AP®︎ Calculus BC: Unit 1 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 1 of AP Calculus BC – Limits and Continuity.
Questions List
Unit 1 (All Topics)
Question 1 Topic 1.1Easy
Let . What is the average rate of change of from to ?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus BC Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.
Unit 1: Limits and Continuity
Explore how limits help analyze change, continuity, and asymptotic behavior.
Topic 1.1: Introducing Calculus: Can Change Occur at 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.
Essential Knowledge: CHA-1.A.1
Calculus uses limits to understand and model dynamic change.
Essential Knowledge: 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.
Essential Knowledge: CHA-1.A.3
The limit concept allows us to define instantaneous rate of change in terms of average rates of change.
Topic 1.2: Defining Limits and Using Limit Notation
Learning Objective: LIM-1.A
Represent limits analytically using correct notation.
Essential Knowledge: 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: 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.
Essential Knowledge: LIM-1.B.1
A limit can be expressed in multiple ways, including graphically, numerically, and analytically.
Topic 1.3: Estimating Limit Values from Graphs
Learning Objective: LIM-1.C
Estimate limits of functions.
Essential Knowledge: LIM-1.C.1
The concept of a limit includes one sided limits.
Essential Knowledge: LIM-1.C.2
Graphical information about a function can be used to estimate limits.
Essential Knowledge: LIM-1.C.3
Because of issues of scale, graphical representations of functions may miss important function behavior.
Essential Knowledge: 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. Illustrative examples: lim (x→0) 1/x^2 = ∞; lim (x→0) |x|/x does not exist; lim (x→0) sin(1/x) does not exist; lim (x→0) 1/x does not exist.
Topic 1.4: Estimating Limit Values from Tables
Learning Objective: LIM-1.C
Estimate limits of functions.
Essential Knowledge: LIM-1.C.5
Numerical information can be used to estimate limits.
Topic 1.5: Determining Limits Using Algebraic Properties of Limits
Learning Objective: LIM-1.D
Determine the limits of functions using limit theorems.
Essential Knowledge: LIM-1.D.1
One-sided limits can be determined analytically or graphically.
Essential Knowledge: LIM-1.D.2
Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.
Topic 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.
Essential Knowledge: LIM-1.E.1
It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.
Topic 1.7: Selecting Procedures for Determining Limits
Learning Objective: 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.
Topic 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. Illustrative examples: The squeeze theorem can be used to show lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1 - cos x)/x = 0.
Topic 1.9: Connecting Multiple Representations of Limits
Learning Objective: 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.
Topic 1.10: Exploring Types of Discontinuities
Learning Objective: LIM-2.A
Justify conclusions about continuity at a point using the definition.
Essential Knowledge: LIM-2.A.1
Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.
Topic 1.11: Defining Continuity at a Point
Learning Objective: LIM-2.A
Justify conclusions about continuity at a point using the definition.
Essential Knowledge: 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).
Topic 1.12: Confirming Continuity over an Interval
Learning Objective: LIM-2.B
Determine intervals over which a function is continuous.
Essential Knowledge: LIM-2.B.1
A function is continuous on an interval if the function is continuous at each point in the interval.
Essential Knowledge: LIM-2.B.2
Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.
Topic 1.13: Removing Discontinuities
Learning Objective: LIM-2.C
Determine values of x or solve for parameters that make discontinuous functions continuous, if possible.
Essential Knowledge: 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.
Essential Knowledge: 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.
Topic 1.14: Connecting Infinite Limits and Vertical Asymptotes
Learning Objective: LIM-2.D
Interpret the behavior of functions using limits involving infinity.
Essential Knowledge: LIM-2.D.1
The concept of a limit can be extended to include infinite limits.
Essential Knowledge: LIM-2.D.2
Asymptotic and unbounded behavior of functions can be described and explained using limits.
Topic 1.15: Connecting Limits at Infinity and Horizontal Asymptotes
Learning Objective: LIM-2.D
Interpret the behavior of functions using limits involving infinity.
Essential Knowledge: LIM-2.D.3
The concept of a limit can be extended to include limits at infinity.
Essential Knowledge: LIM-2.D.4
Limits at infinity describe end behavior.
Essential Knowledge: LIM-2.D.5
Relative magnitudes of functions and their rates of change can be compared using limits.
Topic 1.16: Working with the Intermediate Value Theorem (IVT)
Learning Objective: FUN-1.A
Explain the behavior of a function on an interval using the Intermediate Value Theorem.
Essential Knowledge: 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.
