This document outlines a calculus lecture on the fundamental theorem of calculus. It discusses defining an area function from an integral, proves the first fundamental theorem of calculus, and gives examples of how differentiation and integration are reverse processes. It also provides brief biographies of several important mathematicians like Newton and Leibniz related to the development of calculus. The lecture concludes with an example of differentiating a function defined by an integral using the first fundamental theorem.
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.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which 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.
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.
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.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which 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.
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.
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
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 4 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
We trace the computation of area through the centuries. The process known known as Riemann Sums has applications to not just area but many fields of science.
(Handout version of slideshow from class)
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.
Lesson 22: Optimization II (Section 041 handout)Matthew Leingang
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 more good examples.
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.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Optimization problems are just max/min problems with some additional reading comprehension.
Same content as the slide version, but laid out three to a page with space for notes.
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
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 4 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
We trace the computation of area through the centuries. The process known known as Riemann Sums has applications to not just area but many fields of science.
(Handout version of slideshow from class)
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.
Lesson 22: Optimization II (Section 041 handout)Matthew Leingang
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 more good examples.
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.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Optimization problems are just max/min problems with some additional reading comprehension.
Same content as the slide version, but laid out three to a page with space for notes.
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.
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.
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.
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.
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.
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 25: The Fundamental Theorem of Calculus (handout)
1. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Section 5.4 Notes
The Fundamental Theorem of Calculus
V63.0121.006/016, Calculus I
New York University
April 22, 2010
Announcements
April 29: Movie Day
April 30: Quiz 5 on §§5.1–5.4
Monday, May 10, 12:00noon Final Exam
Announcements
Notes
April 29: Movie Day
April 30: Quiz 5 on
§§5.1–5.4
Monday, May 10, 12:00noon
Final Exam
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 2 / 31
Resurrection policies
Notes
Current distribution of grade: 40% final, 25% midterm, 15% quizzes,
10% written HW, 10% WebAssign
Remember we drop the lowest quiz, lowest written HW, and 5 lowest
WebAssign-ments
If your final exam score beats your midterm score, we will re-weight it
by 50% and make the midterm 15%
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 3 / 31
1
2. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Objectives
Notes
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.
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 4 / 31
Outline
Notes
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 5 / 31
The definite integral as a limit
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 6 / 31
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3. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 7 / 31
The Integral as Total Change
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 8 / 31
The Integral as Total Change
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 8 / 31
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4. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
The Integral as Total Change
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 8 / 31
My first table of integrals
Notes
[f (x) + g (x)] dx = f (x) dx + g (x) dx
x n+1
x n dx = + C (n = −1) cf (x) dx = c f (x) dx
n+1
x x
1
e dx = e + 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
sec2 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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 9 / 31
Outline
Notes
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 10 / 31
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5. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
An area function
Notes
x
Let f (t) = t 3 and define g (x) = f (t) dt. Can we evaluate the integral
0
in g (x)?
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x x 3 x (2x)3 x (nx)3
Rn = · + · + ··· + ·
n n3 n n3 n n3
x4 3
= 4 1 + 23 + 33 + · · · + n 3
n
x4 1 2
= 4 2 n(n + 1)
n
0 x
x 4 n2 (n + 1)2 x4
= 4
→
4n 4
as n → ∞.
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 11 / 31
An area function, continued
Notes
So
x4
g (x) = .
4
This means that
g (x) = x 3 .
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 12 / 31
The area function
Notes
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 ?
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 13 / 31
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6. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Envisioning the area function
Notes
Example
Suppose f (t) is the function graphed below
v
t0 t1 c t2 t3 t
x
Let g (x) = f (t) dt. What can you say about g ?
t0
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 14 / 31
features of g from f
Notes
Interval sign monotonicity monotonicity concavity
of f of g of f of g
[t0 , t1 ] +
[t1 , c] +
[c, t2 ] −
[t2 , t3 ] −
[t3 , ∞) − → none
We see that g is behaving a lot like an antiderivative of f .
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 15 / 31
Notes
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).
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 16 / 31
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7. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Proof. Notes
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and 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).
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 17 / 31
Meet the Mathematician: James Gregory
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 18 / 31
Meet the Mathematician: Isaac Barrow
Notes
English, 1630-1677
Professor of Greek, theology,
and mathematics at
Cambridge
Had a famous student
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 19 / 31
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8. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Meet the Mathematician: Isaac Newton
Notes
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 20 / 31
Meet the Mathematician: Gottfried Leibniz
Notes
German, 1646–1716
Eminent philosopher as well
as mathematician
Contemporarily disgraced by
the calculus priority dispute
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 21 / 31
Differentiation and Integration as reverse processes
Notes
Putting together 1FTC and 2FTC, we get a beautiful relationship between
the two fundamental concepts in calculus.
x
d
f (t) dt = f (x)
dx a
b
F (x) dx = F (b) − F (a).
a
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 22 / 31
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9. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Outline
Notes
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 23 / 31
Differentiation of area functions
Notes
Example
3x
Let h(x) = t 3 dt. What is h (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x 4 , so h (x) = 81x 3 .
4 0 4
Solution (Using 1FTC)
u
We can think of h as the composition g ◦ k, where g (u) = t 3 dt and
0
k(x) = 3x. Then
h (x) = g (k(x))k (x) = (k(x))3 · 3 = (3x)3 · 3 = 81x 3 .
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 24 / 31
Differentiation of area functions, in general
Notes
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)
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 25 / 31
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10. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Notes
Example
sin2 x
Let h(x) = (17t 2 + 4t − 4) dt. What is h (x)?
0
Solution
We have
sin2 x
d
(17t 2 + 4t − 4) dt
dx 0
d
= 17(sin2 x)2 + 4(sin2 x) − 4 · sin2 x
dx
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 26 / 31
Notes
Example
ex
Find the derivative of F (x) = sin4 t dt.
x3
Solution
ex
d
sin4 t dt = sin4 (e x ) · e x − sin4 (x 3 ) · 3x 2
dx x3
Notice here it’s much easier than finding an antiderivative for sin4 .
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 27 / 31
Erf
Notes
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2 2
explicitly, but we do know its derivative: erf (x) = √ e −x .
π
Example
d
Find erf(x 2 ).
dx
Solution
By the chain rule we have
d d 2 2 2 4 4
erf(x 2 ) = erf (x 2 ) x 2 = √ e −(x ) 2x = √ xe −x .
dx dx π π
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 28 / 31
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11. V63.0121.006/016, Calculus I 5.4 : The Fundamental Theorem of Calculus
Section April 22, 2010
Other functions defined by integrals
Notes
The future value of an asset:
∞
FV (t) = π(τ )e −r τ dτ
t
where π(τ ) is the profitability at time τ 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.
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 29 / 31
Surplus by picture
Notes
consumer surplus
price (p)
supply
p∗ equilibrium
demand f (q)
q∗ quantity (q)
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 30 / 31
Summary
Notes
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
V63.0121.006/016, Calculus I (NYU) Section 5.4 The Fundamental Theorem of Calculus April 22, 2010 31 / 31
11