g(x) represents the area under the curve of f(t) from 0 to x. As x increases from 0 to 10, g(x) will increase, representing the accumulating area under f(t) over the interval [0,x].
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 18: Maximum and Minimum Values (Section 021 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 18: Maximum and Minimum Values (Section 041 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 24: Areas, Distances, the Integral (Section 021 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 12: Linear Approximations and Differentials (slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 18: Maximum and Minimum Values (Section 021 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 18: Maximum and Minimum Values (Section 041 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 24: Areas, Distances, the Integral (Section 021 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 12: Linear Approximations and Differentials (slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
Lesson 24: Areas, Distances, the Integral (Section 041 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
JEE Mathematics/ Lakshmikanta Satapathy/ Permutation and Combination QA part 2/ JEE question on four digit numbers permuted from five given digits solved with the related concepts
JEE Mathematics/ Lakshmikanta Satapathy/ Set theory part 3/ JEE question on Power set of the cartesian product of two sets solved with the related concepts
JEE Physics/ Lakshmikanta Satapathy/ Wave Motion QA part 3/ JEE question on fundamental frequencies of Open and Closed pipes solved with the related concepts
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the handout version to take notes on.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
State of ICS and IoT Cyber Threat Landscape Report 2024 preview
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
1. Section 5.4
The Fundamental Theorem of Calculus
V63.0121.041, Calculus I
New York University
December 8, 2010
Announcements
Today: Section 5.4
Monday, December 13: Section 5.5
”Monday,” December 15: Review and Movie Day!
Monday, December 20, 12:00–1:50pm: Final Exam (location still
TBD)
. . . . . . .
2. Announcements
Today: Section 5.4
Monday, December 13:
Section 5.5
”Monday,” December 15:
Review and Movie Day!
Monday, December 20,
12:00–1:50pm: Final Exam
(location still TBD)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 2 / 32
3. Objectives
State and explain the
Fundemental Theorems of
Calculus
Use the first fundamental
theorem of calculus to find
derivatives of functions
defined as integrals.
Compute the average
value of an integrable
function over a closed
interval.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 3 / 32
4. Outline
Recall: The Evaluation Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 4 / 32
5. The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a to b
is the number ∫ b ∑n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 5 / 32
6. Big time Theorem
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F′ for another function F, then
∫ b
f(x) dx = F(b) − F(a).
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 6 / 32
7. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 7 / 32
8. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If v(t) represents the velocity of a particle moving rectilinearly, then
∫ t1
v(t) dt = s(t1 ) − s(t0 ).
t0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 7 / 32
9. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If MC(x) represents the marginal cost of making x units of a product,
then ∫ x
C(x) = C(0) + MC(q) dq.
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 7 / 32
10. The Integral as Total Change
Another way to state this theorem is:
∫ b
F′ (x) dx = F(b) − F(a),
a
or the integral of a derivative along an interval is the total change
between the sides of that interval. This has many ramifications:
Theorem
If ρ(x) represents the density of a thin rod at a distance of x from its
end, then the mass of the rod up to x is
∫ x
m(x) = ρ(s) ds.
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 7 / 32
11. My first table of integrals
.
∫ ∫ ∫
[f(x) + g(x)] dx = f(x) dx + g(x) dx
∫ ∫ ∫
xn+1
xn dx = + C (n ̸= −1) cf(x) dx = c f(x) dx
n+1 ∫
∫
1
ex dx = ex + C dx = ln |x| + C
x
∫ ∫
ax
sin x dx = − cos x + C ax dx = +C
ln a
∫ ∫
cos x dx = sin x + C csc2 x dx = − cot x + C
∫ ∫
2
sec x dx = tan x + C csc x cot x dx = − csc x + C
∫ ∫
1
sec x tan x dx = sec x + C √ dx = arcsin x + C
∫ 1 − x2
1
dx = arctan x + C
1 + x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 8 / 32
12. Outline
Recall: The Evaluation Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 9 / 32
13. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
14. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = .
n n
.
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
15. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
.
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
16. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x x3 x (2x)3 x (nx)3
Rn = · 3+ · + ··· + ·
n n n n3 n n3
.
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
17. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x x3 x (2x)3 x (nx)3
Rn = · 3+ · + ··· + ·
n n n n3 n n3
x4 ( )
= 4 13 + 23 + 33 + · · · + n3
n
.
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
18. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x x3 x (2x)3 x (nx)3
Rn = · 3+ · + ··· + ·
n n n n3 n n3
x4 ( )
= 4 13 + 23 + 33 + · · · + n3
n
x4 [ ]2
.
0 x = 4 1 n(n + 1)
n 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
19. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x4 n2 (n + 1)2
Rn =
4n4
.
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
20. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x4 n2 (n + 1)2
Rn =
4n4
x4
So g(x) = lim Rn =
. x→∞ 4
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
21. Area as a Function
Example
∫ x
3
Let f(t) = t and define g(x) = f(t) dt. Find g(x) and g′ (x).
0
Solution
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x4 n2 (n + 1)2
Rn =
4n4
x4
So g(x) = lim Rn = and g′ (x) = x3 .
. x→∞ 4
0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 10 / 32
22. The area function in general
Let f be a function which is integrable (i.e., continuous or with finitely
many jump discontinuities) on [a, b]. Define
∫ x
g(x) = f(t) dt.
a
The variable is x; t is a “dummy” variable that’s integrated over.
Picture changing x and taking more of less of the region under the
curve.
Question: What does f tell you about g?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 11 / 32
23. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
24. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
25. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
26. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
27. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
28. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
29. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
30. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
31. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
32. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
33. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
34. Envisioning the area function
Example
Suppose f(t) is the function graphed below:
y
g
.
x
2 4 6 8 10f
∫ x
Let g(x) = f(t) dt. What can you say about g?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 12 / 32
35. features of g from f
y
Interval sign monotonicity monotonicity concavity
of f of g of f of g
g
. [0, 2] + ↗ ↗ ⌣
fx
2 4 6 8 10 [2, 4.5] + ↗ ↘ ⌢
[4.5, 6] − ↘ ↘ ⌢
[6, 8] − ↘ ↗ ⌣
[8, 10] − ↘ → none
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 13 / 32
36. features of g from f
y
Interval sign monotonicity monotonicity concavity
of f of g of f of g
g
. [0, 2] + ↗ ↗ ⌣
fx
2 4 6 8 10 [2, 4.5] + ↗ ↘ ⌢
[4.5, 6] − ↘ ↘ ⌢
[6, 8] − ↘ ↗ ⌣
[8, 10] − ↘ → none
We see that g is behaving a lot like an antiderivative of f.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 13 / 32
37. Another Big Time Theorem
Theorem (The First Fundamental Theorem of Calculus)
Let f be an integrable function on [a, b] and define
∫ x
g(x) = f(t) dt.
a
If f is continuous at x in (a, b), then g is differentiable at x and
g′ (x) = f(x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 14 / 32
38. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
g(x + h) − g(x)
=
h
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
39. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
40. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
∫ x+h
f(t) dt
x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
41. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
∫ x+h
f(t) dt ≤ Mh · h
x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
42. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
43. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
So
g(x + h) − g(x)
mh ≤ ≤ Mh .
h
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
44. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
∫
g(x + h) − g(x) 1 x+h
= f(t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
∫ x+h
mh · h ≤ f(t) dt ≤ Mh · h
x
So
g(x + h) − g(x)
mh ≤ ≤ Mh .
h
As h → 0, both mh and Mh tend to f(x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 15 / 32
45. Meet the Mathematician: James Gregory
Scottish, 1638-1675
Astronomer and Geometer
Conceived transcendental
numbers and found
evidence that π was
transcendental
Proved a geometric
version of 1FTC as a
lemma but didn’t take it
further
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 16 / 32
46. Meet the Mathematician: Isaac Barrow
English, 1630-1677
Professor of Greek,
theology, and mathematics
at Cambridge
Had a famous student
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 17 / 32
47. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 18 / 32
48. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 19 / 32
49. Differentiation and Integration as reverse processes
Putting together 1FTC and 2FTC, we get a beautiful relationship
between the two fundamental concepts in calculus.
Theorem (The Fundamental Theorem(s) of Calculus)
I. If f is a continuous function, then
∫ x
d
f(t) dt = f(x)
dx a
So the derivative of the integral is the original function.
II. If f is a differentiable function, then
∫ b
f′ (x) dx = f(b) − f(a).
a
So the integral of the derivative of is (an evaluation of) the original
function.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 20 / 32
50. Outline
Recall: The Evaluation Theorem a/k/a 2nd FTC
The First Fundamental Theorem of Calculus
Area as a Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 21 / 32
51. Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 22 / 32
52. Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x4 , so h′ (x) = 81x3 .
4 4
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 22 / 32
53. Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x4 , so h′ (x) = 81x3 .
4 4
0
Solution (Using 1FTC)
∫ u
We can think of h as the composition g k, where g(u) = ◦ t3 dt and
0
k(x) = 3x.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 22 / 32
54. Differentiation of area functions
Example
∫ 3x
Let h(x) = t3 dt. What is h′ (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x4 , so h′ (x) = 81x3 .
4 4
0
Solution (Using 1FTC)
∫ u
We can think of h as the composition g k, where g(u) = ◦ t3 dt and
0
k(x) = 3x. Then h′ (x) = g′ (u) · k′ (x), or
h′ (x) = g′ (k(x)) · k′ (x) = (k(x))3 · 3 = (3x)3 · 3 = 81x3 .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 22 / 32
55. Differentiation of area functions, in general
by 1FTC
∫ k(x)
d
f(t) dt = f(k(x))k′ (x)
dx a
by reversing the order of integration:
∫ b ∫ h(x)
d d
f(t) dt = − f(t) dt = −f(h(x))h′ (x)
dx h(x) dx b
by combining the two above:
∫ (∫ ∫ )
k(x) k(x) 0
d d
f(t) dt = f(t) dt + f(t) dt
dx h(x) dx 0 h(x)
= f(k(x))k′ (x) − f(h(x))h′ (x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 23 / 32
56. Another Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 24 / 32
57. Another Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
0
Solution
We have
∫ sin2 x
d
(17t2 + 4t − 4) dt
dx 0
( d )
2 2
= 17(sin x) + 4(sin x) − 4 ·
2
sin2 x
( ) dx
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 24 / 32
58. A Similar Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 25 / 32
59. A Similar Example
Example
∫ sin2 x
Let h(x) = (17t2 + 4t − 4) dt. What is h′ (x)?
3
Solution
We have
∫ sin2 x
d
(17t2 + 4t − 4) dt
dx 0
( d )
2 2
= 17(sin x) + 4(sin x) − 4 ·
2
sin2 x
( ) dx
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 25 / 32
60. Compare
Question
Why is
∫ sin2 x ∫ sin2 x
d d
(17t + 4t − 4) dt =
2
(17t2 + 4t − 4) dt?
dx 0 dx 3
Or, why doesn’t the lower limit appear in the derivative?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 26 / 32
61. Compare
Question
Why is
∫ sin2 x ∫ sin2 x
d d
(17t + 4t − 4) dt =
2
(17t2 + 4t − 4) dt?
dx 0 dx 3
Or, why doesn’t the lower limit appear in the derivative?
Answer
Because
∫ sin2 x ∫ 3 ∫ sin2 x
(17t2 + 4t − 4) dt = (17t2 + 4t − 4) dt + (17t2 + 4t − 4) dt
0 0 3
So the two functions differ by a constant.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 26 / 32
62. The Full Nasty
Example
∫ ex
Find the derivative of F(x) = sin4 t dt.
x3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 27 / 32
63. The Full Nasty
Example
∫ ex
Find the derivative of F(x) = sin4 t dt.
x3
Solution
∫ ex
d
sin4 t dt = sin4 (ex ) · ex − sin4 (x3 ) · 3x2
dx x3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 27 / 32
64. The Full Nasty
Example
∫ ex
Find the derivative of F(x) = sin4 t dt.
x3
Solution
∫ ex
d
sin4 t dt = sin4 (ex ) · ex − sin4 (x3 ) · 3x2
dx x3
Notice here it’s much easier than finding an antiderivative for sin4 .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 27 / 32
65. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 28 / 32
66. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
Answer
Some functions are difficult or impossible to integrate in
elementary terms.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 28 / 32
67. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
Answer
Some functions are difficult or impossible to integrate in
elementary terms.
Some functions are naturally defined in terms of other integrals.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 28 / 32
68. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
69. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
It turns out erf is the shape of the bell curve.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
70. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative: erf′ (x) =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
71. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2
explicitly, but we do know its derivative: erf′ (x) = √ e−x .
2
π
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
72. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2
explicitly, but we do know its derivative: erf′ (x) = √ e−x .
2
π
Example
d
Find erf(x2 ).
dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
73. Erf
Here’s a function with a funny name but an important role:
∫ x
2
e−t dt.
2
erf(x) = √
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2
explicitly, but we do know its derivative: erf′ (x) = √ e−x .
2
π
Example
d
Find erf(x2 ).
dx
Solution
By the chain rule we have
d d 2 4
erf(x2 ) = erf′ (x2 ) x2 = √ e−(x ) 2x = √ xe−x .
2 2 4
dx dx π π
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 29 / 32
74. Other functions defined by integrals
The future value of an asset:
∫ ∞
FV(t) = π(s)e−rs ds
t
where π(s) is the profitability at time s and r is the discount rate.
The consumer surplus of a good:
∫ q∗
∗
CS(q ) = (f(q) − p∗ ) dq
0
where f(q) is the demand function and p∗ and q∗ the equilibrium
price and quantity.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 30 / 32
75. Surplus by picture
price (p)
.
quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
76. Surplus by picture
price (p)
demand f(q)
.
quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
77. Surplus by picture
price (p)
supply
demand f(q)
.
quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
78. Surplus by picture
price (p)
supply
p∗ equilibrium
demand f(q)
.
q∗ quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
79. Surplus by picture
price (p)
supply
p∗ equilibrium
market revenue
demand f(q)
.
q∗ quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
80. Surplus by picture
consumer surplus
price (p)
supply
p∗ equilibrium
market revenue
demand f(q)
.
q∗ quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
81. Surplus by picture
consumer surplus
price (p)
producer surplus
supply
p∗ equilibrium
demand f(q)
.
q∗ quantity (q)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 31 / 32
82. Summary
Functions defined as integrals can be differentiated using the first
FTC: ∫ x
d
f(t) dt = f(x)
dx a
The two FTCs link the two major processes in calculus:
differentiation and integration
∫
F′ (x) dx = F(x) + C
Follow the calculus wars on twitter: #calcwars
. . . . . .
V63.0121.041, Calculus I (NYU) Section 5.4 The Fundamental Theorem December 8, 2010 32 / 32