Lesson 1: Functions and their representations (handout)
0 likes295 views
The document is a set of lecture notes for a Calculus I course, focusing on the concept of functions, their definitions, representations, and properties such as monotonicity and symmetry. It includes examples of functions expressed in various forms, including formulas, numerical data, graphs, and verbal descriptions, as well as specific rules regarding domain and range. The notes highlight the importance of understanding functions as the fundamental units of investigation in calculus.
![. V63.0121.001, Calculus I
. Sec on 1.1 : Func ons
. January 24, 2011
Professor Ma hew Leingang
No-no’s for expressions Notes
Cannot have zero in the
denominator of an
expression
Cannot have a nega ve
number under an even
root (e.g., square root)
Cannot have the
logarithm of a nega ve
number
. .
Piecewise-defined functions Notes
Example Solu on
Let The domain is [0, 2]. The graph
{ can be drawn piecewise.
x2 0 ≤ x ≤ 1;
f(x) = 2
3−x 1 < x ≤ 2.
1
Find the domain and range of f
and graph the func on. .
0 1 2
. .
Functions described numerically Notes
We can just describe a func on by a table of values, or a diagram.
. .
. 5
.](https://image.slidesharecdn.com/lesson01-functions001handout-110123211102-phpapp02/85/Lesson-1-Functions-and-their-representations-handout-5-320.jpg)
Lesson 1: Functions and their representations (handout)
- 1. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang . Sec on 1.1 Notes Func ons and their Representa ons V63.0121.001, Calculus I Professor Ma hew Leingang New York University Announcements First WebAssign-ments are due January 31 Do the Get-to-Know-You survey for extra credit! . . Announcements Notes First WebAssign-ments are due January 31 Do the Get-to-Know-You survey for extra credit! . . Objectives Notes Understand the defini on of func on. Work with func ons represented in different ways Work with func ons defined piecewise over several intervals. Understand and apply the defini on of increasing and decreasing func on. . . . 1 .
- 2. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang What is a function? Notes 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. . . Outline Notes Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry . . The Modeling Process Notes Real-world . . model Mathema cal . Problems Model solve test Real-world interpret Mathema cal . . Predic ons Conclusions . . . 2 .
- 3. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Plato’s Cave Notes . . . Outline Notes Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry . . Functions expressed by formulas Notes Any expression in a single variable x defines a func on. In this case, the domain is understood to be the largest set of x which a er subs tu on, give a real number. . . . 3 .
- 4. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Formula function example Notes Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solu on The denominator is zero when x = 2, so the domain is all real numbers except 2. We write: domain(f) = { x | x ̸= 2 } . . Formula function example Notes Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solu on x+1 2y + 1 As for the range, we can solve y = =⇒ x = . So as x−2 y−1 long as y ̸= 1, there is an x associated to y. range(f) = { y | y ̸= 1 } . . How did you get that? Notes . . . 4 .
- 5. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang No-no’s for expressions Notes Cannot have zero in the denominator of an expression Cannot have a nega ve number under an even root (e.g., square root) Cannot have the logarithm of a nega ve number . . Piecewise-defined functions Notes Example Solu on Let The domain is [0, 2]. The graph { can be drawn piecewise. x2 0 ≤ x ≤ 1; f(x) = 2 3−x 1 < x ≤ 2. 1 Find the domain and range of f and graph the func on. . 0 1 2 . . Functions described numerically Notes We can just describe a func on by a table of values, or a diagram. . . . 5 .
- 6. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Functions defined by tables I Notes Example Solu on Is this a func on? If so, what is the range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6 . . Functions defined by tables II Notes Example Solu on Is this a func on? If so, what is the range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6 . . Functions defined by tables III Notes Example Solu on Is this a func on? If so, what is the range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6 . . . 6 .
- 7. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang An ideal function Notes Domain is the bu ons Range is the kinds of soda that come out You can press more than one bu on to get some brands But each bu on will only give one brand . . Why numerical functions matter Notes Ques on Why use numerical func ons at all? Formula func ons are so much easier to work with. . . Numerical Function Example Notes Example Here is the temperature in Boise, Idaho measured in 15-minute intervals over the period August 22–29, 2008. 100 90 80 70 60 50 40 30 20 10 . 8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29 . . . 7 .
- 8. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Functions described graphically Notes Some mes all we have is the “picture” of a func on, by which we mean, its graph. . . . . Functions described verbally Notes O en mes our func ons come out of nature and have verbal descrip ons: The temperature T(t) in this room at me t. The eleva on h(θ) of the point on the equator at longitude θ. The u lity u(x) I derive by consuming x burritos. . . Outline Notes Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry . . . 8 .
- 9. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Monotonicity Notes Example Solu on Let P(x) be the probability that my income was at least $x last year. What might a graph of P(x) look like? . . Monotonicity Notes Defini on A func on f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. A func on f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. . . Examples Notes Example Going back to the burrito func on, would you call it increasing? Answer Example Obviously, the temperature in Boise is neither increasing nor decreasing. . . . 9 .
- 10. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Symmetry Notes Consider the following func ons described as words Example Let I(x) be the intensity of light x distance from a point. Example Let F(x) be the gravita onal force at a point x distance from a black hole. What might their graphs look like? . . Possible Intensity Graph Notes Example Solu on Let I(x) be the intensity of light x distance from y = I(x) a point. Sketch a possible graph for I(x). . x . . Possible Gravity Graph Notes Example Solu on Let F(x) be the gravita onal force at a y = F(x) point x distance from a black hole. Sketch a possible graph for F(x). . x . . . 10 .
- 11. . V63.0121.001, Calculus I . Sec on 1.1 : Func ons . January 24, 2011 Professor Ma hew Leingang Definitions Notes Defini on A func on f is called even if f(−x) = f(x) for all x in the domain of f. A func on f is called odd if f(−x) = −f(x) for all x in the domain of f. . . Examples Notes Example Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, log . . Summary Notes The fundamental unit of inves ga on in calculus is the func on. Func ons can have many representa ons . . . 11 .