# Unit 5 Analytical Applications of Differentiation

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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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 Topic Name Essential Knowledge 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. 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.

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Then there is this – Existence Theorems

Fermat’s Penultimate Theorem

Rolle’s Theorem

Mean Numbers

Mean Tables

The Mean Value Theorem I

The Mean Value Theorem II

Darboux’s Theorem

What’s a Mean Old Average Anyway? (4-29-2014) Helping students understand the difference between the average rate of change of a function, the average value of a function, and the Mean Value theorem

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 5.2 Extreme Value Theorem, Global verses Local Extrema, and Critical Points LEARNING OBJECTIVE 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).

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Concepts Related to Graphs

The Shapes of a Graph  There are only 5

Joining the Pieces of a Graph

Did He or Didn’t He?  How Fermat found extreme values without derivatives.

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ENDURING UNDERSTANDING

FUN-4 A function’s derivative can be used to understand some behaviors of the function.

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 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. 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.

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Open or Closed? (11-2-2012) Include the endpoints or not? See also Going Up?

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 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. FUN-4.A.2 The first derivative of a function can determine the location of relative (local) extrema of the function.

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Extreme Values (10-22-2012)

Curves with Extrema? (10-19-2015)

Far Out!

A Standard Problem (5-14-2013) A max/min problem a further look at the same problem The Marble and the Vase

Extremes without Calculus (10-13-2014) A student’s question.

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 5.5 Using the Candidates 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. FUN-4.A.3 Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

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Extreme Values (10-22-2012)

Curves with Extrema? (10-19-2015)

Far Out!

A Standard Problem (5-14-2013) A max/min problem a further look at the same problem The Marble and the Vase

Extremes without Calculus (10-13-2014) A student’s question.

Did He or Didn’t He?  How Fermat found extreme values without calculus

Optimization – Reflections

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 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. 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.

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Concepts Related to Graphs

The Shapes of a Graph  There are only 5

Joining the Pieces of a Graph

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 5.7 Using the Second Derivative to Determine Extrema LEARNING OBJECTIVE 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.

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Concepts Related to Graphs

The Shapes of a Graph  There are only 5

Joining the Pieces of a Graph

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 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 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.

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Using the Derivative to Graph the Function

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 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. FUN-4.A.11 Key features of the graphs of f, f’, and f” are related to one another.

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Using the Derivative to Graph the Function

 5.10 Introduction to Optimization Problems LEARNING OBJECTIVE 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.

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 5.11 Solving Optimization Problems LEARNING OBJECTIVE 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.

Blog Posts

Extreme Values (10-22-2012)

Curves with Extrema? (10-19-2015)

Far Out!

A Standard Problem (5-14-2013) A max/min problem a further look at the same problem The Marble and the Vase

Extremes without Calculus (10-13-2014) A student’s question.

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 5.12 Exploring Behaviors of Implicit Functions LEARNING OBJECTIVE FUN-4.D Determine critical points of implicit relations. FUN-4.E Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives. 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. 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 $\frac{{dy}}{{dx}}$  .

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