Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.
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.
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.
This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.
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.
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 chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
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 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.
The derivative of a function is another function. We look at the interplay between the two. Also, new notations, higher derivatives, and some sweet wigs
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.
Similar to Lesson 7: The Derivative as a Function (20)
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.
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.
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.
1. Section 2.2
The Derivative as a Function
V63.0121.002.2010Su, Calculus I
New York University
May 24, 2010
Announcements
Homework 1 due Tuesday
Quiz 2 Thursday in class on Sections 1.5–2.5
. . . . . .
2. Announcements
Homework 1 due Tuesday
Quiz 2 Thursday in class
on Sections 1.5–2.5
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 2 / 28
3. Objectives
Given a function f, use the
definition of the derivative
to find the derivative
function f’.
Given a function, find its
second derivative.
Given the graph of a
function, sketch the graph
of its derivative.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 3 / 28
5. Recall: the derivative
Definition
Let f be a function and a a point in the domain of f. If the limit
f(a + h) − f(a) f(x) − f(a)
f′ (a) = lim = lim
h→0 h x→a x−a
exists, the function is said to be differentiable at a and f′ (a) is the
derivative of f at a.
The derivative …
…measures the slope of the line through (a, f(a)) tangent to the
curve y = f(x);
…represents the instantaneous rate of change of f at a
…produces the best possible linear approximation to f near a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 4 / 28
6. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to
find f′ (2).
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28
7. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to x
.
find f′ (2).
Solution
.
′ 1/x − 1/2 2−x .
f (2) = lim = lim x
.
x→2 x−2 x→2 2x(x − 2)
−1 1
= lim =−
x→2 2x 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28
8. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 7 / 28
9. What does f tell you about f′ ?
If f is a function, we can compute the derivative f′ (x) at each point
x where f is differentiable, and come up with another function, the
derivative function.
What can we say about this function f′ ?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28
10. What does f tell you about f′ ?
If f is a function, we can compute the derivative f′ (x) at each point
x where f is differentiable, and come up with another function, the
derivative function.
What can we say about this function f′ ?
If f is decreasing on an interval, f′ is negative (technically,
nonpositive) on that interval
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28
11. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to x
.
find f′ (2).
Solution
.
′ 1/x − 1/2 2−x .
f (2) = lim = lim x
.
x→2 x−2 x→2 2x(x − 2)
−1 1
= lim =−
x→2 2x 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 9 / 28
12. What does f tell you about f′ ?
If f is a function, we can compute the derivative f′ (x) at each point
x where f is differentiable, and come up with another function, the
derivative function.
What can we say about this function f′ ?
If f is decreasing on an interval, f′ is negative (technically,
nonpositive) on that interval
If f is increasing on an interval, f′ is positive (technically,
nonnegative) on that interval
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 10 / 28
14. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
15. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
But if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) − f(x)
f(x + ∆x) > f(x) =⇒ <0
∆x
still!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
16. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
But if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) − f(x)
f(x + ∆x) > f(x) =⇒ <0
∆x
still! Either way,
f(x + ∆x) − f(x) f(x + ∆x) − f(x)
< 0 =⇒ f′ (x) = lim ≤0
∆x ∆x→0 ∆x . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28
17. Another important derivative fact
Fact
If the graph of f has a horizontal tangent line at c, then f′ (c) = 0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28
18. Another important derivative fact
Fact
If the graph of f has a horizontal tangent line at c, then f′ (c) = 0.
Proof.
The tangent line has slope f′ (c). If the tangent line is horizontal, its
slope is zero.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28
19. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 14 / 28
20. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
21. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f(x) − f(a)
lim (f(x) − f(a)) = lim · (x − a)
x→a x→a x−a
f(x) − f(a)
= lim · lim (x − a)
x→a x−a x→a
′
= f (a) · 0 = 0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
22. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f(x) − f(a)
lim (f(x) − f(a)) = lim · (x − a)
x→a x→a x−a
f(x) − f(a)
= lim · lim (x − a)
x→a x−a x→a
′
= f (a) · 0 = 0
Note the proper use of the limit law: if the factors each have a limit at
a, the limit of the product is the product of the limits.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28
23. Differentiability FAIL
Kinks
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
24. Differentiability FAIL
Kinks
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
25. Differentiability FAIL
Kinks
f
.(x) .′ (x)
f
.
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28
26. Differentiability FAIL
Cusps
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
27. Differentiability FAIL
Cusps
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
28. Differentiability FAIL
Cusps
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28
29. Differentiability FAIL
Vertical Tangents
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
30. Differentiability FAIL
Vertical Tangents
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
31. Differentiability FAIL
Vertical Tangents
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28
32. Differentiability FAIL
Weird, Wild, Stuff
f
.(x)
. x
.
This function is differentiable at
0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28
33. Differentiability FAIL
Weird, Wild, Stuff
f
.(x) .′ (x)
f
. x
. . x
.
This function is differentiable at But the derivative is not
0. continuous at 0!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28
34. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 20 / 28
35. Notation
Newtonian notation
f′ (x) y′ (x) y′
Leibnizian notation
dy d df
f(x)
dx dx dx
These all mean the same thing.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 21 / 28
36. Link between the notations
f(x + ∆x) − f(x) ∆y dy
f′ (x) = lim = lim =
∆x→0 ∆x ∆x→0 ∆x dx
dy
Leibniz thought of as a quotient of “infinitesimals”
dx
dy
We think of as representing a limit of (finite) difference
dx
quotients, not as an actual fraction itself.
The notation suggests things which are true even though they
don’t follow from the notation per se
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 22 / 28
37. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 23 / 28
38. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 24 / 28
39. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 25 / 28
40. The second derivative
If f is a function, so is f′ , and we can seek its derivative.
f′′ = (f′ )′
It measures the rate of change of the rate of change!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28
41. The second derivative
If f is a function, so is f′ , and we can seek its derivative.
f′′ = (f′ )′
It measures the rate of change of the rate of change! Leibnizian
notation:
d2 y d2 d2 f
f(x)
dx2 dx2 dx2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28
42. function, derivative, second derivative
y
.
.(x) = x2
f
.′ (x) = 2x
f
.′′ (x) = 2
f
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 27 / 28
43. Summary
A function can be differentiated at every point to find its derivative
function.
The derivative of a function notices the monotonicity of the
function (fincreasing =⇒ f′ ≥ 0)
The second derivative of a function measures the rate of the
change of the rate of change of that function.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 28 / 28
