This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)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.
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.
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.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)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.
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.
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
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.
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!
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.
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
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.
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!
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.
Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.
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.
Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.
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.
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 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.
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.
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 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.
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
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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.
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.
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.
Lesson 15: Exponential Growth and Decay (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.
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Lesson 2: A Catalog of Essential Functions (slides)
1. .
Sec on 2.2
A Catalogue of Essen al Func ons
V63.0121.011, Calculus I
Professor Ma hew Leingang
New York University
Announcements
First WebAssign-ments are due January 31
First wri en assignment is due February 2
First recita ons are February 3
2. Announcements
First WebAssign-ments
are due January 31
First wri en assignment
is due February 2
First recita ons are
February 3
3. Objectives
Iden fy different classes of algebraic
func ons, including polynomial
(linear,quadra c,cubic, etc.), ra onal,
power, trigonometric, and exponen al
func ons.
Understand the effect of algebraic
transforma ons on the graph of a
func on.
Understand and compute the
composi on of two func ons.
4. Recall: What is a function?
Defini on
A func on f is a rela on which assigns to to every element x in a set
D a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.
5. Four ways to represent a function
verbally—by a descrip on in words
numerically—by a table of values or a list of data
visually—by a graph
symbolically or algebraically—by a formula
Today the focus is on the different kinds of formulas that can be
used to represent func ons.
6. Classes of Functions
linear func ons, defined by slope and intercept, two points, or
point and slope.
quadra c func ons, cubic func ons, power func ons,
polynomials
ra onal func ons
trigonometric func ons
exponen al/logarithmic func ons
7. Outline
Algebraic Func ons
Linear func ons
Other polynomial func ons
Other power func ons
General ra onal func ons
Transcendental Func ons
Trigonometric func ons
Exponen al and logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
8. Linear functions
Linear func ons have a y
constant rate of growth and (x2 , y2 )
are of the form
(x1 , y1 ) ∆y = y2 − y1
f(x) = mx + b. (0, b)
∆x = x2 − x1
The slope m represents the ∆y
m=
“steepness” of the graphed ∆x
line, and the intercept b
. x
represents an ini al value of
the func on.
9. Modeling with Linear Functions
Example
Assume that a taxi costs $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a func on of distance x traveled.
10. Modeling with Linear Functions
Example
Assume that a taxi costs $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a func on of distance x traveled.
Answer
The ini al fare is $2.50, and the change in fare per mile is
$0.40/0.2 mi = $2/mi. So if x is in miles and f(x) in dollars, the
equa on is
f(x) = 2.5 + 2x
11. A Biological Example
Example
Biologists have no ced that the chirping rate of crickets of a certain
species is related to temperature, and the rela onship appears to be
very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F
and 173 chirps per minute at 80 ◦ F.
(a) Write a linear equa on that models the temperature T as a
func on of the number of chirps per minute N.
(b) If the crickets are chirping at 150 chirps per minute, es mate the
temperature.
12. Biological Example: Solution
Solu on
The point-slope form of the equa on for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equa on y − y0 = m(x − x0 ).
13. Biological Example: Solution
Solu on
The point-slope form of the equa on for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equa on y − y0 = m(x − x0 ).
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
14. Biological Example: Solution
Solu on
The point-slope form of the equa on for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equa on y − y0 = m(x − x0 ).
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equa on rela ng T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
15. Solution continued
So an equa on rela ng T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
16. Solution continued
So an equa on rela ng T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
If N = 150, then
37
T= + 70 = 76 1 ◦ F
6
6
17. Other polynomial functions
Quadra c func ons take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
18. Other polynomial functions
Quadra c func ons take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
Cubic func ons take the form
f(x) = ax3 + bx2 + cx + d
19. Other power functions
Whole number powers: f(x) = xn .
1
nega ve powers are reciprocals: x−3 = 3 .
1/3
√ x
frac onal powers are roots: x = 3 x.
20. General rational functions
Defini on
A ra onal func on is a quo ent of polynomials.
Example
x3 (x + 3)
The func on f(x) = is ra onal.
(x + 2)(x − 1)
The domain is all real numbers except −2 and 1.
The func on is 0 when x = 0 or x = −3.
21. Outline
Algebraic Func ons
Linear func ons
Other polynomial func ons
Other power func ons
General ra onal func ons
Transcendental Func ons
Trigonometric func ons
Exponen al and logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
22. Trigonometric functions
Sine and cosine
Tangent and cotangent
Secant and cosecant
GeoGebra applets to graph these
23. Exponential and logarithmic
functions
exponen al func ons (for example f(x) = 2x )
logarithmic func ons are their inverses (for example
f(x) = log2 (x))
GeoGebra applets to graph these
24. Outline
Algebraic Func ons
Linear func ons
Other polynomial func ons
Other power func ons
General ra onal func ons
Transcendental Func ons
Trigonometric func ons
Exponen al and logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
25. Transformations of Functions
Take the squaring func on and graph these transforma ons:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
26. Transformations of Functions
Take the squaring func on and graph these transforma ons:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the func on, a
transforma on is applied on the x-axis. A er the func on, to the
y-axis.
27. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
y = f(x − c), shi the graph of y = f(x) a distance c units . . .
y = f(x + c), shi the graph of y = f(x) a distance c units . . .
28. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
upward
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
y = f(x − c), shi the graph of y = f(x) a distance c units . . .
y = f(x + c), shi the graph of y = f(x) a distance c units . . .
29. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
upward
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
downward
y = f(x − c), shi the graph of y = f(x) a distance c units . . .
y = f(x + c), shi the graph of y = f(x) a distance c units . . .
30. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
upward
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
downward
y = f(x − c), shi the graph of y = f(x) a distance c units . . . to
the right
y = f(x + c), shi the graph of y = f(x) a distance c units . . .
31. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shi the graph of y = f(x) a distance c units . . .
upward
y = f(x) − c, shi the graph of y = f(x) a distance c units . . .
downward
y = f(x − c), shi the graph of y = f(x) a distance c units . . . to
the right
y = f(x + c), shi the graph of y = f(x) a distance c units . . . to
the le
32. Why?
Ques on
Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the
le by c?
33. Why?
Ques on
Why is the graph of g(x) = f(x + c) a shi of the graph of f(x) to the
le by c?
Answer
Think about x as me. Then x + c is the me c into the future. To
rec fy the future of the graph of f with that of g, pull the graph of f c
into the past.
36. Illustrating the shift
(x + c, f(x + c))
Adding c moves x to the
right (x, f(x))
But then f is applied
.
x x+c
37. Illustrating the shift
(x, f(x + c)) (x + c, f(x + c))
Adding c moves x to the
right (x, f(x))
But then f is applied
To get the graph of
f(x + c), the value
f(x + c) must be above x
.
x x+c
38. Illustrating the shift
(x, f(x + c)) (x + c, f(x + c))
Adding c moves x to the
right (x, f(x))
But then f is applied
To get the graph of
f(x + c), the value
f(x + c) must be above x
So we translate backward .
x x+c
39. Now try these
y = sin (2x)
y = 2 sin (x)
y = e−x
y = −ex
40. Scaling and flipping
c<0 c>0
|c| > 1 |c| < 1 |c| < 1 |c| > 1
f(cx) . . . .
H compress, flip H stretch, flip H stretch H compress
cf(x) . . . .
V stretch, flip V compress, flip V compress V stretch
41. Outline
Algebraic Func ons
Linear func ons
Other polynomial func ons
Other power func ons
General ra onal func ons
Transcendental Func ons
Trigonometric func ons
Exponen al and logarithmic func ons
Transforma ons of Func ons
Composi ons of Func ons
44. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solu on
(f ◦ g)(x) = sin2 x
(g ◦ f)(x) = sin(x2 )
Note they are not the same.
45. Decomposing
Example
√
Express x2 − 4 as a composi on of two func ons. What is its
domain?
46. Decomposing
Example
√
Express x2 − 4 as a composi on of two func ons. What is its
domain?
Solu on
√
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f. To
insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
47. Summary
There are many classes of algebraic func ons
Algebraic rules can be used to sketch graphs