • 1.7 This topic is intended to focus on the skill of selecting an appropriate procedure for determining limits. Students should be given opportunities to practice when and how to apply all learning objectives relating to determining limits.

1.8 Determining Limits Using the Squeeze Theorem

Syllabus
2020
Topic
1.8
Level
—

LIM-1.E—Determine the limits of functions using equivalent expressions for the function or the squeeze theorem—Topic 1.8

  • LIM-1.E Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.
  • LIM-1.E.2 The limit of a function may be found by using the squeeze theorem.
  • Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

1.9 Connecting Multiple Representations of Limits

Syllabus
2020
Topic
1.9
Level
—

1.9—Connecting Multiple Representations of Limits—Topic 1.9

  • 1.9 This topic is intended to focus on connecting representations. Students should be given opportunities to practice when and how to apply all learning objectives relating to limits and translating mathematical information from a single representation or across multiple representations.

1.10 Exploring Types of Discontinuities

Syllabus
2020
Topic
1.10
Level
—

LIM-2.A—Justify conclusions about continuity at a point using the definition

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

1.11 Defining Continuity at a Point

Syllabus
2020
Topic
1.11
Level
—

LIM-2.A—Justify conclusions about continuity at a point using the definition—Topic 1.11

  • 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, lim x→c f(x) exists, and lim x→c f(x) = f(c).
  • Enduring understanding LIM-2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

1.12 Confirming Continuity over an Interval

Syllabus
2020
Topic
1.12
Level
—

LIM-2.B—Determine intervals over which a function is continuous

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

1.13 Removing Discontinuities

Syllabus
2020
Topic
1.13
Level
—

LIM-2.C—Determine values of x or solve for parameters that make discontinuous functions continuous, if possible

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

1.14 Connecting Infinite Limits and Vertical Asymptotes

Syllabus
2020
Topic
1.14
Level
—

LIM-2.D—Interpret the behavior of functions using limits involving infinity

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

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

Syllabus
2020
Topic
1.15
Level
—

LIM-2.D—Interpret the behavior of functions using limits involving infinity—Topic 1.15

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

1.16 Working with the Intermediate Value Theorem (IVT)

Syllabus
2020
Topic
1.16
Level
—

FUN-1.A—Explain the behavior of a function on an interval using the Intermediate Value Theorem

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