AP®︎ Calculus AB: Unit 5 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 5 of AP Calculus AB – Analytical Applications of Differentiation.
Questions List
Unit 5 (All Topics)
Question 1 Topic 5.1Easy
Suppose , and is continuous on and differentiable on . The Mean Value Theorem guarantees that there is at least one value , , such that which of the following is true?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus AB 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 5: Analytical Applications of Differentiation
Analyze functions using derivatives and solve optimization and graph-analysis problems.
Topic 5.1: Using the Mean Value Theorem
Learning Objective: FUN-1.B
Justify conclusions about functions by applying the Mean Value Theorem over an interval.
Essential Knowledge: FUN-1.B.1
If a function f is continuous over the interval [a, b] and differentiable over the interval (a, b), then the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.
Topic 5.2: Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Learning Objective: FUN-1.C
Justify conclusions about functions by applying the Extreme Value Theorem.
Essential Knowledge: FUN-1.C.1
If a function f is continuous over the interval (a, b), then the Extreme Value Theorem guarantees that f has at least one minimum value and at least one maximum value on [a, b].
Essential Knowledge: FUN-1.C.2
A point on a function where the first derivative equals zero or fails to exist is a critical point of the function.
Essential Knowledge: FUN-1.C.3
All local (relative) extrema occur at critical points of a function, though not all critical points are local extrema.
Topic 5.3: Determining Intervals on Which a Function Is Increasing or Decreasing
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.1
The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.
Topic 5.4: Using the First Derivative Test to Determine Relative (Local) Extrema
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.2
The first derivative of a function can determine the location of relative (local) extrema of the function.
Topic 5.5: Using the Candidates Test to Determine Absolute (Global) Extrema
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.3
Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.
Topic 5.6: Determining Concavity of Functions over Their Domains
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.4
The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.
Essential Knowledge: FUN-4.A.5
The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.
Essential Knowledge: FUN-4.A.6
The second derivative of a function may be used to locate points of inflection for the graph of the original function.
Topic 5.7: Using the Second Derivative Test to Determine Extrema
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.7
The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.
Essential Knowledge: FUN-4.A.8
When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.
Topic 5.8: Sketching Graphs of Functions and Their Derivatives
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.9
Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.
Essential Knowledge: FUN-4.A.10
Graphical, numerical, and analytical information from f' and f'' can be used to predict and explain the behavior of f.
Topic 5.9: Connecting a Function, Its First Derivative, and Its Second Derivative
Learning Objective: FUN-4.A
Justify conclusions about the behavior of a function based on the behavior of its derivatives.
Essential Knowledge: FUN-4.A.11
Key features of the graphs of f, f', and f" are related to one another.
Topic 5.10: Introduction to Optimization Problems
Learning Objective: FUN-4.B
Calculate minimum and maximum values in applied contexts or analysis of functions.
Essential Knowledge: FUN-4.B.1
The derivative can be used to solve optimization problems; that is, finding a minimum or maximum value of a function on a given interval.
Topic 5.11: Solving Optimization Problems
Learning Objective: FUN-4.C
Interpret minimum and maximum values calculated in applied contexts.
Essential Knowledge: FUN-4.C.1
Minimum and maximum values of a function take on specific meanings in applied contexts.
Topic 5.12: Exploring Behaviors of Implicit Relations
Learning Objective: FUN-4.D
Determine critical points of implicit relations.
Essential Knowledge: FUN-4.D.1
A point on an implicit relation where the first derivative equals zero or does not exist is a critical point of the function.
Learning Objective: FUN-4.E
Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.
Essential Knowledge: FUN-4.E.1
Applications of derivatives can be extended to implicitly defined functions.
Essential Knowledge: FUN-4.E.2
Second derivatives involving implicit differentiation may be relations of x, y, and dy/dx.
