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.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Lesson 15: Exponential Growth and Decay (Section 041 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 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Lesson 15: Exponential Growth and Decay (Section 041 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 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
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.
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.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
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.
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.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Mel Anthony Pepito
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 15: Exponential Growth and Decay (Section 041 slides)Mel Anthony Pepito
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 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 041 ...Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Lesson 15: Exponential Growth and Decay (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.
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 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.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Lesson 15: Exponential Growth and Decay (Section 021 handout)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 15: Exponential Growth and Decay (Section 041 handout)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 11: Implicit Differentiation (Section 41 slides)Mel Anthony Pepito
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.
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 (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.
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.
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.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
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The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
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
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Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
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https://arxiv.org/abs/2306.08302
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UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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Lesson 15: Exponential Growth and Decay (handout)
1. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Sec on 3.4
.
Exponen al Growth and Decay
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
March 23, 2011
.
.
Notes
Announcements
Quiz 3 next week in
recita on on 2.6, 2.8, 3.1,
3.2
.
.
Notes
Objectives
Solve the ordinary
differen al equa on
y′ (t) = ky(t), y(0) = y0
Solve problems involving
exponen al growth and
decay
.
.
. 1
.
2. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
Derivatives of exponential and Notes
logarithmic functions
y y′
ex ex
ax (ln a) · ax
1
ln x
x
1 1
loga x ·
ln a x
.
.
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
. 2
.
3. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
What is a differential equation?
Defini on
A differen al equa on is an equa on for an unknown func on
which includes the func on and its deriva ves.
Example
Newton’s Second Law F = ma is a differen al equa on, where
a(t) = x′′ (t).
In a spring, F(x) = −kx, where x is displacement from
equilibrium and k is a constant. So
k
−kx(t) = mx′′ (t) =⇒ x′′ (t) + x(t) = 0.
. m
.
Notes
Showing a function is a solution
Example (Con nued)
Show that x(t) = A sin ωt + B cos ωt sa sfies the differen al
k √
equa on x′′ + x = 0, where ω = k/m.
m
Solu on
We have
x(t) = A sin ωt + B cos ωt
x′ (t) = Aω cos ωt − Bω sin ωt
x′′ (t) = −Aω 2 sin ωt − Bω 2 cos ωt
.
.
Notes
The Equation y′ = 2
Example
Find a solu on to y′ (t) = 2.
Find the most general solu on to y′ (t) = 2.
Solu on
A solu on is y(t) = 2t.
The general solu on is y = 2t + C.
Remark
. If a func on has a constant rate of growth, it’s linear.
.
. 3
.
4. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
The Equation y′ = 2t
Example
Find a solu on to y′ (t) = 2t.
Find the most general solu on to y′ (t) = 2t.
Solu on
A solu on is y(t) = t2 .
The general solu on is y = t2 + C.
.
.
Notes
The Equation y′ = y
Example
Find a solu on to y′ (t) = y(t).
Find the most general solu on to y′ (t) = y(t).
Solu on
A solu on is y(t) = et .
The general solu on is y = Cet , not y = et + C.
(check this)
.
.
Notes
Kick it up a notch: y′ = 2y
Example
Find a solu on to y′ = 2y.
Find the general solu on to y′ = 2y.
Solu on
y = e2t
y = Ce2t
.
.
. 4
.
5. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
In general: y′ = ky
Example
Find a solu on to y′ = ky.
Find the general solu on to y′ = ky.
Solu on Remark
y=e kt What is C? Plug in t = 0:
y = Cekt y(0) = Cek·0 = C · 1 = C,
so y(0) = y0 , the ini al value
of y.
.
.
Constant Relative Growth =⇒ Notes
Exponential Growth
Theorem
A func on with constant rela ve growth rate k is an exponen al
func on with parameter k. Explicitly, the solu on to the equa on
y′ (t) = ky(t) y(0) = y0
is
y(t) = y0 ekt
.
.
Notes
Exponential Growth is everywhere
Lots of situa ons have growth rates propor onal to the current
value
This is the same as saying the rela ve growth rate is constant.
Examples: Natural popula on growth, compounded interest,
social networks
.
.
. 5
.
6. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
Notes
Bacteria
Since you need bacteria
to make bacteria, the
amount of new bacteria
at any moment is
propor onal to the total
amount of bacteria.
This means bacteria
popula ons grow
exponen ally.
.
.
Notes
Bacteria Example
Example
A colony of bacteria is grown under ideal condi ons in a laboratory.
At the end of 3 hours there are 10,000 bacteria. At the end of 5
hours there are 40,000. How many bacteria were present ini ally?
Solu on
.
.
. 6
.
7. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Bacteria Example Solution
Solu on (Con nued)
We have
.
.
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
Notes
Modeling radioactive decay
Radioac ve decay occurs because many large atoms spontaneously
give off par cles.
This means that in a sample of a
bunch of atoms, we can assume a
certain percentage of them will “go
off” at any point. (For instance, if all
atom of a certain radioac ve element
have a 20% chance of decaying at any
point, then we can expect in a
sample of 100 that 20 of them will be
decaying.)
.
.
. 7
.
8. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Radioactive decay as a differential equation
The rela ve rate of decay is constant:
y′
=k
y
where k is nega ve. So
y′ = ky =⇒ y = y0 ekt
again!
It’s customary to express the rela ve rate of decay in the units of
half-life: the amount of me it takes a pure sample to decay to one
which is only half pure.
.
.
Notes
Computing the amount remaining
Example
The half-life of polonium-210 is about 138 days. How much of a
100 g sample remains a er t years?
.
.
Notes
Carbon-14 Dating
The ra o of carbon-14 to carbon-12 in
an organism decays exponen ally:
p(t) = p0 e−kt .
The half-life of carbon-14 is about 5700
years. So the equa on for p(t) is
p(t) = p0 e− 5700 t = p0 2−t/5700
ln2
.
.
. 8
.
9. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Computing age with Carbon-14
Example
Suppose a fossil is found where the ra o of carbon-14 to carbon-12
is 10% of that in a living organism. How old is the fossil?
Solu on
.
.
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
Notes
Newton’s Law of Cooling
Newton’s Law of Cooling states
that the rate of cooling of an
object is propor onal to the
temperature difference between
the object and its surroundings.
This gives us a differen al
equa on of the form
dT
= k(T − Ts )
dt
(where k < 0 again).
.
.
. 9
.
10. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
General Solution to NLC problems
′ ′
To solve this, change the variable y(t) = T(t) − Ts . Then y = T and
k(T − Ts ) = ky. The equa on now looks like
dT dy
= k(T − Ts ) ⇐⇒ = ky
dt dt
Now we can solve!
y′ = ky =⇒ y = Cekt =⇒ T − Ts = Cekt =⇒ T = Cekt + Ts
Plugging in t = 0, we see C = y0 = T0 − Ts . So
Theorem
The solu on to the equa on T′ (t) = k(T(t) − Ts ), T(0) = T0 is
. T(t) = (T0 − Ts )ekt + Ts
.
Notes
Computing cooling time with NLC
Example
A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. A er 5
minutes, the egg’s temperature is 38 ◦ C. Assuming the water has
not warmed appreciably, how much longer will it take the egg to
reach 20 ◦ C?
Solu on
We know that the temperature func on takes the form
T(t) = (T0 − Ts )ekt + Ts = 80ekt + 18
To find k, plug in t = 5 and solve for k.
.
.
Notes
Finding k
Solu on (Con nued)
.
.
. 10
.
11. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Finding t
Solu on (Con nued)
.
.
Notes
Computing time of death with NLC
Example
A murder vic m is discovered at
midnight and the temperature of the
body is recorded as 31 ◦ C. One hour
later, the temperature of the body is
29 ◦ C. Assume that the surrounding
air temperature remains constant at
21 ◦ C. Calculate the vic m’s me of
death. (The “normal” temperature of
a living human being is approximately
37 ◦ C.)
.
.
Solu on Notes
.
.
. 11
.
12. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Outline
Recall
The differen al equa on y′ = ky
Modeling simple popula on growth
Modeling radioac ve decay
Carbon-14 Da ng
Newton’s Law of Cooling
Con nuously Compounded Interest
.
.
Notes
Interest
If an account has an compound interest rate of r per year
compounded n mes, then an ini al deposit of A0 dollars
becomes ( r )nt
A0 1 +
n
a er t years.
For different amounts of compounding, this will change. As
n → ∞, we get con nously compounded interest
( r )nt
A(t) = lim A0 1 + = A0 ert .
n→∞ n
Thus dollars are like bacteria.
.
.
Notes
Continuous vs. Discrete Compounding of interest
Example
Consider two bank accounts: one with 10% annual interested compounded
quarterly and one with annual interest rate r compunded con nuously. If they
produce the same balance a er every year, what is r?
Solu on
The balance for the 10% compounded quarterly account a er t years
is
A1 (t) = A0 (1.025)4t = P((1.025)4 )t
The balance for the interest rate r compounded con nuously
account a er t years is
A2 (t) = A0 ert
.
.
. 12
.
13. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Solving
Solu on (Con nued)
A1 (t) = A0 ((1.025)4 )t
A2 (t) = A0 (er )t
For those to be the same, er = (1.025)4 , so
r = ln((1.025)4 ) = 4 ln 1.025 ≈ 0.0988
So 10% annual interest compounded quarterly is basically equivalent
to 9.88% compounded con nuously.
.
.
Computing doubling time with Notes
exponential growth
Example
How long does it take an ini al deposit of $100, compounded
con nuously, to double?
Solu on
.
.
Notes
I-banking interview tip of the day
ln 2
The frac on can also
r
be approximated as
either 70 or 72 divided by
the percentage rate (as a
number between 0 and
100, not a frac on
between 0 and 1.)
This is some mes called
the rule of 70 or rule of
72.
72 has lots of factors so
. it’s used more o en.
.
. 13
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14. . V63.0121.001: Calculus I Sec on 3.4: Exponen al Growth and. Decay
. March 23, 2011
Notes
Summary
When something grows or decays at a constant rela ve rate,
the growth or decay is exponen al.
Equa ons with unknowns in an exponent can be solved with
logarithms.
.
.
Notes
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.
Notes
.
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. 14
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