Discussion of Functions and Function Notation (Chapter 1) Show/Hide Discussion of Functions and Function Notation

Introduction to Limits (Book Section 1.1)

A printed Definition of Limit was distributed.

Students worked on Group Work GW01 ( Limits for a Function Given by a Graph ).

Students worked on Group Work GW02 ( Guessing Limits for a Function Given by a Formula by Plugging in Numbers ).

Introduction to Limits (Book Section 1.1)

Finding Limits Analytically (Book Section 1.3)

A printed Class Activity Class Activity

We discussed a Student's Solution to Homework Exercise 1.1#11 that they sent to Mark B when asking for help. (Mark B. would like to profoundly thank the student for letting us discuss their solution during class. Their solution provided the inspiration for the entire meeting outline.)

Continuing Discussion of Finding Limits Analytically (Book Section 1.3)

We discussed a Student's Solution to Homework Exercises from Section 1.1 that they sent to Mark B when asking for help. (Mark B. would like to thank the student for letting us discuss their solution during class..)

A printed Worksheet about the Squeeze Theorem was distributed.

Finding Limits Analytically (Book Sections 1.3)

Students worked Group Work GW04 ( Finding Limits ).

Quiz Q01 During Tue Aug 30 Meeting Covering Section 1.1

• 20 Minutes at the end of class
• No Calculators
• Similar to GW01 and Suggested Exercises 1.1#7-13 about computing \(y\) values and estimating limits for function given by a formula .

One Sided Limits (Book Section 1.3)

Class Presentations for Fri Sep 2 Show/Hide Presentations

Mark will briefly discuss Continuity (Book Section 1.4)

Handed out Reference R03 The Intermediate Value Theorem

Continuity (Book Section 1.5)

Class Presentations Involving Continuity for Tue Sep 6 Show/Hide Presentations

Limits Involving Infinity (Book Section 1.6)

Discussed Reference R04 (Limits Involving Infinity).

Students worked Group Work GW05 (Limits Involving Infinity for a function given by a graph).

Limits Involving Infinity (Book Section 1.6)

Quiz Q02 on Wed Sep 7 Covering Sections 1.3, 1.4

Limits Involving Infinity (Book Section 1.6)

Introduction

Today, we're discussing infinite limits . That, is, limits such as

The book gives only a precise definition of infinite limits , a definition involving N and delta . This is the same style of definition as the book's initial definition of regular limits in Section 1.2, a definition involving epsilon and delta . There is nothing wrong with the book's precise definitions, but what is lacking in the book's presentation of limits is an inf ormal definition of limits, both regular limits and infinite limits .

For our course, there are two Reference Pages about limits, posted on the Course Web Page , that show the book's precise definitions of regular limits and of infinite limits , along with informal definitions of those limits. Here are links to those two Reference Pages .

• R01: Definition of Limit was distributed on Tue Aug 23.
• R04: Limits Involving Infinity was discussed Fri Sep 9.

The precise definitions of limits are above the level of MATH 2301, but the informal definition of limits (both kinds) are appropriate for a class at the level of MATH 2301. Furthermore, he gave an example of computing limits, both regular limits and infinite limits , using what could be called an informal method.

The informal method of computing the limit of an expression involves examining the trends in the various terms in the expression, and making a conclusion about the trend in the value of the expression.

[Example 1] we will revisit the following function, which was studied in Group Work GW02: Guessing Limits for a Function Given by a Formula by Plugging in Numbers on Tue Aug 23.

(a) find \(lim_{x \rightarrow 3^-}f(x)\) by using the informal method of computing the limit . That is, by considering the trends in the size of the factors. Use the Expanded Definition of Limit, Involving Infinity , if appropriate. Show/Hide Solution

(b) find \(lim_{x \rightarrow 2^-}f(x)\) by using the informal method of computing the limit . That is, by considering the trends in the size of the factors. Use the Expanded Definition of Limit, Involving Infinity , if appropriate. Show/Hide Solution

End of [Example 1]

[Example 2] we will revisit the following function, which was studied in Group Work GW03: Function Values and Limits for a Rational Function on Fri, Aug 26.

(a) find \(lim_{x \rightarrow -3^+}f(x)\) by using the informal method of computing the limit . That is, by considering the trends in the size of the factors. Use the Expanded Definition of Limit, Involving Infinity , if appropriate. Show/Hide Solution

(b) find \(lim_{x \rightarrow -2^+}f(x)\) by using the informal method of computing the limit . That is, by considering the trends in the size of the factors. Use the Expanded Definition of Limit, Involving Infinity , if appropriate. Show/Hide Solution

End of [Example 2]

Limits Involving Infinity (Book Section 1.6)

On Fri Sep 9, Mark discussed infinite limits . He presented two Reference Pages about limits, posted on the Course Web Page . These Reference Pages show the book's precise definitions of regular limits and of infinite limits , along with informal definitions of those limits. Here are links to those two Reference Pages .

• R01: Definition of Limit was distributed on Tue Aug 23.
• R04: Limits Involving Infinity was discussed Fri Sep 9.

Mark also discussed an informal method of computing the limit of an expression. This method involves examining the trends in the various terms in the expression, and making a conclusion about the trend in the value of the expression. The method works equally well for limits that turn out to be numbers and limits that turn out to be infinite .

See the Calendar entry for Fri Sep 9 to see the discussion of the informal method of computing the limit and to see some examples of its use.

We'll start today with three presentations involving using that method to find limits of very similar-looking rational functions.

Class Presentations Involving the Informal Method for Computing Limits for Mon Sep 12 Show/Hide Presentations

Observe that all three of the limits in the presentations were of \(\frac{0}{0}\) indeterminate form . We see that their limits turned out very differently!

Mark will Present Conclusions about General Trends in limits of the form \( \lim_{x\rightarrow c}f(x)\), where \(c\) is a real number and \(f(x)\) is a rational function .

Mark will Discuss limits of the form \( \lim_{x\rightarrow \infty}f(x)\), where \(f(x)\) is a rational function .

Topic: Limits Involving Infinity for a Rational Function (Book Section 1.6)

Group Work: Students worked on Group Work GW06 (Limits Involving Infinity for a Rational Function). ( GW06 Solutions )

Instantaneous Rates of Change: The Derivative (Book Section 2.1)

New Concepts for Today

Definition of the Derivative

• Words: the derivative of \(f(x)\)
• Symbol: \(f'(x)\)
• Meaning: \(f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)

Definition of the Tangent Line

• Words: the line tangent to the graph of \(f(x)\) at \(x=a\)
• Meaning: the line that has these two properties:
  • The line contains the point \((a,f(a))\).
  • The line has slope \(m=f'(a)\).

The Point Slope Form of the Equation of a Line

If a line that has these two properties:

• The line contains the point \((a,b)\).
• the line has slope \((m\).

The Point Slope Form of the Equation of the Tangent Line

Since the tangent line that has these two properties:

• The line contains the point \((a,f(a))\).
• the line has slope \(m=f'(a)\).

Class Presentations Involving Computing Derivatives for Fri Sep 16 Show/Hide Presentations

Instantaneous Rates of Change: The Derivative (Book Section 2.1)

Concepts from Fri Sep 16.

Definition of the Derivative

• Words: the derivative of \(f(x)\)
• Symbol: \(f'(x)\)
• Meaning: \(f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\)

Definition of the Tangent Line

• Words: the line tangent to the graph of \(f(x)\) at \(x=a\)
• Meaning: the line that has these two properties:
  • The line contains the point \((a,f(a))\).
  • The line has slope \(m=f'(a)\).

The Point Slope Form of the Equation of a Line

If a line that has these two properties:

• The line contains the point \((a,b)\).
• the line has slope \((m\).

The Point Slope Form of the Equation of the Tangent Line

Since the tangent line that has these two properties:

• The line contains the point \((a,f(a))\).
• the line has slope \(m=f'(a)\).

New Concept: the Normal Line.

Definition of the Normal Line

• Words: the line normal to the graph of \(f(x)\) at \(x=a\)
• Meaning: the line that has these two properties:
  • The line contains the point \((a,f(a))\).
  • The line is perpendicular to the line that is tangent to the graph of \(f(x)\) at \(x=a\).
• Remark: By this definition, realize that if the tangent line has slope \(m=f'(a)\), where \(f'(a) \neq 0\), then the normal line will have slope \(m=-\frac{1}{f'(a)}\). But also realize that if the tangent line has slope \(m=f'(a)\), where \(f'(a) = 0\), then this means that the tangent line is horizontal . That will mean that the normal line will be vertical . Remember that vertical lines have undefined slope . One can still write a line equation for a vertical line. But it is in the form \(x=a\). That is, it is not in slope intercept form.

Class Presentations Involving Tangent Lines and Normal Lines for Mon Sep 19 Show/Hide Presentations

Topic: Group Works About Interpretations Of The Derivative (Book Section 2.2)

• GW07: Representations of Slopes , Tue Sep 20
• GW08: Finding Derivatives Graphically Using a Ruler , Tue Sep 20
• GW09: Finding Derivatives Graphically Using a Ruler , Tue Sep 20
• GW10: Which is the Function; Which is the Derivative? , Tue Sep 20

Interpretations of the Derivative (Book Section 2.2) (#1-18)

Class Presentations for Wed Sep 21 Show/Hide Presentations

Quiz Q03 Covering Section 2.1

Section 2.3 Basic Differentiation Rules (#1-38)

Class Presentations for Fri Sep 23 Show/Hide Presentations

Section 2.4 The Product and Quotient Rules

Exercises: Section 2.4 #1 - 14, 15 - 47 odd

Class Presentations for Mon Sep 26 Show/Hide Presentations

Section 2.4 The Chain Rule

Exercises: Section 2.5: #1-6, 7-39odd, 41, 42

Discussed nested functions . That is, functions of the form

Question: How to find the derivative of such a function?

Answer: Theorem 2.5.3 The Chain Rule

Examples

[Example 1]: For \(f(x)=\left(\cos{(x)}\right)^2\), find \(f�(x)\).

[Example 2]: For \(f(x)=\cos{\left(x^2\right)}\), find \(f�(x)\).

[Example 3: For \(f(x)=\cos{\left(\ln{(x)}\right)}\), find \(f�(x)\).

[Example 4]: For \(f(x)=\ln\left(\cos{(x)}\right)\), find \(f�(x)\).

Discuss Basic Derivative Rules from Section 2.3

Discuss these Derivative Rules :

• Constant Function Rule: If \(f(x)=c\) then \(f�(x)=0\).
• Sine Function Rule: If \(f(x)=\sin{(x)}\) then \(f�(x)=\cos{(x)}\).
• Exponential Function Rule: If \(f(x)=e^{(x)}\) then \(f�(x)=e^{(x)}\).
• Logarithm Function Rule: If \(f(x)=\ln{(x)}\) then \(f�(x)=\frac{1}{x}\).

For each Derivative Rule , do an construction similar to those done in Group Works GW08 and GW09 , Finding Derivatives Graphically Using a Ruler .

• Draw basic graph of \(f(x)\).
• Pin some tangent lines to the graph of \(f(x)\).
• Estimate the slopes of those tangent lines.
• Use the slope date to draw a crude graph of \(f�(x)\).
• Observe that the drawn graph of \(f�(x)\) agrees with the function \(f�(x)\) given by the Derivative Rule .

Section 2.5 The Chain Rule

Exercises: Section 2.5: #1 - 6,1 - 39 odd, 41, 42

More Examples involving the Chain Rule

Class Presentations for Wed Sep 28 Show/Hide Presentations

Section 2.6 Implicit Differentiation

Exercises: Section 2.6: #1-4, 5-25odd, 26, 27-41odd

Topic 1: Some Derivative Calculations Using Old Rules (Sections 2.3, 2.4, 2.5)

Mark B will do an example, and we�ll have two presentations .

Topic 2: The Power Rule for Derivatives, Valid for All Real Number Exponents Show/Hide Topic 2

Class Presentations for Mon Oct 3 Show/Hide Presentations

Section 2.6 Implicit Differentiation

Exercises: Section 2.6: #1-4, 5-25odd, 26, 27-41odd

Topic 1: The Method of Implicit Differentiation Show/Hide Topic 1

Class Presentations for Tue Oct 4 Show/Hide Presentations

Group Work: GW11: Rewrite Function, then Find the Derivative.

Topic 1: Logarithmic Differentiation (Section 2.6) Show/Hide Topic 1

Topic 2: Leftover from Sections 2.2 and 2.3: Approximation with the Derivative

Class Presentations for Wed Oct 5 Show/Hide Presentations

Book Sections and (Homework Exercises): Section 3.1 Extreme Values (1-6, 7-25odd)

Discussed these topics from Section 3.1:

• Absolute Extrema (Absolute Max and Absolute Min)
• The Extreme Value Theorem
• Relative Extrema (Relative Max and Relative Min)

Students did Group Work GW12: The Extreme Value Theorem .

Book Sections and (Homework Exercises): Section 3.1 Extreme Values (1-6, 7-25odd)

Discuss these topics from Section 3.1:

• Definition 3.1.11 Critical Numbers and Critical Points
• Theorem 3.1.12 Relative Extrema and Critical Points
• Key Idea 3.1.14 Finding Extrema on a Closed Interval ( presentations are about this)

Class Presentations for Tue Oct 11 Show/Hide Presentations

Discussed more examples of Finding Extrema on a Closed Interval .

Students did Group Work GW13: Finding Absolute Extrema on a Closed Interval .

Students did Group Work GW14: Comparing Two Solutions to an Absolute Extrema Problem .

Book Sections and (Homework Exercises): 3.2 The Mean Value Theorem (1, 2, 3-20odd)

Reviewed these old concepts (from book Section 2.1)

• The Average Rate of Change , \(m=\frac{f(b)-f(a)}{b-a}\), which is the slope of a secant line on the graph of the function
• The Instantaneous Rate of Change , \(m=\lim_{h\rightarrow 0}\frac{f(c+h)-f(C)}{h}\), which is the slope of a secant line on the graph of the function

Discussed the Mean Value Theorem (from book Section 3.2)

Theorem 3.2.3 The Mean Value Theorem

Let \(f(x)\) be a continuous function on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\). There exists an \(x\)value, \(x=c\), such that $$f�(c)=\frac{f(b)-f(a)}{b-a}$$ That is, there is at least one \(x\) value, \(x=c\), in the interval \((a,b)\) such that the instantaneous rate of change of \(f(x)\) at \(x=c\) is equal to the average rate of change of \(f(x)\) on the interval \([a,b]\) .

In terms of the graph, this meant that there is at least one \(x\) value, \(x=c\), in the interval \((a,b)\) such that the line tangent to the graph of \(f(x)\) at \(x=c\) has the same slope as the secant line that touches the graph of \(f(x)\) at \(x=a\) and \(x=b\).

Saw Four Examples Involving the Mean Value Theorem

• Tim Jaskiewicz CP3: (book exercise 3.2#12) Can the Mean Value Theorem be applied to the function \(f(x)=5x^x-6x+8\) on the interval \([0,5]\)? If so, find a number \(c\) in the interval \((0,5)\) that is guaranteed by the Mean Value Theorem.
• Olivia Keener CP3: (book exercise 3.2#14) Can the Mean Value Theorem be applied to the function \(f(x)=\sqrt{25-x}\) on the interval \([0,9]\)? If so, find a number \(c\) in the interval \((0,9)\) that is guaranteed by the Mean Value Theorem.
• Paul Thorp CP3: (book exercise 3.2#16) Can the Mean Value Theorem be applied to the function \(f(x)=\ln{(x)}\) on the interval \([1,5]\)? If so, find a number \(c\) in the interval \((1,5)\) that is guaranteed by the Mean Value Theorem.
• Mark B: For the function \(f(x)=\frac{1}{x}\) on the interval \([-1,1]\), it does not seem possible to find a number \(c\) in the interval \((-1,1)\) such that $$f�(c)=\frac{f(1)-f(-1)}{1-(-1)}$$ Why not?!?

Book Sections and (Homework Exercises): Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

Discussed these topics from Section 3.3:

• Definition 3.3.2 Definition of Increasing and Decreasing Functions
• Theorem 3.3.4 Test for Increasing/Decreasing

Discussed method of making a Sign Chart to determine the sign behavior of a function.

[Increasing/Decreasing Example] For the function $$f(x)=x^4-6x^2+5$$ find the intervals where \(f(x)\) is increasing and the intervals where \(f(x)\) is decreasing . Show/Hide Results

Book Sections and (Homework Exercises): Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

Reviewed past topics from Section 3.1 and 3.3 that were discussed last week

• Definition 3.1.1 Extreme Values (absolute max and absolute min)
• Theorem 3.1.3 The Extreme Value Theorem If \(f(x)\) is a continuous function defined on a closed interval \(I=[a,b]\), then \(f(x)\) has both an absolute max and an absolute min on the interval \(I\).
• Definition 3.1.6 Relative Max and Relative Min
• Definition 3.1.11 Critical Numbers and Critical Points
• Theorem 3.1.11 Critical Numbers and Critical Points Relative extrema can only occur at \(x\) values that are critical numbers .
• Fact Absolute extrema can only occur at \(x\) values that are endpoint of the domain or critical numbers .
• Key Idea 3.1.4 Method for finding absolute extrema for a function \(f(x)\) that is continuous on a closed interval .
• Definition 3.3.2 Definition of Increasing and Decreasing Functions
• Theorem 3.3.4 Test for Increasing/Decreasing

Discussed new topic from Section 3.3:

• Theorem 3.3.10 First Derivative Test

Amy Evers Presentation CP3: Find the critical numbers for the function \(f(x)=xe^{(-x)}\).

[Example 1] (done by Mark B) Find the relative extrema for the function \(f(x)=xe^{(-x)}\).

Ben Oldiges Presentation CP3: Find the critical numbers for the function \(f(x)=\frac{1}{x^2}\).

[Example 2] (done by Mark B) Find the relative extrema for the function \(f(x)=\frac{1}{x^2}\).

Book Sections and (Homework Exercises): Section 3.3 Increasing and Decreasing Functions (1-6, 7-23odd)

Discussed Partition Numbers and Critical Numbers

A partition number for a function \(g(x)\) is an \(x\) value where \(g(x)=0\) or \(g\) is discontinuous .
Remark: A function \(g(x)\) can only change sign at its partition numbers .

A critical number for a function \(f(x)\) is an \(x\) value \(x=c\) that has these two properties:

1. \(x=c\) is a partition number for \(f'(x)\). That is, \(f'(c)=0\) or \(f'\) is discontinuous at \(x=c\).
2. \(f\) is continuous at \(x=c\).

Class Presentations for Tue Oct 18 Show/Hide Presentations

Students did three Group Works .

• GW15: Analyzing a Polynomial , Tue Oct 18
• GW16: Analyzing a Rational Function with a Vertical Asymptote , Tue Oct 18
• GW17: Analyzing a Rational Function with a Horizontal Asymptote , Tue Oct 18

Book Sections and (Homework Exercises)

• Section 3.4 Concavity and the Second Derivative (1-4, 5-56odd)
• Section 3.5 Curve Sketching (1-5, 6-25odd, 26-28)

Discussed Section 3.4 Concavity and the Second Derivative

Discussed Section 3.5 Curve Sketching

• Reference R05: Graphing Strategy

Students did a Group Work .

• GW18: Using Given Information to Sketch a Graph , Wed Oct 19

Book Sections and (Homework Exercises): Section 3.5 Curve Sketching (1-5, 6-25odd, 26-28)

• Reference R05: Graphing Strategy

Class Presentations for Fri Oct 21 Show/Hide Presentations

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Dalana Goddard Presentation CP3: Given the following in formation about \(f,f�,f�\):

• \(f(x)\)is continuous for all x
• \(f(7)=10\)
• \(f�(x)\) is positive on the interval \((-\infty,2)\).
• \(f�(2)=0\)
• \(f�(x)\) is negative on the interval \((2,12)\).
• \(f�(12)=0\)
• \(f�(x)\) is positive on the interval \((12,\infty)\).
• \(f��(x)\) is negative on the interval \((-\infty,7)\).
• \(f��(7)=0\)
• \(f��(x)\) is positive on the interval \((7,\infty)\).

Sketch a possible graph of \(f(x)\).

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Lauren Hartel Presentation CP3 The graph of a function \(f(x)\) is given without gridlines and without coordinate axes.

The formula for \(f(x)\) is not given, but the formulas for its first and second derivatives are

$$\begin{eqnarray} f�(x) &=& \frac{x}{x^2+1} \\ f�(x) &=& -\frac{2(x^2-1)}{(x^2+1)^2}=-\frac{2(x+1)(x-1)}{(x^2+1)^2} \end{eqnarray}$$

1. It looks like there is a relative min on the graph of \(f(x)\). What is the \(x\) coordinate of that point? Explain how you know.
2. It looks like there is are two inflection points on the graph of \(f(x)\). What are the \(x\) coordinates of those point? Explain how you know.

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Hide Fri Oct 21 Presentations

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Curve Sketching (Section 3.5)

• Reference R05: Graphing Strategy

Mark Did Some more examples involving curve sketching.

Class Presentations for Mon Oct 24, having to do with Curve Sketching Show/Hide Presentations

Newton's Method (Section 4.1) (Suggested Exercises 4.1# 3, 5, 7, 17)

Students worked on Group Work GW19: The Idea Behind Newton�s Method

Then Mark discussed Newton�s Method

Topic for Today: Analyzing a function and then graphing it, using the Graphing Strategy

(Concepts from Section 3.5)

We will be discussing the following function:

Refer to the Reference R05: Graphing Strategy , handed out in class on Wed Oct 19.

First Presentation , presented by these students:

• In Section 173 (Tue 9:30), Nana Asare CP4
• In Section 174 (Tue 11:00), Gavin Wolfe CP3
• In Section 175 (Tue 2:00), Kierston Harper CP4
• In Section 176 (Tue 3:30), Kelly Koenig CP3

Second Presentation , presented by these students:

• In Section 173 (Tue 9:30), Tyler Boldon CP4
• In Section 174 (Tue 11:00), Drew Conway CP4
• In Section 175 (Tue 2:00), Evan Green CP4
• In Section 176 (Tue 3:30), Reggie Shaffer CP3

Groups: Complete GW20: Using the Graphing Strategy to Analyze and Graph a Polynomial

Topic for Today: Newton's Method (Section 4.1) (Suggested Exercises 4.1# 3, 5, 7, 17)

Students finished the Group Work that was begun on Monday: GW19: The Idea Behind Newton�s Method .

Mark Discussed Newton�s Method.

Students worked on Group Work GW21: Newton�s Method

Class Presentations for Wed Oct 26, having to do with Newton�s Method Show/Hide Presentations

Topic for Today: Related Rates problems involving Right Triangles

(Concepts from Section 4.2)

First Presentation , presented by these students:

• In Section 173 (Tue 9:30), Amy Evers CP4
• In Section 174 (Tue 11:00), Carly Doros CP4
• In Section 175 (Tue 2:00), Austin Kiggins CP4
• In Section 176 (Tue 3:30), Paul Gbadebo CP4

Second Presentation , presented by these students:

• In Section 173 (Tue 9:30), Dalana Goddard CP4
• In Section 174 (Tue 11:00), Nicole Grant CP4
• In Section 175 (Tue 2:00), Dylan Pohovey CP4
• In Section 176 (Tue 3:30), Alan Romerero Herrera CP3

Third Presentation , presented by these students:

• In Section 173 (Tue 9:30), Lauren Hartel CP4
• In Section 174 (Tue 11:00), Tim Jaskiewicz CP4
• In Section 175 (Tue 2:00), Colin Sorge CP3
• In Section 176 (Tue 3:30), Emily Wilkerson CP3

Groups: Complete the three group works that were started by your classmates

• GW22: Related Rates #1
• GW23: Related Rates #2
• GW24: Related Rates #3

Topic for Today: Antiderivatives

(Concepts from Section 5.1)

Book Sections and (Homework Exercises)

• Section 5.1 Antiderivatives and Indefinite Integration (9-27odd, 28, 29, 31-39odd)

In book Section 5.1, in Definition 5.1.1 at the beginning of the section, the authors present the definitions of Antiderivatives and Indefinite Integral together. I think it is useful to spend some time learning about antiderivatives before learning about indefinite integrals. In today�s meeting, you�ll learn about, and discuss, antiderivatives.

Definition of Antiderivative

• Words: \(F\) is an antiderivative of \(f\). (Note the uppercase and lower case letters!)
• Meaning: \(f\) is the derivative of \(F\). That is, \(f=F�\).
• Arrow diagram: \(F \xrightarrow[\text{take derivative}] {}f\)

In coming days, you will be learning some techniques for finding antiderivatives. We�ll start discussing those techniques on Wed Nov 9. But in today�s meeting, you will focus on the following kind of problem:

• Question: Is one given function \(f(x)\) and antiderivative of another given function \(g(x)\).
• Solution Strategy: Find \(f�(x)\) and see if it equals \(g(x)\).

[Example 1] Question: $$\text{Is }F(x)=\frac{x^3}{3}\text{ an antiderivative of }f(x)=x^2?$$ Explain why or why not. (Show the steps!)

Solution: Strategy: Find \(F�(x)\) and see if it equals \(f(x)\).

First, rewrite $$F(x)=\frac{x^3}{3}=\left(\frac{1}{3}\right)x^3$$

Now, find the derivative. $$F�(x)=\frac{d}{dx}\left(\frac{1}{3}\right)x^3=\left(\frac{1}{3}\right)\frac{d}{dx}x^3=\left(\frac{1}{3}\right)\left(3x^{3-1}\right)=x^2=f(x)$$

The answer is yes , \(F(x)=\frac{x^3}{3}\) an antiderivative of \(f(x)=x^2\), because \(F�(x)=f(x)\)

End of [Example 1]

First Presentation , presented by these students:

• In Section 173 (Tue 9:30), Ben Oldiges CP4
• In Section 174 (Tue 11:00), Bilal Tahir CP4
• In Section 175 (Tue 2:00), Olivia Keener CP4
• In Section 176 (Tue 3:30), Carlotta Datillo CP4

Second Presentation , presented by these students:

• In Section 173 (Tue 9:30), Sarah Weller CP4
• In Section 174 (Tue 11:00), Andrew Champagne CP5
• In Section 175 (Tue 2:00), Jonah Lewis CP4
• In Section 176 (Tue 3:30), Kelly Koenig CP4

Third Presentation , presented by these students:

• In Section 173 (Tue 9:30), Nana Asare CP5
• In Section 174 (Tue 11:00), Drew Conway CP5
• In Section 175 (Tue 2:00), Colin Sorge CP4
• In Section 176 (Tue 3:30), Alan Romero Herrera CP4

Fourth Presentation , presented by these students:

• In Section 173 (Tue 9:30), Tyler Boldon CP5
• In Section 174 (Tue 11:00), Carly Doros CP5
• In Section 175 (Tue 2:00), Evan Green CP5
• In Section 176 (Tue 3:30), Reggie Shaffer CP4

Groups: Work on Group Work GW25: Antiderivatives .

First Topic for Today: Indefinite Integrals (Concepts from Section 5.1)

Exercises: Section 5.1# 9-27odd, 28, 29, 31-39odd

Mark will do some Examples

Second Topic for Today: The Definite Integral (Concepts from Section 5.2)

Exercises: Section 5.2# 5-17odd, 19-22

Mark Discussed the informal definition of the Definite Integral presented in the book:

Informal Definition of the Definite Integral

• Symbol: $$\int_{x=a}^{x=b}f(x)dx$$
• Spoken: The definite integral of \(f(x)\) from \(x=a\) to \(x=b\).
• Usage: The function \(f(x)\) is continuous on the interval \([a,b]\).
• Informal Definition: the signed area of the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\).
• Remark: This is an informal definition because we have only have a notion of area for certain basic geometric shapes. For now, this definition of definite integral can only be used in situations where the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is made up of basic geometric shapes. In those situations, the value of the definite integral can be found by using familiar geometric formulas to compute the areas of the shapes that make up the region. (Note, however, that the book does not call this an informal definition. Rather, the book just presents this as the definition of the Definite Integral . That is a significant mistake in the book.)

Mark will do some Examples of Definite Integrals Gound Using Geometry

Students worked on Group Works involving concepts from these two sections:

• Section 5.1 Indefinite Integrals (Exercises: 5.1# 9-27odd, 28, 29, 31-39odd)
• Section 5.2 Definite Integrals (Exercises: 5.2# 5-17odd, 19-22)

The Group Works:

• GW26: Good and Bad Indefinite Integral Solutions (Section 5.1 Concepts)
• GW27: Definite Integrals for a Graph Made up of Geometric Shapes (Section 5.2 Concepts)
• GW28: Computing Definite Integrals By Using Geometry (Section 5.2Concepts)

Approximating Areas Using Riemann Sums

Sections for Today

• Section 5.3: Riemann Sums (Exercises: 17-39odd)

Mark Discussed the informal definition of the Definite Integral presented in the book:

Informal Definition of the Definite Integral

• Symbol: $$\int_{x=a}^{x=b}f(x)dx$$
• Spoken: The definite integral of \(f(x)\) from \(x=a\) to \(x=b\).
• Usage: The function \(f(x)\) is continuous on the interval \([a,b]\).
• Informal Definition: the signed area of the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\).
• Remark: This is an informal definition because we have only have a notion of area for certain basic geometric shapes. For now, this definition of definite integral can only be used in situations where the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is made up of basic geometric shapes. In those situations, the value of the definite integral can be found by using familiar geometric formulas to compute the areas of the shapes that make up the region. (Note, however, that the book does not call this an informal definition. Rather, the book just presents this as the definition of the Definite Integral . That is a significant mistake in the book.)

Natural Question: What do we do when the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is not made up of basic geometric shapes?

That is the essence of The Area Problem .

The Area Problem

When the region between the graph of \(f(x)\) and the \(x\) axis, from \(x=a\) to \(x=b\), is not made up of basic geometric shapes,

• What does the area of the region even mean?
• How do we compute a value for the area of the region?

We start by thinking about a short wish list for Area , things that we expect to be true for Area. We don�t yet have a definition of what Area

Area Axioms

1. The area of a region should be \(\geq 0\), and should \(=0\) only when the region does not contain any discs. (That�s a cheesy, but actually mathematically fine, way of waying that the region does not enclose any space.)
2. The area of a region should equal the sum of the areas of the sub-regions that make up the larger region.
3. (This is a consequence of axiom 2) If one region is entirely contained in some larger region, then the area of the smaller, inner region should be less than the area of the larger region.

With that third axiom in mind, we can think of a way of getting an underestimate and an overestimate for the unknown area of some curvy region by constructing a sort of sandwich involving the curvy region and two simple geometric regions. A region made up of rectangles is built in a way that it is entirely contained in the curvy region. A second region made up rectangles is built in a way that the curvy region is entirely contained in the second region.

This led to the introduction of the Left Riemann Sum with \(n\) subintervals , denoted by \(L_n\), and the Right Riemann Sum with \(n\) subintervals , denoted by \(R_n\). These sums are the area of regions made up of left rectangles and right rectangles .

Students worked on this Group Work involving calculating Left and Right Riemann Sums for a function given by a graph .

• GW29: Estimating the Area Under a Graph by Using Riemann Sums

Then Mark B worked on an analogous problem involving a function given by a formula not a graph. That problem was framed as a Quest .

Quest

For the function \(f(x)\) given by the formula $$f(x)=5+\frac{x^2}{10}$$ the region between the graph of \(f(x)\) and the \(x\) axis from \(x=2\) to \(x=12\) is a curvy region, not made up of simple geometric shapes. We would like to find the area of that region.

First Work on the Quest

We don�t have a method for calculating the area of that region. We don�t even have a definition of what area even means for a region like that. But it is possible to find an underestimate and an overestimate using Riemann Sums . Mark did this with a Left Sum with \(5\) subintervals and a Right Sum with \(5\) subintervals . The result was $$L_5=94 \lt \text{ unknown area } \lt R_5=122$$

Sections for Today

• Section 5.3: Riemann Sums (Exercises: 17-39odd)
• Useful video Approximating areas with Sums
  • Video
  • Accompanying Notes

Meeting Part 1: Group Work: Computing Riemann Sums

Using Reference R06: Steps for Computing Riemann Sums as a guide, students worked on Group Work GW30: Computing Riemann Sums .

Meeting Part 2: The Definite Integral as a Limit of Riemann Sums

Recall our Quest from Wednesday�s meeting:

Quest

For the function \(f(x)\) given by the formula $$f(x)=5+\frac{x^2}{10}$$ the region between the graph of \(f(x)\) and the \(x\) axis from \(x=2\) to \(x=12\) is a curvy region, not made up of simple geometric shapes. We would like to find the area of that region.

We don�t have a method for calculating the area of that region. We don�t even have a definition of what area even means for a region like that. But it is possible to find an underestimate and an overestimate using Riemann Sums . Mark did this with a Left Sum with \(5\) subintervals and a Right Sum with \(5\) subintervals . The result was $$L_5=94 \lt \text{ unknown area } \lt R_5=122$$

In this part of the meeting, we will try to improve the underestimate and overestimate by using more subintervals. We use a computer tool to do the repetitive work.

Here is a handy Riemann Sum Calculator .

Rather than lecturing, Mark B will be summarizing the video Approximating areas with Sums

• Video
• Accompanying Notes

Meeting Part 3: Quiz Q08

The Quiz will be over Indefinite Integrals (concepts from Section 5.1).

Book Sections and (Homework Exercises)

• Section 5.4 The Fundamental Theorem of Calculus (5-29odd, 35-57odd)
• Video that may be helpful: ( video ) ( notes )

Meeting Part 1: Intro to the Fundamental Theorem of Calculus

Meeting Part 2: Examples using the Fundamental Theorem of Calculus

Book Sections and (Homework Exercises)

• Section 5.4 The Fundamental Theorem of Calculus (5-29odd, 35-57odd)
• Video that may be helpful: ( video ) ( notes )

Tue Nov 22:

• Student presentations involve showing how to find a definite integral using the Fundamental Theorem of Calculus.
• All students will have presentations
• Extra Credit Presentations (for 10 points):
  • If a student is absent, another student can do their presentation for extra credit.
  • Also, notice that there are 10 presentations. But no section has that many students. If there is a presentation with no student assigned to it, a student do that presentation for extra credit.
  • Each student can do at most one extra credit presentation.
  • The substitute student must have prepared the presentation.
  • Therefore, students hoping to earn extra credit points by doing an extra presentation should prepare by studying all of the presentations and knowing how to do them.

The Fundamental Theorem of Calculus: $$\int_{x=a}^{x=b}f(x)dx\underset{\text{FTC}}{=}\left. \left(\int f(x)dx\right)\right\vert_{x=a}^{x=b}$$ Mark B introduced this theorem in lecture on Mon Nov 21. He also discusses it at length in this video and its accompanying notes . And of course the Theorem is discussed in the online Apex Calculus book, in Section 5.4.

The Presentations: (Student Numbers are shown below.)

1. Show why $$\int_0^6(x+7)^2dx=618$$ That is, clearly show the steps that lead to this result.
2. Show why $$\int_0^\frac{\pi}{2}\cos{(x)}dx=1$$ That is, clearly show the steps that lead to this result.
3. Show why $$\int_{1}^2e^{(x)}dx=e^2-e$$ That is, clearly show the steps that lead to this result.
4. Show why $$\int_{1}^{2}3^{(x)}dx=\frac{6}{\ln{(3)}}$$ That is, clearly show the steps that lead to this result.
5. Show why $$\int_{16}^{49}\sqrt{x}dx=186$$ That is, clearly show the steps that lead to this result.
6. Show why $$\int_{16}^{49}\frac{1}{\sqrt{x}}dx=6$$ That is, clearly show the steps that lead to this result.
7. Show why $$\int_0^{\pi}(3\cos{(x)}-2\sin{(x)})dx=-4$$ That is, clearly show the steps that lead to this result.
8. Show why $$\int_{75}^{100}dx=25$$ That is, clearly show the steps that lead to this result.
9. Show why $$\int_1^{e^2}\frac{1}{x}dx=2$$ That is, clearly show the steps that lead to this result.
10. Show why $$\int_1^{2}\frac{1}{x}dx=\ln{(2)}$$ That is, clearly show the steps that lead to this result.

Student Numbers for Section 173 (Tue 9:30)

1. Nana Asare
2. Tyler Boldon
3. Ellie Bower
4. Amy Evers
5. Dalana Goddard
6. Lauren Hartel
7. Sara Weller

Student Numbers for Section 174 (Tue 11:00)

1. Andrew Champagne
2. Drew Conway
3. Carly Doros
4. Nicole Grant
5. Tim Jaskiewicz
6. Bilal Tahir
7. Gavin Wolfe

Student Numbers for Section 175 (Tue 2:00

1. Evan Green
2. Kierston Harper
3. Olivia Keener
4. Austin Kiggins
5. Jonah Lewis
6. Dylan Pohovey
7. Colin Sorge

Student Numbers for Section 176 (Tue 3:30)

1. Carlotta Dattilo
2. Paul Gbadebo
3. Kelly Koenig
4. Alan Romero Herrera
5. Reggie Shaffer
6. Paul Thorp
7. Emily Wilkerson