Unit 5: Analytical Applications of Differentiation
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5.1 Using the Mean Value Theorem
FUN-1.B—Justify conclusions about functions by applying the Mean Value Theorem over an interval
• FUN-1.B Justify conclusions about functions by applying the Mean Value Theorem over an interval. • 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. • Enduring understanding FUN-1: Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
FUN-1.C—Justify conclusions about functions by applying the Extreme Value Theorem
• FUN-1.C Justify conclusions about functions by applying the Extreme Value Theorem. • 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]. • 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. • FUN-1.C.3 All local (relative) extrema occur at critical points of a function, though not all critical points are local extrema. • Enduring understanding FUN-1: Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior.
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • 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. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.4
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • FUN-4.A.2 The first derivative of a function can determine the location of relative (local) extrema of the function. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.5
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • FUN-4.A.3 Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.6 Determining Concavity of Functions over Their Domains
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.6
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • 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. • 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. • 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. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.7 Using the Second Derivative Test to Determine Extrema
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.7
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • 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. • 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. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.8 Sketching Graphs of Functions and Their Derivatives
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.8
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • FUN-4.A.9 Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations. • FUN-4.A.10 Graphical, numerical, and analytical information from f′ and f″ can be used to predict and explain the behavior of f. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.9
• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives. • FUN-4.A.11 Key features of the graphs of f, f′, and f″ are related to one another. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.10 Introduction to Optimization Problems
FUN-4.B—Calculate minimum and maximum values in applied contexts or analysis of functions
• FUN-4.B Calculate minimum and maximum values in applied contexts or analysis of functions. • 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. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.11 Solving Optimization Problems
FUN-4.C—Interpret minimum and maximum values calculated in applied contexts
• FUN-4.C Interpret minimum and maximum values calculated in applied contexts. • FUN-4.C.1 Minimum and maximum values of a function take on specific meanings in applied contexts. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
5.12 Exploring Behaviors of Implicit Relations
FUN-4.D—Determine critical points of implicit relations
• FUN-4.D Determine critical points of implicit relations. • 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. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.
FUN-4.E—Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives
• FUN-4.E Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives. • FUN-4.E.1 Applications of derivatives can be extended to implicitly defined functions. • FUN-4.E.2 Second derivatives involving implicit differentiation may be relations of x, y, and dy/dx. • Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.