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.

Blog Posts

Then there is this – Existence Theorems

Foreshadowing the Mean Value Theorem

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).

Blog Posts

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.

Blog Posts

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.

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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5.5 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.

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.

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.

Blog Posts

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.

Blog Posts

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.

Blog Posts

Reading the Derivative’s Graph  My all-time most read post.

Using the Derivative to Graph the Function

Real “Real Life” Graph Reading

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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.

Blog Posts

Reading the Derivative’s Graph  My all-time most read post.

Using the Derivative to Graph the Function

Real “Real Life” Graph Reading

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.

Blog Posts

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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}}  .

Blog Posts

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