This document contains notes for a lesson on the chain rule from a Calculus 1 class. It defines the chain rule formula and provides an example of applying the chain rule to find the derivative of a function. It also includes another example problem and its step-by-step solution using the chain rule. The document concludes with a metaphor to help understand applying the chain rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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.
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What follows is a collection of snippets from the podcast. To hear the full interview and more, check out the podcast on all podcast platforms and at www.dsmsports.net
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Mats André Zuccarello Aasen, commonly known as Mats Zuccarello, was born on September 1, 1987, in
Oslo, Norway. He grew up in the bustling neighborhood of Løren, where his passion for ice hockey began
at a young age. His mother, Anita Zuccarello, is of Italian descent, and his father, Glenn Aasen, is
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Narrated Business Proposal for the Philadelphia Eaglescamrynascott12
Slide 1:
Welcome, and thank you for joining me today. We will explore a strategic proposal to enhance parking and traffic management at Lincoln Financial Field, aiming to improve the overall fan experience and operational efficiency. This comprehensive plan addresses existing challenges and leverages innovative solutions to create a smoother and more enjoyable experience for our fans.
Slide 2:
Picture this: It’s a crisp fall afternoon, driving towards Lincoln Financial Field. The atmosphere is electric—tailgaters grilling, fans in Eagles jerseys creating a sea of green and white. The air buzzes with camaraderie and anticipation. You park, join the throng, and make your way to your seat. The stadium roars as the Eagles take the field, sending chills down your spine. Each play is a thrilling dance of strategy and skill. This is what being an Eagles fan is all about—the joy, the pride, and the shared experience.
Slide 3:
But now, the day is marred by frustration. The excitement wanes as you struggle to find a parking spot. The congestion is overwhelming, and tempers flare. The delays mean you miss the pre-game excitement, the tailgate camaraderie, and even the opening kick-off. After the game, the joy of victory or the shared solace of defeat is overshadowed by the stress of navigating out of the parking lot. The gridlock, honking horns, and endless waiting drain the energy and joy from what should have been an unforgettable experience.
Our proposal aims to eliminate these frustrations, ensuring that from arrival to departure, your experience is extraordinary. Efficient parking and smooth traffic flow are key to maintaining the high spirits and excitement that make game days special.
Slide 4:
The Philadelphia Eagles are not just a premier NFL team; they are an integral part of the community, hosting games, concerts, and various events at Lincoln Financial Field. Our state-of-the-art stadium is designed to provide a world-class experience for every attendee. Whether it's the thrill of game day, the excitement of a live concert, or the camaraderie of community events, we pride ourselves on delivering a fan-first experience and maintaining operational excellence across all our activities. Our commitment to our fans and community is unwavering, and we continuously strive to enhance every aspect of their experience, ensuring they leave with unforgettable memories.
Slide 5:
Recent trends show an increasing demand for efficient event logistics. Our customer feedback has consistently highlighted frustrations with parking and traffic. Surveys indicate that a significant number of fans are dissatisfied with the current parking situation. Comparisons with other venues like Citizens Bank Park and Wells Fargo Center reveal that we lag in terms of parking efficiency and convenience. These insights underscore the urgent need for innovation to meet and exceed fan expectations.
Slide 6:
As we delve into the intricacies of our operations, one glaring issue emer
4. Analogy
Think about riding a bike. To
go faster you can either:
pedal faster
change gears
5. Analogy
Think about riding a bike. To
go faster you can either:
pedal faster
change gears
The angular position of the back wheel depends on the position of
the front wheel:
Rθ
ϕ(θ) =
r
And so the angular speed of the back wheel depends on the
derivative of this function and the speed of the front wheel.
6.
7. Theorem of the day: The chain rule
Theorem
Let f and g be functions, with g differentiable at a and f
differentiable at g (a). Then f ◦ g is differentiable at a and
(f g ) (a) = f (g (a))g (a)
◦
In Leibnizian notation, let y = f (u) and u = g (x). Then
dy dy du
=
dx du dx
9. Example
Example
3x 2 + 1. Find h (x).
let h(x) =
Solution
First, write h as f g.
◦
10. Example
Example
3x 2 + 1. Find h (x).
let h(x) =
Solution √
u and g (x) = 3x 2 + 1.
First, write h as f g . Let f (u) =
◦
11. Example
Example
3x 2 + 1. Find h (x).
let h(x) =
Solution √
First, write h as f ◦ g . Let f (u) = u and g (x) = 3x 2 + 1. Then
f (u) = 1 u −1/2 , and g (x) = 6x. So
2
h (x) = 1 u −1/2 (6x)
2
12. Example
Example
3x 2 + 1. Find h (x).
let h(x) =
Solution √
First, write h as f ◦ g . Let f (u) = u and g (x) = 3x 2 + 1. Then
f (u) = 1 u −1/2 , and g (x) = 6x. So
2
3x
h (x) = 1 u −1/2 (6x) = 2 (3x 2 + 1)−1/2 (6x) = √
1
2
3x 2 + 1
13. Example
2
3
x5 − 2 + 8
Let f (x) = . Find f (x).
14.
15. Example
2
3
x5 − 2 + 8
Let f (x) = . Find f (x).
Solution
d d
2
3 3 3
x5 − 2 + 8 x5 − 2 + 8 x5 − 2 + 8
=2
dx dx
16. Example
2
3
x5 − 2 + 8
Let f (x) = . Find f (x).
Solution
d d
2
3 3 3
x5 − 2 + 8 x5 − 2 + 8 x5 − 2 + 8
=2
dx dx
d
3 3
x5 − 2 + 8 x5 − 2
=2
dx
19. A metaphor
Think about peeling an onion:
2
3
x 5 −2 +8
f (x) =
5
√
3
+8
2
− 2)−2/3 (5x 4 )
3 15
x5 − 2 + 8
f (x) = 2 3 (x
20. Question
The area of a circle, A = πr 2 , changes as its radius changes. If the
radius changes with respect to time, the change in area with
respect to time is
dA
A. = 2πr
dr
dA dr
B. = 2πr +
dt dt
dA dr
C. = 2πr
dt dt
D. not enough information
21. Question
The area of a circle, A = πr 2 , changes as its radius changes. If the
radius changes with respect to time, the change in area with
respect to time is
dA
A. = 2πr
dr
dA dr
B. = 2πr +
dt dt
dA dr
C. = 2πr
dt dt
D. not enough information