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# Tuesday – AB - PowerPoint PPT Presentation

Tuesday – AB. Morning (Part 1) Developing Understanding of the Derivative Upload TI 84 Programs Break Morning (Part 2) Ideas That Can Be Explored Before Working with Formulas Connecting Graphs of f, f’, and f” Connecting Differentiability with Continuity Local Linearity. Lunch

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Tuesday – AB

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### Tuesday – AB

• Morning (Part 1)

• Developing Understanding of the Derivative

• Break

• Morning (Part 2)

• Ideas That Can Be Explored Before Working with Formulas

• Connecting Graphs of f, f’, and f”

• Connecting Differentiability with Continuity

• Local Linearity

• Lunch

• Afternoon (Part 1)

• Share an Activity

• Calculus Games

• Discussion of Homework Problems

• Break

• Afternoon (Part 2)

• Slope Fields

• Reasoning with Tabular Data

### Tuesday Assignment - AB

• Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Free Response:

• 2014: AB2, AB3/BC3

• 2013: AB3

### Tuesday-AB/BC

• Morning (Part 1)

• Developing Understanding of the Derivative

• Break

• Morning (Part 2)

• AB:

• Ideas That Can Be Explored Before Working with Formulas

• Connecting Graphs of f, f’, and f”

• Connecting Differentiability with Continuity

• Local Linearity

• Derivative “Lesson”

• BC:

• Connecting Graphs of f, f’, and f”

• Series

• Lunch

• Afternoon (Part 1)

• Share an Activity

• Discussion of Homework Problems

• Break

• Afternoon (Part 2)

• AB:

• Slope Fields

• Reasoning with Tabular Data

• BC:

• Series

### Tuesday Assignment – AB/BC

• Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Free Response for AB Track

• 2014: AB2, AB3/BC3

• 2013: AB3

• Free Response for BC Track

• 2014: AB3/BC3, BC2

• 2013: BC3

### Tuesday Files

• The Derivative

• Understanding the Derivative Graphically

• Understanding the Derivative Numerically

• A Function and Its Derivative

• Connecting Graphs of f and f ’ and Descriptions

• Connecting Continuity and Differentiability

• Local Linearity

• How f ’(a) fails to Exist

• Derivative Lesson

• Homework Discussion

• Slope Fields

• Reasoning with Tabular Data

Topics listed in the course description relating to the introduction of the derivative and the definition of the derivative are:

• Presenting the derivative numerically, graphically, and analytically

• The derivative as an instantaneous rate of change, the limit of the average rate of change (day 1)

• The definition of derivative as the limit of a difference quotient

• The slope of a curve at a point, vertical tangents, points where there are no tangents

### Numerical Approach

• Students should explore understanding the forward difference quotient, the backwards difference quotient, and the symmetric difference quotient

Forward Difference Quotient

Backward Difference Quotient

Symmetric Difference Quotient

All lead to the derivative of a function at a point x=a.

Activities with the graphing calculator can numerically and graphically develop understanding for the algebraic approach.

### Understanding The Derivative Graphically Using Difference Quotients

or

• The values of a derivative are not random. They are values of a function defined by

• As we saw in the first two activities this limit defines a function of x, not a number.

### Example

Do you see how this relates to the activity we did on the difference quotients?

### Building upon these activities it is now appropriate to explore the analytical approach to the definition of a derivative

• You may want to go on and learn other properties of the derivative and uses of the derivative before you actually derive the formulas for the derivatives.

• Once you get into the formulas that’s where the emphasis will be and not on the concept.

### Making Observations about the Function and Its Derivative

When y1 is increasing, what do you notice about the values of y2?

When y1 is decreasing, what do you notice about the values of y2?

When y1 reaches a maximum, what do you notice about the value of y2?

Making Observations about the Function and Its Derivative

When y2 is equal to zero, what do you notice about the behavior of y1?

Would you describe y1 as concave down or concave up? How would you describe the slope of y2?

Using any of the difference quotients (with small h values)obtain graphical (and sometimes numerical)information that can be generalized.

### Ideas That Can Be Explored Without the Knowing Derivative Formulas

The graph of f ’

using the difference

quotient with f

The graph of f “

using the difference

quotient with f ‘

The graph of f

Notice that

• when the derivative of f is positive the original function f is increasing and tangent lines to f have positive slopes

• when the derivative of f is negative the original function f is decreasing and tangent lines to f have negative slopes

• when the derivative of f is zero after being positive and then negative the original function f has reached a minimum;

• when the derivative of f is zero after being negative and then positive the original function f has reached a maximum. ;

• The slope of a tangent line to f at a maximum or minimum is zero.

• If the derivative of f is positive and decreasing the slope of the original function f must be decreasing (or f is concave down)

• If the derivative of f is negative and increasing the slope of the original function f must be increasing or (or f is concave up)

• A minimum of f occurs when the derivative of f goes from negative to positive; A maximum of f occurs when the derivative of f goes from positive to negative

• When the sign changes on the second derivative of f the concavity of f is changing and a point of inflection of f has been located

• Differentiability of f implies Continuity of f but continuity of f does not imply differentiability of f.

Extreme Value Theorem

• A function f, continuous on a closed interval, must have both an absolute minimum and maximum value

• The location for an extrema is found where the function changes from increasing to decreasing or visa versa

• We also need to check the value at either endpoint

• A derivative of a derivative is the second derivative

• The 2nd derivative provides the same information about the first derivative that the first derivative provides about the function

• When the second derivative of a function is positive-the first derivative of the function is increasing –the slope is getting steeper

### Concavity

• A function f is concave up if

• f’ is increasing

• f” is positive or

• A tangent line to f lies below the graph(except at the point of tangency)

### Concavity

• A function f is concave down if

• f’ is decreasing

• f” is negative or

• A tangent line of f lies above the graph(except at the point of tangency)

• Concavity is defined on an interval not at a point

### A Point of Inflection

• A point where the second derivative of a function (f”) changes sign (therefore changing the concavity of function f) is called a point of inflection

• First find where the second derivative (f”) is zero or undefined. Check on both sides of that point to see if the second derivative (f”) changes sign

• Points of inflection correspond to the extreme values of the first derivative (f’) equal zero.

### Remember

• Functions are not differentiable at the endpoints of a closed interval.

• The limit only exists from one side

### Connecting Graphs of f, f’ and descriptions

Match graphs of f, f ‘ and descriptions of f and f’

### Connecting Continuity and Differentiability

• Because a limit is used to define the derivative

• If the derivative exists at a point, the function is continuous at that point

• Differentiability implies continuity

• If a function is differentiable at a point, it is continuous there

• If a function is differentiable on an interval, it is continuous on the interval

### Local Linearity

• Local linearity is a property of differentiable functions that says – roughly – that if you zoom in on a point on the graph of the function (with equal scaling horizontally and vertically), the graph will eventually look like a straight line with a slope equal to the derivative of the function at that point.

• Local linearity is the graphical approach to the derivative

• Functions that are differentiable are locally linear, and, conversely, functions that are locally linear are differentiable.

• Unfortunately, there is not sure way of determining whether a function is locally linear until you know if it’s differentiable.

• Locally linear is a good, informal, way to introduce the concept of the derivative and to let your students see what differentiable means.

• Local linearity and the secant line approximations can be explored in precalculus without reference to differentiability.

• Local linearity can be introduced through zooming out and zooming in

• Differentiable functions are smoothFunctions that are not differentiable have sharp bends or discontinuities in them

### Introduction to Local Linearity

Write a rule for each of the three lines. Give justification for why you

wrote each equation.

J.T. Sutcliff

• Which line is

• Enter each of these functions in your graphing calculator in a zoom 4 Decimal window. Record your sketch below

• Zoom in on the origin by resetting the window to [-0.004, 0.004, 0.001, -0.003, 0.003, 0.001].

• What has happened to each of the graphs when you look at a very small window around the origin?

• We say that a function is locally linear when we can make a curved line appear linear. Each straight line equation that your wrote is called a linear approximation for these graphs at the point x = 0.

• Graph the equation in on a zoom 4 decimal window.

Zoom in to a small window and write the equation of the line that can be used as the linear approximation for this function atx = 0.

### Sample Differentiation Lessons

• The Rules for Differentiation

• Thinking about the Derivative of a Function

## Discussion of Monday Homework

### Monday - AB

• Multiple Choice Questions on the 2013 test: 1, 2, 4, 5, 7, 9, 12, 13, 14, 15, 16, 17, 18, 19, 22, 24

• Free Response:

• 2014: AB1/BC1, AB6

• 2013: AB1

### Monday – AB/BC

• Multiple Choice Questions on the 2013 test: 1, 2, 4, 5, 7, 9, 12, 13,14, 15, 16, 17, 18, 19, 22, 24

• Free Response for AB Track

• 2014: AB1/BC1, AB6

• 2013: AB1

• Free Response for BC Track

• 2014: AB1/BC1, BC6

• 2013: BC1

### Slope Fields

• There are two different types of problems in an AP Calculus course. In one type, you are given a function and then asked about its rate of change; in the other type, you are given how the function changes and then asked to identify the function. Thus derivative and antiderivative permeate the course.

• The term differential equation may seem formidable at first, but since a differential equation is nothing more than an equation that involves a derivative, differential equations occur throughout the course. A solution to a differential equation is simply a function that satisfies the equation.

### Introducing the a Slope Field

• Most people think that if they are handed a differential equation the task will be to solve it.

• But what is a differential equation really describing?

• Students can be asked to describe the behavior or tangent lines based on the differential equation.

• Introducing Slope Fields (Smartboard)

Create a slope field from a differential equation

• Creating Basic Slope Fields (Smartboard)

Practice Reading Information from a Slope Field

• Using Technology to Create Slope Fields

Viewing Slope Fields on a TI-84

### Nancy Stephenson’s Materialson AP Central

• Slope Field Handout

• Slope Field Card Match

### What to include in your study

Build activities so that student

• become familiar with the terminology of differential equations

• recognize what is meant by a solution to a differential equation

• use differential equations in modeling applications

• understand the relationship between a slope field and a solution curve for the differential equation

### What might students be asked to do

• verify whether or not a given function is a solution to a differential equation

• manually construct a portion of a slope field for a given differential equation

• choose from among many differential equations which one is associated with a given slope field

• Choose from among many slope fields which one is associated with a given differential equation

• Recognize exponential growth and decay, the governing differential equation and its solution

• Solve a given separable differential equation

### Reasoning with Tabular Data

2008 Curriculum Module

### Instantaneous Rate of Change

Pages 1 and 2

Approximate y’(12) and explain the meaning of y’(12) in terms of the population of the town.

### Average Value of a Function

Pages 1 and 2

Approximate, with a trapezoidal rule, the average population of the town over the 20 years.

### Approximate an Integral

Pages 3 and 4

Use a midpoint Riemann sum with three subintervals to approximate

Explain the meaning of this definite integral in terms of the water flow, using correct units.

### Evaluate an Average Rate

Pages 4 and 5

Use P(t) to find the average rate of water flow during the 12-hour period. Indicate units of measure.

### Approximate a Total Distance Traveled

Pages 5 and 6

Approximate the distance traveled over

Using a right Riemann sum with four intervals.

Use P(t) to find the average rate of water flow during the 12-hour time period. Indicate units of measure.

Other AP Free Response questions that reference tabular data

### Tuesday Assignment - AB

• Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Free Response:

• 2014: AB2, AB3/BC3

• 2013: AB3

### Tuesday Assignment – AB/BC

• Multiple Choice Questions on the 2013 test: 3, 6, 8, 10, 11, 13, 17, 20, 21, 23, 28, 76, 78, 82, 84

• Free Response for AB Track

• 2014: AB2, AB3/BC3

• 2013: AB3

• Free Response for BC Track

• 2014: AB3/BC3, BC2

• 2013: BC3